Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference

Historical reassessment

• The widespread story that Bohr and Heisenberg resolved the interpretive disputes at the 1927 Solvay Conference is contradicted by the continued disagreement of de Broglie, Einstein, and Schrödinger.The conference instead featured conflicting positions, and several major participants remained unconvinced at its conclusion.Source: ISBN: 9780521814218 · p. 9

Motivation

• The authors argue that quantum interpretation remains controversial today, with no established interpretation, making it important to return to and re-evaluate the historical sources.They specifically question whether Bohr and Heisenberg’s opponents may have raised important points that were later overlooked.Source: ISBN: 9780521814218 · pp. 9–10

Conference scope

• The 1927 meeting presented de Broglie’s pilot-wave theory, Born and Heisenberg’s quantum mechanics, and Schrödinger’s wave mechanics as distinct theories discussed on an equal footing.The debates addressed issues including measurement, determinism, nonlocality, subjectivity, and interference.Source: ISBN: 9780521814218 · p. 10

Physical questions

• The conference treated interpretation as a physical question, including whether electrons are particles, wave packets, or neither and when quantum outcomes occur.Related disputes concerned compatibility between nonlocality and relativity and whether indeterminism is fundamental or arises from coarse-graining.Source: ISBN: 9780521814218 · p. 10

Aftermath

• After 1927, the Copenhagen interpretation became established and rival views were marginalized, although later work by Bohm, Everett, and Bell kept foundational questioning alive.The prevailing attitude treated interpretive issues as essentially settled and encouraged a “shut up and calculate” approach.Source: ISBN: 9780521814218 · pp. 10–11

• The published proceedings omit the famous Einstein–Bohr uncertainty-principle discussions, and Bohr did not give a conference report despite the later inclusion of his Como lecture.The resulting editorial arrangement may have encouraged the misunderstanding that Bohr reported at the conference.Source: ISBN: 9780521814218 · p. 12
• Born and Heisenberg presented unfamiliar views of the wave function, time, and probability, with no apparent collapse postulate and a seemingly phenomenological time-dependent Schrödinger equation.Their “quantum mechanics” therefore appears markedly different from the Dirac–von Neumann formulation familiar today.Source: ISBN: 9780521814218 · p. 12
• De Broglie’s pilot-wave theory received extensive discussion, including nine pages after his report and discussion across 15 of the 42 pages of general debate.Reactions came from Pauli, Born, Brillouin, Einstein, Kramers, Lorentz, Schrödinger, and others, contradicting claims that the theory was hardly discussed.Source: ISBN: 9780521814218 · p. 12
• De Broglie proposed a many-body pilot-wave dynamics in which the total configuration is guided by a wave in configuration space, not merely a one-body theory in three-dimensional space.The account is presented as essentially the same as Bohm’s 1952 theory, differing in formulating the dynamics in terms of velocity rather than momentum.Source: ISBN: 9780521814218 · pp. 12–13
• The book translates and comments on the 1927 proceedings, situating them within current quantum-foundations research rather than attempting an exhaustive historical analysis.It aims to make the material accessible to physicists and stimulate serious debate, while its English publication may encourage historians and philosophers to reconsider the proceedings’ significance.Source: ISBN: 9780521814218 · p. 13, §7.1 · p. 212
• The volume includes historical and interpretive essays on pilot-wave theory, quantum mechanics, and wave mechanics, followed by translations of reports, their discussions, and the general discussion.Part II reviews contemporary foundations and extracts conference highlights, while Part III provides the translated proceedings using original texts where possible.Source: ISBN: 9780521814218 · p. 13

Acknowledgments

• The authors thank archives staff in Brussels and Konstanz for archival access and continuing assistance with the history of quantum physics archive.They specifically acknowledge Didier Devriese, Brigitte Parakenings, and the Konstanz archive staff.Source: ISBN: 9780521814218 · p. 14
• Valentini acknowledges colleagues and institutions that supported his study of the proceedings from his 1994–96 Rome postdoctoral period through later work at Perimeter Institute.He also thanks collaborators and interlocutors for discussions, suggestions, correspondence, references, support, and encouragement.Source: ISBN: 9780521814218 · p. 14
• The authors thank Cambridge University Press and Augustus College for support and encouragement during the book’s gestation and final stages.They specifically acknowledge Tamsin van Essen and other publishing support.Source: ISBN: 9780521814218 · p. 14

• The section includes abbreviations for institutional and library locations such as the Hebrew University of Jerusalem and the Bohr Arkiv in Copenhagen.Source: Abbreviations · p. 15
• The abbreviations identify archival collections and source repositories, including AHQP, AEA, AHQP-LTZ, AHQP-RDN, AHQP-OHI, IIPCS, and AHQP-BSC.Source: Abbreviations · p. 15
• They also abbreviate historical journals, academies, proceedings, and publication series cited in the bibliography.Source: Abbreviations · pp. 15–16
• Editorial notation is specified: square brackets mark editorial amendments or translation changes, while curly and angle brackets mark additions and cancellations.Source: Abbreviations · p. 16

Note on the bibliograph y

• The bibliography lists references cited in Parts I and II, the editorial and endnotes, and footnotes to Part III, while original-volume references remain in the Part III proceedings translation.The surrounding material identifies the proceedings as the main source of original material from the fifth Solvay conference.Source: Note on the bibliograph y · pp. 16, 21

1.1 Ernest Solv a y and the Institute of Ph ysi s

• Ernest Solvay used his industrial fortune to support public-benefit initiatives, scientific research, education, and the Cité Scientifique, culminating in the Institutes of Physics and Chemistry founded in 1912 and 1913.The Cité Scientifique developed through collaboration among Solvay, the Université Libre de Bruxelles, and the City of Brussels.Source: §1.1 · pp. 21–22
• The first Solvay physics conference arose from Nernst and Planck’s call for an international meeting on urgent radiation and specific-heat problems, with Solvay providing the opportunity and Lorentz eventually presiding.After the 1911 conference, Solvay and Lorentz also developed a proposed scientific foundation supporting physics, physical chemistry, international researchers, and scholarships for young Belgian scientists.Source: §1.1 · p. 22
• The proposed foundation would support physics and physical chemistry, provide international research assistance and Belgian scholarships, and include administrative and scientific governance linked to existing institutes.Lorentz’s January 1912 proposals were implemented after Solvay sent Paul Héger to Leiden to work with him on the statutes, which Solvay approved with few modifications.Source: §1.1 · p. 23
• The Solvay International Institute of Physics was officially established on 1 May 1912, supported scientific research, and was followed by a similarly organized Institute of Chemistry in 1913.Its establishment preceded comparable Belgian state institutions, founded in 1920 and 1928.Source: §1.1 · p. 24

1.2 W ar and in ternational relations

• World War I disrupted the Institute through the German invasion of Belgium, communication difficulties, requisition fears, and delays in publishing the second conference proceedings.The proceedings were printed in three languages in 1915, largely destroyed, and published in French only in 1921.Source: §1.2 · p. 24
• Postwar tensions excluded German scientists from the 1921 and 1924 Solvay conferences, while Einstein and Bohr declined participation amid the continued exclusion.Schrödinger, who was Austrian and working in Switzerland, was invited to the 1921 conference.Source: §1.2 · p. 25
• After the 1925 Treaty of Locarno, the committee began considering renewed German inclusion, and by April 1926 unanimously decided to invite Einstein and include German scientists again.The King approved the initiative after Lorentz’s interview, and Lorentz informed Einstein of the decision on 6 April.Source: §1.2 · p. 26
• German participation remained politically sensitive, even as Lorentz promoted reconciliation, readmission, and broader international intellectual cooperation around the Solvay Institute.Lorentz advocated not formally excluding Germans after the war and was active in international organizations, while invitations and publicity remained delicate in 1926–1927.Source: §1.2 · p. 27

1.3 S ien ti planning and ba kground

• Ehrenfest objected that describing the proposed topic as possible “reconciliation” between classical theories and quantum theory could obscure the conflict’s productive sharpness.Lorentz accepted the criticism and removed references to reconciliation from the title and later descriptions of the meeting’s focus.Source: §1.3 · p. 28
• Schrödinger’s wave mechanics became central to the planning after Einstein suggested replacing Lorentz as a speaker, while Lorentz continued engaging critically with it.Schrödinger was later considered as a substitute for Heisenberg, and Lorentz discussed wave and matrix mechanics in lectures and correspondence.Source: §1.3 · p. 29
• Lorentz invited Dirac and Pauli as quantum mechanics developed rapidly and their contributions became especially notable.He asked Born and Heisenberg to include Dirac’s work in their report and later invited both Dirac and Pauli to the conference.Source: §1.3 · p. 30
• Recent matter-wave experiments prompted Lorentz to broaden de Broglie’s planned report to cover light quanta, material particles, and evidence from Elsasser, Dymond, and Davisson–Germer.The experimental developments were treated as supporting the notion of matter waves.Source: §1.3 · p. 30
• The final program included foundational reports on a new mechanics by de Broglie, Born and Heisenberg, and Schrödinger.This reflected the growing importance of quantum mechanics during the conference’s planning.Source: §1.3 · p. 30
• Bragg and Compton’s talks preserved the conference’s initial emphasis on radiation phenomena, experiments, and competing radiation theories.Compton compared the photon hypothesis, BKS theory, and classical radiation theory, while Bragg focused on X-ray analysis and related phenomena.Source: §1.3 · pp. 30–31
• The BKS proposal described radiation through virtual oscillators and fields, but energy and momentum conservation held only statistically rather than for individual events.The theory linked atomic transitions through virtual radiation fields and violated conservation laws for single emission and absorption events.Source: §1.3 · p. 32
• Bothe–Geiger and Compton–Simon experiments established conservation laws for individual processes, ruling out BKS by the time of the conference.The experimental evidence led the short note on BKS to be dropped from the programme; Bothe and Geiger had reported clear results against the theory by April 1925 (Bothe and Geiger 1925a,b).Source: §1.3 · p. 32
• The Compton effect remained a problem for Bohr and contributed to his developing views on wave–particle dualism and complementarity.Bohr discussed the motivations behind BKS after Compton’s report, even though the theory itself had been experimentally disconfirmed.Source: §1.3 · p. 32
• Einstein withdrew his planned Solvay report, citing insufficient engagement with modern quantum theory and opposition to its purely statistical way of thinking.He had accepted an invitation to discuss quantum statistics but concluded that he could not provide a report corresponding to the state of the field.Source: §1.3 · p. 33
• The final programme included reports by Bragg, Compton, de Broglie, Born and Heisenberg on matrix mechanics, and Schrödinger on wave mechanics, with Born and Heisenberg submitting a joint report.The matrix-mechanics report was to include Dirac’s theory, while de Broglie was asked to address free electrons and wave mechanics was assigned to Schrödinger.Source: §1.3 · p. 34

1.4 F urther details of planning

• Heisenberg’s correspondence with Einstein presented his views on theories containing particle trajectories, including Einstein’s hidden-variables proposal and Heisenberg’s response.The exchange was prompted by Einstein’s communication, and the discussion is identified as material treated later in the book.Source: §1.4 · p. 34
• Bohr contributed significantly to the general discussion and reported his Como views, but could not prepare an edited version in time, so a Naturwissenschaften translation was included.This substitution gave rise to a common belief that Bohr gave a report.Source: §1.4 · p. 34

1.5 The Solv a y meeting

• The conference took place in Brussels from 24 to 29 October 1927, with participants housed at the Hôtel Britannique and travel within Europe covered.Meals, catering, and reserved theatre seats were also arranged for participants.Source: §1.5 · p. 38
• The reconstructed schedule placed Bragg and Compton on Monday, de Broglie on Tuesday, Born and Heisenberg on Wednesday morning, and Schrödinger on Wednesday afternoon.The reconstruction assumes that talks followed the order described in planning documents and the printed volume.Source: §1.5 · pp. 38–39
• The general discussion extended over at least two days, as Dirac referred to Bohr’s comments from the previous day in his main contribution.The schedule also included a Tuesday university reception and Thursday travel to Paris for the Fresnel centenary.Source: §1.5 · pp. 38–39
• The conference used English, German, and French, while informal discussions—including exchanges between Einstein and Bohr—must largely be reconstructed from later or external sources.Schrödinger volunteered to speak in English, and language difficulties hindered communication; some of the Einstein–Bohr discussions were recalled in later sources.Source: §1.5 · pp. 39–40

1.6 The editing of the pro e e dings

• The proceedings were edited from translated reports and stenographed discussion notes that speakers used to prepare drafts before Verschäfelt produced the final version.By 6 January 1928, the reports had been translated and several proofs were circulating; Kramers helped complete two of Lorentz’s contributions.Source: §1.6 · pp. 40–41
• Bohr’s discussion contributions were especially incomplete: his notes contained many gaps, and material from the general discussion survives only in the Bohr archives.Bohr’s contributions after Compton’s and Born and Heisenberg’s reports were edited in March, whereas general-discussion material remained only as archival notes.Source: §1.6 · p. 41

1.6 The editing of the proceedings

• The remaining material was sent to Gauthier-Villars in September 1928, and the volume was published in early December.Source: §1.6 · p. 42

1.7 Conclusion

• Historical assessments diverged: Heisenberg described the conference as completing the quantum theory’s foundations (Heisenberg 1929, p. 495), whereas Langevin called it the peak of conceptual confusion.100Source: §1.7 · p. 42
• The authors argue that the conference’s exchanges among different research lines contributed extraordinarily to clarifying quantum theory’s physical foundations.Source: §1.7 · p. 42
• The subsequent analysis revisits the conference through de Broglie’s pilot-wave theory, Born and Heisenberg’s quantum mechanics, and Schrödinger’s wave mechanics.Source: §1.7 · pp. 42–43

1.7 Con lusion

• The references document extensive correspondence among Einstein, Lorentz, Ehrenfest, de Broglie, Born, Heisenberg, Schrödinger, Bragg, and others surrounding the 1927 conference.Sources include letters in German, French, English, and Dutch held in AHQP, IIPCS, and AEA collections.Source: §1.7 · pp. 45–46
• The bibliography also cites archival materials on conference organization, invitations, replies, sessions, receptions, meals, and related documentary records.These materials include IIPCS files, AHQP documents, a seating plan, and references to Pelseneer’s account.Source: §1.7 · pp. 46–47
• The notes cite archival documents from AHQP, IISPCS, and related collections, including correspondence involving Schrödinger, Lorentz, Born, Ehrenfest, Bragg, Planck, Bohr, and others.The references include materials in German, English, French, Danish, and Dutch.Source: §1.7 · pp. 45, 47
• The notes also refer to Pelseneer’s pages 50–51 and a microfilm in AHQP-BMSS-11.These references are given as supporting secondary or archival materials.Source: §1.7 · p. 47
• Later references document additional correspondence and publication records, including Born and Heisenberg’s report and IISPCS items 2761–2762.The notes identify sources dated from late 1927 through 1928, often with language or document-location information.Source: §1.7 · p. 47

2.1 Ba kground

• De Broglie predicted that matter would display diffraction and interference, extending Einstein’s wave-particle duality to electrons through E = hν and p = h/λ.The prediction was experimentally confirmed a few years later, although such phenomena were often treated as evidence against particle trajectories.Source: §2.1 · pp. 48–49
• In pilot-wave dynamics, particles follow trajectories orthogonal to equal-phase surfaces of an associated guiding wave.For N spinless particles, the formulation uses a Schrödinger equation for Ψ(x_1, x_2, ..., x_N, t) and a de Broglie guidance equation for trajectories x_i(t), with S = ℏIm ln Ψ.Source: §2.1 · pp. 48, 50, 55
• De Broglie’s work was largely underestimated and ignored, partly because he worked in relative isolation outside the mainstream centers of theoretical physics.The section contrasts this neglect with the later presentation of pilot-wave theory in textbooks and its acceptance as an alternative, if little-used, formulation of quantum theory (Bohm and Hiley 1993; Holland 1993).Source: §2.1 · pp. 49–50
• De Broglie’s central innovation was a new dynamics in which particle velocities are determined by guiding waves rather than classical laws.The guiding-wave construction unified the variational principles of Maupertuis and Fermat, and replaced Newtonian or Einsteinian laws with new dynamical laws.Source: §2.1 · p. 49
• The 1927 de Broglian formulation differs from Bohm’s 1952 presentation, which treats Newton’s law with a quantum-potential term as the law of motion.For Bohm, the guidance equation constrains initial momenta, whereas de Broglie presented the dynamics differently.Source: §2.1 · pp. 50–51
• De Broglie’s equations deterministically evolve an initial multiparticle wave function and configuration into a wave function and particle trajectory at every later time.The theory uses a velocity-based dynamics for a single multiparticle system.Source: §2.1 · p. 51
• With initial configurations distributed as P(x1, x2, ..., xN, 0) = |Ψ(x1, x2, ..., xN, 0)|2, pilot-wave predictions agree statistically with standard quantum theory.Treating the apparatus and measured system as one multiparticle system yields the Born-rule distribution for later pointer positions and measurement outcomes.Source: §2.1 · pp. 51–52
• The account extends the same agreement to later particle-position distributions and experimental pointer outcomes, although de Broglie’s 1927 report did not provide a general measurement-equivalence demonstration.Bohm’s 1952 work supplied the detailed pilot-wave account of measurement cited here.Source: §2.1 · p. 51
• De Broglie developed the theory from earlier work through his 1927 publications and Solvay presentation, applying it to interference, atomic transitions, and electron scattering.The dynamics treats electrons and photons as both particles and waves, with qualitative antecedents in Newton’s account of light.Source: §2.1 · p. 52
• De Broglie’s pilot-wave theory extended a particle-and-wave description from light to ordinary matter, with Hamiltonian mechanics recovered in the short-wavelength limit.For independent particles guided by waves in three-dimensional space, the early theory did not yet specify a wave equation.Source: §2.1 · pp. 53–54
• De Broglie proposed replacing Newtonian dynamics with a framework unifying Maupertuis’ and Fermat’s principles, where particle velocity follows the phase gradient of an accompanying wave.This approach explained atomic energy quantisation and single-photon interference, and predicted electron diffraction and interference.Source: §2.1 · p. 55

2.2.1 First p ap ers on pilot-wave the ory (1923)

• In his 1922 blackbody-radiation paper, de Broglie treated radiation as a gas of light quanta and assumed photons had a very small but non-zero rest mass m0.The assumption enabled him to apply E = hν and p = h/λ to massive particles, treating light quanta analogously to ordinary matter.Source: §2.2.1 · p. 56
• De Broglie extended wave–particle duality to ordinary matter, proposing an internal periodic phenomenon and a non-material phase wave for every massive particle.His initial motivation included optical–mechanical analogy and the appearance of whole numbers in quantisation conditions.Source: §2.2.1 · pp. 56–57
• The proposed dynamics explained diffraction by requiring particle trajectories to curve like diffracted-wave rays when openings are small relative to the phase wavelength.The particle velocity was identified with the wave’s group velocity, while its phase remained matched to the particle’s internal oscillation.Source: §2.2.1 · pp. 57–58
• De Broglie argued that diffraction requires abandoning Newton’s first law and replacing it with motion guided by the phase wave’s rays.A free body was postulated to move uniformly along the normal to equal-phase surfaces, explaining trajectory curvature near openings.Source: §2.2.1 · p. 58
• De Broglie noted that the postulate preserves energy but generally not momentum, because the pilot wave or quantum potential acts as an external momentum source.Constant speed occurs only in special cases, rather than generally for free particles.Source: §2.2.1 · p. 58
• He predicted that any moving body could undergo diffraction, including electron streams passing through sufficiently small openings.He suggested diffraction experiments as a possible test of the proposal.Source: §2.2.1 · p. 59
• De Broglie characterized the phase wave as a guiding wave that synthesizes wave and quantum descriptions of motion.He presented this synthesis as linking the historical development of dynamics and optics.Source: §2.2.1 · p. 59
• Interfering phase waves were used to explain bright and dark fringes in Young’s experiment, even for arbitrarily weak incident light.The phase-wave interference state determined the position-dependent probability of photon detection.Source: §2.2.1 · p. 59
• For gases, de Broglie assumed stationary waves for all atoms and obtained the Maxwell distribution for nonrelativistic atoms and the Planck distribution for photons.He assigned atoms to stationary modes with probabilities determined by the Boltzmann factor, although later discussion notes errors in the statistical-mechanical application.Source: §2.2.1 · p. 60
• De Broglie’s 1923 variational treatment identified Maupertuis’ and Fermat’s principles and emphasized their fundamental link between dynamics and geometrical optics.He later expressed the dynamical postulate in covariant four-vector form, while his thesis consolidated these papers and acknowledged gaps in the theory.Source: §2.2.1 · pp. 60–61, §2.2.2 · p. 61

2.2.2 Thesis (1924)

• His doctoral thesis (de Broglie 1924e) was largely based on the preceding papers, completed in 1924, and published in 1925 despite acknowledged theoretical gaps.The thesis was defended at the Sorbonne in November 1924 and published early in 1925 (de Broglie 1925).Source: §2.2.2 · p. 61
• The thesis aimed to unify corpuscular and wave theories, presenting new ideas, successes, and remaining gaps through a proposed synthesis.De Broglie argued that apparently opposed theories might reflect inadequate synthesis rather than a fundamental contradiction.Source: §2.2.2 · p. 62
• The phase wave has phase velocity v_ph = c^2/v, while its group velocity equals the particle’s velocity.De Broglie characterized this superluminal phase wave as representing phase distribution rather than energy transport.Source: §2.2.2 · p. 63
• For a fixed total energy and phase-wave frequency, dynamically possible particle trajectories coincide with the phase wave’s possible rays.This extends the phase-wave picture beyond uniform rectilinear motion to non-uniform motion.Source: §2.2.2 · p. 64
• De Broglie related Hamilton’s and Maupertuis’ principles, showing that Hamilton’s principle reduces to Maupertuis’ principle when p0 is constant.He further connected the wave formulation to Fermat’s principle, with the Hamiltonian form reducing to the Maupertuisian form when ν is constant.Source: §2.2.2 · pp. 64–65
• The thesis assigned symmetric roles to the particle energy-momentum 4-vector pµ and the phase-wave 4-vector.This extended the quantum relation E = hν within a relativistic spacetime formulation.Source: §2.2.2 · p. 65
• De Broglie identified Fermat’s principle for phase waves with Maupertuis’ principle for moving bodies, making particle trajectories coincide with possible wave rays.He presented this correspondence as a route toward synthesizing waves and quanta.Source: §2.2.2 · p. 65
• The extended quantum relation became a relativistic guidance equation in which the guiding-wave phase determines particle velocity, constituting de Broglie’s new first-order dynamics.The formulation expresses particle velocity through the phase gradient and incorporates electromagnetic potentials when fields are present.Source: §2.2.2 · p. 66
• Wave resonance reproduced the required stability condition for quantized orbits, yielding m0vdl = nh and recovering Bohr’s angular-momentum condition m0ωR2 = nℏ.De Broglie thus treated atomic quantization as arising from wave properties.Source: §2.2.2 · p. 67
• For the hydrogen atom, de Broglie analyzed the two-body problem using separate phase waves for the electron and nucleus, and he proposed a wave-based interpretation of light quanta.He also claimed to generalize the results from closed to quasi-periodic or multiperiodic motion, but that derivation was faulty.Source: §2.2.2 · p. 67
• De Broglie proposed a unified explanation of diffraction in which the wave ray curves and the associated moving body undergoes the same deviation, abandoning Newton’s first law for that motion.This connected particle behavior near obstacles to wave-theoretical propagation.Source: §2.2.2 · p. 68
• De Broglie abandoned Newton’s first law and based his preferred dynamics on velocity rather than acceleration.He nevertheless acknowledged that the theory could be expressed in Newtonian terms using a quantum potential.Source: §2.2.2 · p. 68
• His phase-wave framework provided a qualitative account of optical interference by linking light-quantum detectability to phase-wave interference.De Broglie suggested that phase waves would determine where light quanta interact with matter, while admitting that the detailed explanation of bright and dark fringes was unclear.Source: §2.2.2 · pp. 68–69
• Stationary phase waves in a confined gas yielded resonance conditions and quantized particle velocities, reproducing the familiar quantization of momentum.The resonance condition was l = nλ, with allowed speeds equal to integer multiples of v0.Source: §2.2.2 · pp. 69–70
• De Broglie claimed that his statistical treatment of stationary phase waves derived the Planck distribution, although the application contained errors.The analysis treated stationary phase waves as elementary objects and required an additional hypothesis for overlapping light-quantum phase waves.Source: §2.2.2 · p. 70
• Interpreting atomic quantum conditions as phase-wave resonance along trajectories was presented as the first physically plausible explanation of Bohr–Sommerfeld stability conditions.The thesis also developed a qualitative theory of interference and associated phase waves with uniformly moving bodies through relativistic considerations.Source: §2.2.2 · p. 71
• De Broglie’s thesis presented phase waves as unifying wave and particle conceptions, with material-point dynamics understood through wave propagation.He claimed the approach could derive Planck’s phase-volume element and the blackbody spectrum.Source: §2.2.2 · p. 71
• The 1924 theory remained incomplete: its phase waves and light quantum lacked fully specified physical definitions, a new electromagnetic theory, and a particular wave equation.De Broglie described it as an incompletely specified physical theory rather than a consistent, definitive doctrine.Source: §2.2.2 · p. 71
• De Broglie proposed electron diffraction by crystals as an experimental test of his ideas, and later recalled that this test was subsequently discovered in America by Davisson.His thesis itself did not explicitly discuss electron diffraction or interference experiments.Source: §2.2.2 · pp. 71–72

2.2.3 Opti al interfer en e fringes: Novemb er 1924

• In his November 1924 account, de Broglie explained optical interference through the spatial distribution of photon trajectories rather than solely through photon–matter interaction probabilities.Bright and dark fringes corresponded to high and low trajectory densities, respectively.Source: §2.2.3 · pp. 72–73
• De Broglie argued that interfering phase waves would bunch photon trajectories into regions matching observed bright fringes, reproducing the familiar interference pattern.The phase velocity and de Broglie’s particle velocity were also reported to agree.Source: §2.2.3 · p. 74
• De Broglie treated phase waves as a direct representation of the electromagnetic field, but acknowledged that their connection to the electromagnetic field was still unknown.This treatment simplified the relation between guiding waves and light waves.Source: §2.2.3 · p. 74
• Einstein interpreted the wave-like term in Bose–Einstein fluctuations through de Broglie’s matter waves, and his paper largely made de Broglie’s work known outside France.Born and Heisenberg later gave the impression that Einstein first recognized material-particle diffraction, although de Broglie had made the suggestion earlier.Source: §2.2.3 · p. 75
• De Broglie’s thesis also became a starting point for Schrödinger’s attempt to formulate a wave equation for the phase waves.Schrödinger applied de Broglie’s frequency and phase-velocity formulas to the hydrogen atom and represented the phase wave as ψ = ψ(x, t).Source: §2.2.3 · p. 76
• Schrödinger’s original relativistic equation predicted energy eigenvalues hν that disagreed with experiment, whereas its nonrelativistic approximation yielded the correct low-energy levels.The low-energy limit reduces to the modern nonrelativistic, time-independent Schrödinger equation for a particle in potential V.Source: §2.2.3 · p. 77
• By recasting de Broglie’s electron-wave expressions as a wave equation, Schrödinger obtained eigenvalues that agreed remarkably well with observed hydrogen-spectrum features.This transformed de Broglie’s elementary quantisation argument into a rigorous and powerful technique.Source: §2.2.3 · p. 77
• Schrödinger later derived the time-dependent equation by replacing Eψ with iℏ∂ψ/∂t, requiring a complex wave function and yielding an equation first order in time.His fourth wave-mechanics paper, completed in June 1926, supplied the time-dependent formulation.Source: §2.2.3 · pp. 77–78
• Schrödinger adopted de Broglie’s wave-based quantisation and classical-limit intuition but rejected de Broglie’s localized particle within an extended wave, working instead with nonsingular extended waves.Schrödinger used the optical-mechanical analogy, whereas de Broglie framed the corresponding analogy through Maupertuis and Fermat principles.Source: §2.2.3 · p. 78

2.3.1 `Stru tur e': Journal de Physique, May 1927

• De Broglie’s 1927 work treated particles as small singular regions of large amplitude within extended waves and sought a deterministic synthesis of particle dynamics with wave theory.The approach aimed to reconcile full determinism with wave–corpuscle dualism.Source: §2.2.3 · p. 79

2.2.3 Opti al interfer en e fringes: Novemb er 1924

• De Broglie's “Structure” paper (1927b) represents particles as moving singularities accompanied by continuous wave solutions, with pilot-wave dynamics emerging from stated assumptions.The paper ultimately presents the pilot-wave equations provisionally, without deriving them from a deeper theory.Source: §2.2.3 · pp. 79–80
• Earlier notes developed related probability-density ideas, but unlike “Structure” they did not propose adopting pilot-wave theory as a provisional view.The earlier treatments linked photon probability density to classical wave intensity and material-particle density to wave-function amplitude-squared.Source: §2.2.3 · p. 80
• For a free-particle ensemble, the continuous wave corresponds to a distribution of singularities, and its amplitude-squared measures the probability of finding a particle in d^3x.The ensemble is described as a cloud whose phase distribution and density correspond to the continuous wave representation.Source: §2.2.3 · p. 81
• In an external potential, de Broglie derives coupled equations for wave amplitude and phase, with particle velocity tied to the phase gradient under an essential hypothesis.The assumed velocity–phase relation is intended to remain valid beyond the classical limit and supports agreement with the relativistic velocity–momentum relation.Source: §2.2.3 · pp. 81–82
• When amplitude-dependent terms are negligible, the equations recover classical mechanics for moving singularities and geometrical optics for continuous waves.In this limit, both descriptions are governed by the same Jacobi function.Source: §2.2.3 · pp. 82–83
• De Broglie’s double-solution principle pairs a point-like singular wave with a continuous Ψ-wave sharing the same phase, with Ψ suggesting a means to determine trajectories.The trajectories are orthogonal to the continuous wave’s phase surfaces, while the density ρ defines single-particle position probabilities in terms of Ψ.Source: §2.2.3 · pp. 83–84
• For time-dependent electromagnetic fields, de Broglie derives singularity velocities and a continuity equation, while retaining ρ ∝ a2 in the nonrelativistic limit.The probability at time t is ρ(x,t)d3x and is expressed through Ψ.Source: §2.2.3 · pp. 84–85
• De Broglie treats the guidance equation v ∝ ∇φ as fundamental and recasts the dynamics using a variable proper mass or, nonrelativistically, an additional quantum potential.The additional term coincides with Bohm’s quantum potential Q, with the second relativistic term negligible as c →∞.Source: §2.2.3 · pp. 85–86
• For N particles, de Broglie uses a wave u in 3N-dimensional configuration space but argues that the physical picture requires N waves propagating in ordinary three-dimensional space.He criticizes Schrödinger’s extended wave-packet interpretation because it lacks definite particle positions or trajectories.Source: §2.2.3 · p. 86
• In the two-particle case, each singular wave’s propagation depends on the other singularity’s position, leaving de Broglie with coupled partial differential equations whose solution is formidable.The equations are intended to yield the motions of the two singularities, but de Broglie does not solve them in this passage.Source: §2.2.3 · pp. 86–87
• De Broglie’s claimed existence of a two-particle guiding function relied on an incorrect argument and generally requires the quantum potential to depend only on interparticle separation.The required condition Q(x1, x2) = Q(|x1 − x2|) is generally false.Source: §2.2.3 · pp. 87–88
• For a two-particle system, the configuration-space wave function determines ensemble probability density through its amplitude and individual motion through its phase.The wave function is defined over the six-dimensional configuration space of the two particles.Source: §2.2.3 · p. 89
• The pilot-wave construction was provisional because the double-solution principle remained hypothetical in the general case.De Broglie instead proposed taking the velocity law as a postulate, while acknowledging that this attitude could only be provisional.Source: §2.2.3 · pp. 89–90
• De Broglie introduced the pilot wave by postulating a physically real Ψ whose phase guides the motion of a distinct material point.This is identified as the first appearance in the literature of pilot-wave or de Broglian dynamics, stated explicitly for a single particle.Source: §2.2.3 · p. 89
• The theory predicted photon density proportional to the guiding-wave amplitude squared and hydrogen electronic charge density proportional to Ψ-based assumptions, matching wave optics and Schrödinger’s treatment.For interference with reflected waves, de Broglie attributed pressure to stresses in the interference zone rather than direct particle impacts on the wall.Source: §2.2.3 · p. 90

2.3.2 Signi ance of de Br o glie's `Stru tur e' p ap er

• De Broglie recognized that the singularity model could be dropped while retaining the velocity and probability-density formulas as postulates, yielding pilot-wave theory as a provisional measure.He nevertheless maintained that an underlying theory was still needed, probably along the lines of the discarded model.Source: §2.3.2 · p. 91
• Although later regarded as a consistent account of quantum phenomena, de Broglie’s “Structure” paper had been essentially ignored by physicists and historians and faced Bohr’s criticism.Bohr argued that its particle-wave reconciliation relied on classical concepts and was therefore unsuited to resolving fundamental difficulties.Source: §2.3.2 · pp. 90–91
• De Broglie’s “Structure” paper introduced pilot waves in configuration space to guide material systems, with particle motion and ensemble distributions determined by Ψ.The theory was formulated as a first-order dynamics with velocity law m_i v_i = ∇_i φ, unifying Maupertuis’s and Fermat’s principles.Source: §2.3.2 · pp. 91–92

2.3.2 Signi an e of de Br o glie's `Stru tur e' p ap er

• De Broglie challenges Schrödinger’s formulation by questioning configuration space without a real configuration and emphasizing that Ψ differs from an ordinary three-dimensional wave.The dimensionality of the abstract space is fixed by the system’s degrees of freedom.Source: §2.3.2 · pp. 93–94
• De Broglie’s pilot-wave theory extends from a single particle to N particles, using a wave function in configuration space to determine particle velocities.For many-body systems, the guiding wave propagates in configuration space rather than ordinary three-dimensional space.Source: §2.3.2 · pp. 93–94
• Given the wave function and an initial position, de Broglie’s first-order guidance law completely determines the particle’s motion and trajectory.The guidance equation is postulated by extending the classical Hamilton-Jacobi velocity formula beyond the classical limit.Source: §2.3.2 · pp. 94–95
• The report explicitly uses neither singular wave solutions nor the principle of the double solution, instead simply postulating pilot-wave dynamics.This contrasts with de Broglie’s earlier motivation involving particles as moving singularities within an extended wave.Source: §2.3.2 · p. 94
• For ensembles guided by the velocity field, the probability distribution is preserved in time, yielding the usual position probability when the initial position is ignored.De Broglie treats this as the probability of the particle being somewhere, but the distribution is an equilibrium distribution rather than an arbitrary one.Source: §2.3.2 · p. 95

2.3.2 Significance of de Broglie's 'Structure' paper

• His single-particle equations reduce to the standard pilot-wave equations in a nonrelativistic external electromagnetic field and reproduce familiar interference and diffraction patterns.The predicted patterns are the same for an intense beam over a short time or a feeble beam over a long time.Source: §2.3.2 · p. 96
• De Broglie treats v ∝∇φ as the fundamental equation of motion, preserving deterministic trajectories while the wave also yields particle-position probabilities.Ignoring the initial particle position gives a probability for finding the corpuscle in a volume element dτ at time t.Source: §2.3.2 · pp. 95–96
• For photons, de Broglie's guiding wave is distinct from the electromagnetic field, yet its dynamics account for interference, diffraction, and radiation pressure on a mirror.The photon trajectories need not strike the mirror; stresses associated with the wave are invoked to explain the reflected-light pressure.Source: §2.3.2 · pp. 96–97
• A neglected feature of the proposal is its explicit generalization to nonrelativistic many-body dynamics in configuration space, commonly attributed instead to Bohm.De Broglie describes N interacting waves whose propagation depends on the other N−1 particles, while a single configuration-space wave offers simpler autonomous dynamics.Source: §2.3.2 · pp. 97–98
• In this many-body formulation, each particle's conditional wave determines its velocity through its phase gradient while depending on the actual positions of the other particles.The proposal addresses Schrödinger's concerns by retaining real configurations and connecting particle motion to a configuration-space guiding wave.Source: §2.3.2 · p. 97

2.3.2 Signi an e of de Br o glie's `Stru tur e' p ap er

• The many-body probability formula appeared to agree with Born’s electron-scattering results and Fermi’s rotator-scattering results.The passage identifies agreement with both comparison cases but gives no numerical values.Source: §2.3.2 · p. 99
• The theory faced a fundamental difficulty in constructing a Lorentz-invariant pilot-wave formulation for relativistic many-body systems and remained incomplete regarding electromagnetic fields and electron spin.De Broglie had already recognized the relativistic many-body problem from the beginning of the theory’s development.Source: §2.3.2 · pp. 99–100
• De Broglie formulated pilot-wave theory for many-body systems with a guiding wave in configuration space, whose dynamics determine the representative point’s motion and probability distribution.This extends the single-particle pilot-wave framework to N particles in the nonrelativistic approximation.Source: §2.3.2 · pp. 98–99
• He nevertheless treated the many-body configuration-space wave as an effective mathematical theory, not a physically real pilot wave, and called pilot-wave theory provisional.De Broglie suggested that a deeper theory should incorporate corpuscles into the wave phenomenon through singular solutions.Source: §2.3.2 · pp. 99–100
• De Broglie applied the dynamics to general time-dependent wave functions, including hydrogen transitions, whose calculated dipole moments contained the correct transition frequencies.He also used the resulting charge and current densities with Maxwell’s equations to calculate mean atomic radiation, reproducing semiclassical radiation theory.Source: §2.3.2 · p. 100

2.3.2 Signi an ce of de Br o glie's `Stru tur e' p ap er

• Electron diffraction and inelastic electron–helium scattering provided numerical or qualitative support for de Broglie’s predictions, but he judged agreement with experiment premature and theory far behind experiment.The inelastic-scattering results showed maxima in the cross section qualitatively consistent with predictions, whereas de Broglie emphasized that a full agreement had not been established.Source: §2.3.2 · p. 101
• The conference discussion of de Broglie’s theory was extensive and raised unresolved questions about quantisation, the guidance equation, velocity fields, photon energy-momentum, and electron orbits.Participants including Lorentz, Born, Pauli, Schrödinger, Kramers, Ehrenfest, and Brillouin challenged or probed distinct aspects of the proposal.Source: §2.3.2 · p. 101

2.5 Signi an ce of de Br o glie's w ork from 1923 to 1927

• In 1927, de Broglie proposed pilot-wave dynamics for many-body systems, using a guiding wave in configuration space and presenting the first-order form still commonly used today.He regarded the theory as provisional and hoped it would emerge from a more fundamental theory in which particles were singularities of three-dimensional waves.Source: §2.5 · p. 103
• De Broglie’s earlier wave-guided dynamics explained single-particle interference, crystal-diffraction scattering, and electron scattering by atoms, but many applications remained undeveloped in 1927.The theory accounted for these phenomena while lacking a treatment of measurement for arbitrary quantum observables, which requires applying the dynamics to the measuring device itself.Source: §2.5 · p. 103

2.5 Signi an e of de Broglie's w ork from 1923 to 1927

• De Broglie progressed from particle–wave intuitions to a complete deterministic dynamics for nonrelativistic systems, later shown empirically equivalent to quantum theory under a Born-rule distribution.The theory’s first-order formulation is naturally expressed in configuration space rather than classical phase space.Source: §2.5 · p. 104
• The common characterization of pilot-wave theory as merely adding particle trajectories to Schrödinger’s equation is grossly inaccurate, since Schrödinger removed trajectories from de Broglie’s theory.This mischaracterization contributed to an undue reduction of de Broglie’s stature as a major contributor to quantum theory.Source: §2.5 · p. 104
• Pilot-wave theory is a consistent, viable approach without a measurement problem, showing that objectivity and determinism are compatible with quantum physics and motivating Bell’s inequalities.These points support a reassessment of de Broglie’s place in twentieth-century physics, regardless of whether the theory is closer to truth than other interpretations.Source: §2.5 · pp. 104–105

3.1 Summary of Born and Heisen b erg's rep ort

• Born and Heisenberg’s report begins with matrix-mechanical formalism, extends it toward Schrödinger’s wave mechanics, and develops a statistical interpretation using axiomatic and uncertainty-based arguments.The report covers formalism, interpretation, axiomatics and uncertainty, and applications, while presenting the probabilistic aspect as fundamental in its conclusion.Source: §3.1 · p. 108

3.1 Summary of Born and Heisenberg's report

• Born and Heisenberg present Schrödinger theory as a special case of operator quantum mechanics, with matrix mechanics recovered by choosing energy eigenstates as a basis.The Hamiltonian’s diagonalization yields the time-independent Schrödinger equation, while Hamiltonian eigenfunctions define the associated unitary transformation.Source: §3.1 · p. 109
• Their physical interpretation denies that matrix mechanics directly specifies actual states or when state changes occur, instead introducing probabilities for transitions and positions.For resonant coupled systems, quantum jumps between energy levels receive explicit transition probabilities; squared wave-function moduli provide position probability densities.Source: §3.1 · p. 109
• The report extends its generalized formalism to interference, open systems, spin, identical particles, quantum statistics, and spontaneous-emission transition probabilities.It treats interference through wave-function amplitudes, connects statistical choices to Bose-Einstein or Fermi-Dirac statistics, and emphasizes applications beyond matrix or wave mechanics proper.Source: §3.1 · p. 109
• Time dependence can be introduced through the time-dependent Schrödinger equation, whose transition expressions and energy-basis coefficient moduli receive probabilistic interpretations.The squared modulus of a wave function’s energy-basis coefficient is interpreted as the probability of the corresponding stationary state occurring.Source: §3.1 · p. 109
• Born and Heisenberg’s report treated indeterminism as consistent with experience and argued that electrodynamics would not alter this conclusion.They identified the remaining difficulties primarily with developing a fully relativistic quantum electrodynamics, while noting progress in that direction.Source: §3.1 · p. 111

3.2 Writing of the report

• Born and Heisenberg planned a joint report emphasizing the static conception of quantum mechanics and seeking agreement with Schrödinger on its physical interpretation.The report was assembled through exchanged drafts, with Born covering the introduction and sections I–II and Heisenberg sections III–IV and the conclusion.Source: §3.2 · pp. 111–112

3.3.1 Before matrix mechanics

• The old quantum theory was criticized as artificial because its stationary-state transitions did not naturally connect orbital and spectral frequencies, creating the radiation problem.The BKS theory offered a different picture by associating atoms with virtual oscillators at spectral frequencies, while conserving energy and momentum only statistically.Source: §3.3.1 · pp. 112–113

3.3.1 Befor e matrix me hani s

• After BKS, Heisenberg’s 1925 formulation [1] abandoned kinematical spatial coordinates while retaining classical-form motion equations under reformulated quantum conditions.The new variables retained formal continuities with earlier theories, including virtual oscillators and, initially, quantum jumps.Source: §3.3.1 · pp. 113–114

3.3.2 Matrix me hani s

• The early formulation was incomplete: its equations were Newtonian rather than Hamiltonian, and energy conservation was not established to all orders.The equations of motion use matrix-valued q and p, quantum conditions, and a Hamiltonian function H, but the original presentation had not yet established all later properties.Source: §3.3.2 · pp. 114–115
• Matrix mechanics represents observables as doubly infinite arrays indexed by transition frequencies satisfying Ritz’s combination principle, with products preserving the same time-dependent phase structure.The treatment follows Born and Jordan (1925 [2]), Born, Heisenberg and Jordan (1926 [4]), and Born (1926d,e [58]).Source: §3.3.2 · p. 114
• Solving the motion equations essentially reduces to diagonalizing the energy matrix, which yields frequencies, energies, conserved quantities, stationary-state averages, and transition-related amplitudes.In the non-degenerate case, H is diagonal; a unitary transformation can preserve Hermiticity of the new coordinates, while squared amplitudes |q_ij|^2 relate to spontaneous-emission intensities or transition probabilities.Source: §3.3.2 · pp. 116–117
• Matrix mechanics identified allowed transitions between states through relations among amplitudes and conserved quantities, including selection rules.Its historical formulation nevertheless lacked general expectation values for physical quantities and was regarded by Born and Heisenberg as inadequate and needing extension.Source: §3.3.2 · p. 118

3.3.3 F ormal extensions of matrix me hani s

• The formal extensions generalized matrices into q-numbers and operators, enabling treatment of aperiodic quantities and singular cases such as the free particle (Dira 1926a [7]; Born and Wiener 1926a [21]).Born and Wiener's operators act on a space of functions rather than being defined through matrix elements in a basis.Source: §3.3.3 · p. 119
• The operator approach allowed explicit solutions of equations of motion for systems more general than those previously treated, especially the aperiodic one-dimensional free particle.Born and Wiener's main example was the free particle, while Born and Heisenberg presented von Neumann's Hilbert-space operator formalism and connected it to Schrödinger's theory.Source: §3.3.3 · p. 119
• Despite their formal usefulness, q-numbers and operators did not provide further insight into the theory's physical interpretation.The subsequent discussion turned to describing actual states and processes in suitably extended matrix mechanics.Source: §3.3.3 · p. 120

3.4 In terpretation

• Born and Heisenberg identified the original matrix mechanics' main defect as describing possible states and processes rather than actual phenomena.This interpretational problem appears at the beginning of their section on physical interpretation.Source: §3.4 · p. 120
• Matrix mechanics describes closed periodic systems and does not by itself specify when a given state is present or when a change should be expected.Accounting for actual states and their time development therefore raises the problem of incorporating an interacting subsystem into the formalism.Source: §3.4 · p. 120
• The report contains at least two, possibly three, disparate approaches to quantum-mechanical states, and its physical interpretation remains unsettled.These include stationary states, wave-function-based states, and a possibly purely effective notion of state.Source: §3.4 · pp. 120–121

3.4.1 Matrix me hani s, Born and Wiener

• Original matrix mechanics lacks the modern notion of state, retaining an ostensibly stationary-state and quantum-jump interpretation while formally containing only matrices.The matrices can at most be understood as collective representations of all stationary states.Source: §3.4.1 · p. 122
• Born and Wiener associate operator columns with stationary states, and their free-particle example identifies the real part of a generalized column sum with classical inertial motion of energy W.The generalized column sum is given as µ + F(W), with µ the particle’s mass and F(W) independent of t.Source: §3.4.1 · p. 122
• The analysis suggests that Born and Wiener’s framework associates a system’s state with a position-operator column and permits a limited spatiotemporal particle picture.The passage presents this as an indication of their view rather than a fully developed interpretation.Source: §3.4.1 · p. 123
• Born and Wiener allow trajectories in the absence of interactions, extending limited spatiotemporal descriptions of atomic states.Source: §3.4.1 · p. 123

3.4.2 Born and Jor dan on guiding elds, Bohr on ol lisions

• Born and Jordan developed a guiding-field proposal combining wave emission during stationary states with light-quantum emission during instantaneous quantum jumps.The proposal was applied successfully to a few simple examples, but Bohr criticized its trajectory and absorption assumptions.Source: §3.4.2 · p. 123
• Bohr’s collision scheme faced difficulty accommodating the Ramsauer effect, contributing to doubts about statistical conservation and spacetime collision pictures.The Ramsauer effect involved anomalously low cross sections for slow electrons in certain gases.Source: §3.4.2 · p. 124

3.4.3 Born 's ol lision p ap ers

• Born treated collisions as transitions between stationary states, using atom–electron scattering and asymptotic behavior to infer initial-to-final transition probabilities.The combined system was assumed to approach an atomic stationary state plus uniform electron motion asymptotically.Source: §3.4.3 · p. 125
• Born’s wave function defines a probability distribution over the atom’s and electron’s stationary states, distinct from those stationary states themselves.Source: §3.4.3 · p. 125
• Born solves the time-independent Schrödinger equation for atom–electron collisions and interprets squared wave-function coefficients as transition probabilities.The initial wave function represents a determined stationary state and inertial motion, so the scattering probability also represents the probability of a quantum jump to the final state.Source: §3.4.3 · pp. 125–126
• Born treats de Broglie–Schrödinger waves as guiding fields whose particle trajectories are determined only probabilistically, making the guiding-field picture fundamentally indeterministic.Born states that a deterministic completion, if possible, would have no practical use.Source: §3.4.3 · p. 126

3.4.4 Heisenb er g on ener gy u tuations

• Heisenberg derives transition probabilities for resonant subsystems within matrix mechanics without introducing the wave function as an external aid.His reasoning considers two weakly coupled systems, whose uncoupled degeneracy is lifted and whose coupled-system eigenstates are related to the product basis by a transformation matrix.Source: §3.4.4 · pp. 127–128
• Heisenberg treats transition probabilities between stationary subsystem states as conditional probabilities, with an apparently instantaneous, correlated transfer of energy between the systems.The account avoids the BKS theory’s energy non-conservation but appears to require explicit correlation at a distance.Source: §3.4.4 · pp. 128–129

3.4.5 T r ansformation the ory

• Dirac’s transformation theory [38] generalizes Heisenberg’s approach by representing observables in arbitrary conjugate-variable representations and interpreting diagonal elements as conditional probabilities.For uniformly distributed conjugate variables, the diagonal elements yield the conditional probability or density for an observable given a fixed value of its conjugate.Source: §3.4.5 · pp. 129–131
• Dirac’s interpretation assumes that diagonal matrix elements represent time averages, but its stated justification is only that this should hold in the limiting case of large quantum numbers [38].The passage explicitly characterizes this justification as weak.Source: §3.4.5 · p. 130
• Jordan independently developed a similar transformation theory [39], while the text situates these developments within the broader realization that quantum-mechanical operators act on a Hilbert space.Born and Heisenberg reportedly regarded the two methods as equivalent, although later analysis points to subtle differences.Source: §3.4.5 · pp. 131–132
• Dirac’s transformation theory did not resolve quantum mechanics’ meaning: his paper lacked a state-vector notion, although probabilities arose from probabilistic initial information (Dirac 1927a [38]).The equations remain deterministic, while probability enters through assumptions about the initial state.Source: §3.4.5 · p. 132

3.4.6 Development of the `statisti al view' in the r ep ort

• Born and Heisenberg derived quantitative probabilities for quantum jumps between energy eigenstates, showing that matrix mechanics could describe time-dependent phenomena without wave mechanics.Their treatment used coupled systems and transition probabilities between energy eigenstates.Source: §3.4.6 · p. 133
• The time-dependent Schrödinger equation appears to function as an effective, convenient description, while the wave function supports probabilities and interference within the statistical view.Born and Heisenberg introduce it after their matrix-mechanical discussion and use it in discussing probability and interference.Source: §3.4.6 · p. 134
• Born and Heisenberg’s presentation first brings measurements into the physical interpretation of the theory, alongside their discussion of interference.This marks a distinct role for measurement within the statistical view.Source: §3.4.6 · p. 135
• Born and Heisenberg treat |S_nk(t)|^2 as transition probabilities, whereas |c_n|^2 are state probabilities only when energies have been measured non-selectively.This distinction limits the interpretation of |S_nk(t)|^2 as quantum-jump probabilities to special cases and makes the terminology somewhat ambiguous.Source: §3.4.6 · pp. 135–136
• Born and Heisenberg generalize the statistical discussion to arbitrary observables using relative-state probabilities, while retaining interference as a general expectation.The physical interpretation presupposes that quantities such as |ϕ|^2 become well-defined through measurement, but their terminology is ambiguous.Source: §3.4.6 · p. 136
• The report identifies a conceptual shift from probabilistic transitions between possessed energy values to probabilistic transitions from one measurement to the next.Frequencies appear well-defined only upon measurement, in a role compared with von Neumann’s projection postulate, although it remains unclear whether this was Born and Heisenberg’s intended view.Source: §3.4.6 · p. 137

3.4.7 Justi ation and over al l onclusions

• Jordan’s axiomatic formulation is presented as justifying a statistical view of quantum mechanics through Heisenberg’s notion of uncertainty.The supplied discussion begins by motivating this view through uncertainty, including an analogy with limitations already present in classical mechanics.Source: §3.4.7 · p. 137

3.4.7 Justi ation and over al l on lusions

• Born and Heisenberg argue that quantum mechanics’ statistical element can be reconciled with macroscopic determinism.Their discussion distinguishes statistically constrained future evolution from the classical evolution of probability densities and classical particle trajectories.Source: §3.4.7 · p. 138
• Born and Heisenberg describe quantum laws as governing wave-like propagation of probability functions, with classical probability evolution recovered as a very good approximation in suitable cases.They use this relation to justify classical treatments of α- and β-particle trajectories.Source: §3.4.7 · p. 138
• The comparison between quantum probability spreading and a Liouville distribution may be an early decoherence consideration, but the text is too brief to determine this.It may instead simply compare the spreading of quantum probabilities with Liouville evolution; Ballentine (2003) is cited for a modern treatment.Source: §3.4.7 · p. 138
• Quantum mechanics, excluding electrodynamics, is presented as a closed theory whose physical alternatives must agree with experience.The report treats divergent assumptions about quantum mechanics’ physical meaning, including Schrödinger’s interpretation of |ϕ|2 as charge, as empirically constrained.Source: §3.4.7 · p. 139

Schrödinger's w a v e me hani s

• Schrödinger’s 1926 wave mechanics pursued a purely wave-theoretical description of matter, treating particles as wave packets and eliminating point particles altogether.This program contrasted with de Broglie’s work and became a central point of conflict with proponents of quantum mechanics.Source: §3.4.7 · p. 141

4.1 Planning of S hrö dinger's rep ort

• Schrödinger planned to distinguish waves in spacetime from the highly successful, nonrelativistic waves in configuration space, while focusing on principles rather than applications.The spacetime theory faced difficulties especially for many-electron systems, whereas the configuration-space theory raised questions about interpreting the wave function.Source: §4.1 · p. 143
• Schrödinger’s preferred individual-system interpretation conflicted with the widespread ensemble view and faced difficulties including electron separation, time symmetry, and probabilistic amplitudes.He questioned how a time-symmetric Schrödinger equation could describe ensembles whose statistical behavior is not time-symmetric, and objected to resulting probability problems.Source: §4.1 · p. 144

4.2 Summary of the rep ort

• The eventual report treated configuration-space theory, spacetime theory, and the many-electron problem in three main sections, identifying the multidimensional theory as most successful.It described configuration space as a mathematical representation of events in space and time and emphasized its analytical power in relation to Heisenberg and Born’s matrix mechanics.Source: §4.2 · p. 144
• Schrödinger frames the central interpretive problem as explaining what a configuration-space wave function represents in three-dimensional space.He considers an individual system as continuously filling space weighted by |ψ|2, while acknowledging that the resulting density is not classical charge density.Source: §4.2 · p. 145
• Schrödinger interprets the wave function through spatial densities, but rejects treating them as classical charge densities acting back on particles.The densities can nevertheless partially account for calculating the classical radiation field, although this is only an approximation because radiation involves further interactions.Source: §4.2 · p. 145
• In the four-dimensional theory, coupling the Klein-Gordon equation to Maxwell’s field yields charge and current densities, but the electron’s self-field gives wrong hydrogen results.Schrödinger therefore argues that interacting-particle theories must distinguish fields generated by individual particles, and notes that spin requires modifying the scalar equation.Source: §4.2 · pp. 145–146
• For many-electron systems, Schrödinger expands the full solution in products of single-electron wave functions with time-dependent coefficients.Before approximation, the coefficient equations depend only on potentials calculated from purely spatial charge densities, reinforcing a spatial interpretation of the wave function.Source: §4.2 · p. 146
• Schrödinger’s wave-packet picture aims to localize particles through superposed plane waves, but such packets spread over increasingly large volumes after a focal point.This spreading motivates seeking a quantum-theoretical modification of classical wave laws to address the light-quantum dilemma (1926a, p. 101).Source: §4.2 · pp. 146–147
• Wave-packet centroids follow classical equations of motion, but generic spreading makes electron trajectories and positions lose their meaning.Schrödinger initially hoped material points could be represented by small wave packets, yet packet spreading proved more general than expected.Source: §4.2 · p. 147
• Although packets remain localized for the harmonic oscillator, Schrödinger’s hoped-for generalization to quasi-periodic motion admitted possible dissolution for a free electron.The oscillator result therefore did not remove the broader localization problem.Source: §4.2 · p. 148
• A Lorentz calculation showed that wave packets on high hydrogen orbits spread out, undermining hopes of connecting such packets to Bohr’s correspondence principle.This result weakened the prospect that wave packets could explain the transition from microscopic to macroscopic mechanics along high-quantum-number orbits.Source: §4.2 · p. 148
• Scattering sharpened the tension: wave packets spread in all directions at obstacles, while Wilson cloud-chamber experiments indicated continued well-defined trajectories after collision.Davisson–Germer interference supported part of the spreading-wave picture, whereas cloud-chamber observations supported a particle-like trajectory.Source: §4.2 · p. 149
• Schrödinger regarded combining waves with point electrons as a provisional compromise rather than a satisfactory resolution of the micro–macrophysics relation.The underlying difficulty was linked to the wave equation’s linearity and its production of highly nonclassical states.Source: §4.2 · p. 149
• Nonlinear terms were proposed as necessary for combination tones and radiation-related effects, including Schrödinger’s speculation that the equations might be only approximately linear.Fermi (1927) later proposed modifying Schrödinger’s equation so that the expectation value ⟨x⟩ obeys the Abraham–Lorentz radiation-reaction equation.Source: §4.2 · pp. 149–150
• Wave packets can represent the classical regime only if Schrödinger’s equation is modified stochastically, since straightforward wave-packet localization fails.The report distinguishes this limitation from the broader question of whether Schrödinger’s wave picture can describe apparently particulate or discontinuous behavior.Source: §4.2 · p. 150
• Schrödinger also explored whether continuous waves could describe phenomena such as the Compton effect, quantum jumps, blackbody radiation, and the photoelectric effect.These efforts were part of a broader attempt to account for apparently particulate or discontinuous quantum behavior using continuous waves.Source: §4.2 · p. 150

4.4 The problem of radiation

• The radiation problem remained conceptually difficult because the frequency of emitted light differed from the electron’s oscillation frequency in the Bohr model.Schrödinger welcomed the BKS theory partly because it supplied a virtual mechanism for radiation and allowed energy and momentum conservation to hold only statistically.Source: §4.4 · p. 151
• Wave mechanics promised a continuous radiation description, but linear time-dependent wave equations yield beats without providing a known radiation mechanism.Schrödinger regarded this continuous-wave picture as more appealing than quantum jumps, although the proposal remained speculative.Source: §4.4 · pp. 151–152
• Schrödinger interpreted ψ as a superposition of classical configurations weighted by |ψ|2, associating it with a corresponding charge density in three-dimensional space.For several particles, each particle’s contribution can be obtained by integrating over the coordinates of the other particles (Przibram 1967, p. 56).Source: §4.4 · p. 153
• Schrödinger’s wave-function interpretation focused on how a multiparticle wave function manifests in three-dimensional space, but its “snapshot” picture prompted extensive discussion.The interpretation associated all possible classical configurations with a three-dimensional charge density, while Schrödinger acknowledged that the snapshot description had been misunderstood.Source: §4.4 · pp. 153–154
• The associated charge density was not an ordinary electric charge: it sourced semiclassical fields but did not experience electromagnetic forces.Schrödinger regarded the field–charge coupling as provisional because radiation reaction is omitted and semiclassical coupling is inconsistent for interacting charged particles in a closed system.Source: §4.4 · p. 153
• Schrödinger’s classical dipole-moment formula does not in general yield the correct radiation intensities.The discrepancy arises because the calculation produces coefficients involving both initial- and final-state amplitudes, whereas radiation intensity should not depend on the final-state coefficient.Source: §4.4 · p. 154
• Interpreting the wave function as a charge density creates principled problems, although under some circumstances its spatial density can function as a radiation source.This proposal does not resolve wave-packet spreading, Schrödinger’s cat, or related difficulties, but offers a platform for investigating them.Source: §4.4 · p. 155

4.5 Schrödinger and de Broglie

• Schrödinger and de Broglie applied the optical-mechanical analogy in opposite directions: de Broglie retained particle trajectories, whereas Schrödinger treated the electron as a pure wave.Schrödinger’s wave mechanics developed from de Broglie’s phase-wave ideas but rejected the trajectory picture.Source: §4.5 · pp. 155–156
• Schrödinger may have abandoned trajectories because well-behaved trajectories appeared incompatible with well-behaved waves, especially when degeneracy permitted arbitrary linear combinations and complicated orbits.The passage presents this as a possible reason rather than a definitive explanation of Schrödinger’s decision.Source: §4.5 · p. 156
• De Broglie’s trajectory theory reproduced Bohr orbits from wave considerations but yielded the wrong radiation frequency if the electron acted as a classical source.The discrepancy arises because the electron’s revolution frequency remains the Bohr-model frequency, and degeneracy would make the situation still more complicated.Source: §4.5 · p. 157

4.5 S hrö dinger and de Broglie

• Misgivings about Born’s treatment of waves and particles could also apply to de Broglie’s theory.Source: §4.5 · p. 157

4.6.1 Early days

• In the early debate, Schrödinger treated spatiotemporal intelligibility as central, sought a physical interpretation, and viewed wave mechanics as offering a continuous alternative to matrix mechanics’ quantum jumps.Schrödinger also held that mathematical equivalence did not guarantee physical equivalence, because theories could support different generalizations and further developments.Source: §4.6 · pp. 157–158

4.6 The on i t with matrix me hani s

• The conflict between wave and matrix mechanics concerned physical issues—not only philosophy—including scattering, radiation, quantum jumps, and continuous atomic transitions.Neither scattering nor radiation resolved the debate for either theory; both were modified or reinterpreted, although the resulting statistical and Copenhagen interpretations did not satisfy Schrödinger.Source: §4.6 · pp. 157–158

4.6.1 Early days

• Heisenberg regarded Schrödinger’s claim of physical Anschaulichkeit as ludicrous, partly because the electron was spread out and the waves extended into a fourth and fifth dimension.The criticism targeted the physical meaning of Schrödinger’s theory rather than its mathematical formulation alone.Source: §4.6.1 · p. 160

4.6.2 F r om Muni h to Cop enhagen

• Schrödinger’s radiation mechanism drew enthusiasm from Wien but criticism from Sommerfeld and Heisenberg, producing a heated discussion.The dispute concerned wave mechanics’ account of radiation and its relation to quantum jumps and coherent scattering.Source: §4.6.2 · pp. 160–161
• The discussion identified a crucial experiment that could decide between a continuous wave mechanism and quantum-jump explanations of radiation.Sommerfeld and Heisenberg were reportedly prepared to bet on the existence of the predicted coherent-scattering effect, while experimental physicists opposed it and Schrödinger remained cautious.Source: §4.6.2 · pp. 160–161
• The Copenhagen discussions left the structure of matter without an adequate contradiction-free interpretation, despite wave theory’s success for interference and diffraction and corpuscular theory’s necessity for other phenomena.Heisenberg characterized the prevailing interpretation as a dualism whose contradictions remained unsatisfactory.Source: §4.6.2 · pp. 162–163
• Bohr and Heisenberg argued that quantum jumps were necessary, while Schrödinger maintained that wave mechanics lacked a satisfactory mechanism for radiation and resisted discontinuous probabilistic transitions.Heisenberg later reported that the Copenhagen discussions recognized wave mechanics without quantum jumps as impossible, whereas Schrödinger regarded both quantum jumps and their probabilistic replacement as unsatisfactory.Source: §4.6.2 · pp. 163–164
• Heisenberg’s uncertainty work and Bohr’s complementarity sought a missing visualizable interpretation, but Heisenberg treated apparent wave aspects as recoverable in matrix terms.The uncertainty formulation also displaced stationary states as the privileged basis and replaced their transitions with probabilistic transitions between measurements.Source: §4.6.2 · p. 164

4.6.3 Continuity and dis ontinuity

• Schrödinger continued developing wave-based approaches from late 1926 to the Solvay conference, moving between a four-dimensional theory and a many-dimensional theory.His related work included papers on the Compton effect (1927a) and the energy-momentum tensor (1927b), followed by a return to the many-dimensional theory shortly before Solvay.Source: §4.6.3 · pp. 164–165
• The continuous wave treatment of the Compton effect relied on stationary waves and did not directly describe an individual collision.Source: §4.6.3 · p. 165
• Schrödinger’s formal energy-momentum development intensified difficulties with the four-dimensional view, including incorrect results when an electron’s self-field was included.Despite being described as purely formal, the four-dimensional theory later provided a basis for quantum field theory.Source: §4.6.3 · p. 165
• Schrödinger argued that wave mechanics could eliminate quantum jumps from the description of resonant energy exchange between atoms.He further suggested interpreting quantised energy in terms of wave frequency rather than definite exchanged quanta.Source: §4.6.3 · p. 166
• Schrödinger linked squared wave amplitudes to the strength of excitation of an eigenvalue and proposed that resonance with a heat bath could recover Planck’s radiation formula and old quantum statistics.Source: §4.6.3 · p. 166
• The wave picture treats crucial wave-function properties as properties of the entire wave, introducing nonlocalisable properties as a price for describing matter continuously.Source: §4.6.3 · p. 167

5.1 What is quan tum theory?

• The pilot-wave theory of de Broglie and Bohm (de Broglie 1928; Bohm 1952a,b; Bohm and Hiley 1993; Holland 1993) is one of several actively pursued alternatives.Other approaches include Everett interpretations, collapse models, and consistent or decoherent histories.Source: §5.1 · p. 172
• The meaning of quantum theory remains an open question rather than a settled Bohr–Heisenberg orthodoxy, much as it arguably was in October 1927.The interpretation debate remains controversial, and Bohr’s and Heisenberg’s views are not regarded as standard or canonical.Source: §5.1 · p. 173

5.2 The measuremen t problem to da y

• The measurement problem concerns where measurement and observation begin and end, with Bell arguing that physicists’ observational language can become conceptually empty (Bell 1986, p. 48).This recurring puzzle encompasses related questions grouped under the general heading of the measurement problem.Source: §5.2 · p. 173

5.2.1 A fundamental ambiguity

• Textbook quantum theory is practically successful but fundamentally ambiguous because it presumes an imprecise boundary between microscopic systems and macroscopic apparatus.Bell’s formulation divides the world into observer and observed without specifying where that division lies, while the theory must explain how definite macroscopic states emerge at smaller scales.Source: §5.2.1 · p. 173
• The fundamental ambiguity is the absence of a sharp boundary between classical systems with real states and a quantum domain where that concept is unavailable.Standard textbook quantum theory has not resolved this ambiguity, motivating proposals for unified descriptions of microscopic and macroscopic phenomena, including de Broglie's pilot-wave theory (1928).Source: §5.2.1 · pp. 174–175
• The section notes that realist alternatives extend quantum theory’s notion of real state, while some proposed hidden-variable approaches fail to reproduce standard quantum dynamics except for restricted wave functions.The stochastic hidden-variable theory of Fényes (1952) and Nelson (1966) treats the wave function as an emergent probability description, but obtains the Schrödinger equation only for exceptional wave functions (Wallstrom 1994; Pearle and Valentini 2006).Source: §5.2.1 · p. 175, §5.2.2 · p. 175

5.2.2 Me asur ement as a physi al pr o cess: quantum the ory

• Quantum theory’s application to measurement remains controversial because apparatus and observers should, in principle, be treated as physical systems, yet doing so can produce paradoxes and confusion.The Wigner’s-friend scenario illustrates the tension between continuous Schrödinger evolution for the whole box and the definite result experienced by the internal observer; environmental decoherence may also limit the scenario’s realism.Source: §5.2.2 · p. 175

5.2.2 Me asur ement as a physi al pr o ess: quantum the ory

• Treating humans and apparatus as physical systems makes measurement analyzable through their interaction with microsystems and raises whether interference effects should remain statistically observable.Denying this possibility suggests that a box containing a human violates quantum theory.Source: §5.2.2 · p. 177

5.2.2 Me asur ement as a physi al pr o cess: quantum the ory

• A later interference experiment may destroy B’s memory, but it cannot remove the prior matter of fact that B obtained a definite result.The section distinguishes facts about completed experiments from memories, opinions, or appearances.Source: §5.2.2 · pp. 177–178

5.2.2 Me asur ement as a physi al pr o ess: quantum the ory

• Wigner’s paradox contrasts the claim that B has no definite state, because A can test for interference, with the claim that B has a definite state as a human experimenter who can report what he saw.The contradiction concerns B’s physical state before A decides whether to perform the interference experiment.Source: §5.2.2 · pp. 177–178
• Accepting that experimental facts could later cease to be facts creates a paradox, whereas treating measurement physically preserves the definite results of past laboratory actions.Wigner’s argument also appeals to the reality of other minds: A treats B’s answer as evidence of B’s earlier mental state, even before A chooses an interference test.Source: §5.2.2 · p. 178
• A coherent account of quantum measurement requires extending quantum theory from microsystems to a universal physical theory encompassing everyday macroscopic realism.Such a theory could analyze measurement as an ordinary physical process, yielding a quantum theory without observers as extra-physical elements.Source: §5.2.2 · pp. 179–180
• The physical account of measurement depends on the chosen theory: Everett permits two outcomes and later interference, whereas de Broglie-Bohm selects one outcome while retaining a potentially re-overlapping empty wave packet.In de Broglie-Bohm theory, suitable Hamiltonians can make the empty and occupied packets interfere again.Source: §5.2.2 · p. 180

5.2.3 Quantum osmolo gy

• Inflationary cosmology made quantum accounts of the early universe practically relevant, linking primordial quantum fluctuations to density perturbations whose imprint appears in the cosmic microwave background.This constitutes a quantum-to-classical transition before life or galaxies developed, with details measurable in the sky.Source: §5.2.3 · pp. 180–181
• Quantum cosmology exposes the tension between Copenhagen theory and a closed universe, which lacks an external classical vantage point or anything outside the system to produce state transitions.Everett’s motivation included describing quantum physics internally to an isolated system, though his original formulation faced criticisms concerning worlds and probability.Source: §5.2.3 · p. 181
• When the whole universe is treated as a quantum object without a classical observational background, physics becomes unintelligible unless the quantum object is assigned a real state or ontology.The Everett interpretation provides one way, among others, to supply this ontology.Source: §5.2.3 · p. 181
• Generalised quantum mechanics based on consistent or decoherent histories aims to provide a theory without observers or a presumed classical background for quantum cosmology.This motivation arises because conventional quantum mechanics relies on external observation and wave-function collapse to prescribe outcome probabilities.Source: §5.2.3 · p. 182

5.2.4 The me asur ement pr oblem in `statisti al'

• The statistical interpretation makes the measurement problem a pseudo-problem only if ψ describes ensemble frequencies rather than physical superpositions of individual systems.On this view, a mathematical superposition in ψ does not represent a real superposition of states for an individual system.Source: §5.2.4 · p. 182
• Even the statistical interpretation remains incomplete because it refers only to ensembles and does not fully describe individual quantum systems, measurement events, or the macroscopic boundary.Source: §5.2.4 · p. 183
• Bell’s theorem shows that quantum physics is nonlocal under assumptions excluding backward causation or many worlds, while locality prevents deterministic completions from reproducing all entangled-state correlations.Consequently, Einstein’s locality-based argument cannot establish anything about quantum theory’s completeness or the nature of ψ.Source: §5.2.4 · p. 184
• Because locality is no longer a reasonable premise, Einsteinian arguments cannot establish that the quantum state is subjective or epistemic.This conclusion concerns the measurement problem in the statistical interpretation.Source: §5.2.4 · p. 185

6.1 Probabilit y and in terferen e

• Standard probability calculus is consistent with the two-slit experiment because P12, P1, and P2 describe distinct conditional conditions, not related outcomes.The claim that quantum interference requires adding probability amplitudes, associated with Feynman (1965), is therefore rejected.Source: §6.1 · p. 186
• De Broglie's pilot-wave theory explains single-particle interference, with conference discussions showing no objection to its consistency.The theory reproduces the observed interference pattern while retaining definite particle trajectories.Source: §6.1 · p. 187
• Born and Heisenberg presented views differing from contemporary associations of quantum mechanics, while Schrödinger's contributions contain no specific discussion of interference.The section contrasts their account of probability calculus with Schrödinger's wave-mechanical treatment.Source: §6.1 · p. 187

6.1.1 Interfer en e in de Br o glie's pilot-wave the ory

• De Broglie's model derives the usual interference and diffraction patterns from a guiding wave, matching classical wave-optics fringes for intense or feeble sources.Its preserved distribution π = const·a2 yields the same bright and dark fringes regardless of source intensity.Source: §6.1.1 · p. 188
• Pilot-wave trajectories guided by the quantum current produce an arrival distribution |ψ|2, including interference or diffraction patterns determined by the evolving wave.The construction is presented as simple: trajectories follow current lines, and their distribution necessarily becomes |ψ|2 at the backstop.Source: §6.1.1 · p. 188
• Pilot-wave theory rejects Feynman’s and Heisenberg’s inference that definite trajectories require the non-interfering result P1 + P2.Their argument assumes a particle’s motion is unaffected by opening or closing the other slit, whereas the guiding wave depends on whether both slits are open.Source: §6.1.1 · p. 189
• Born and Heisenberg interpret |cn(t)|2 as state probabilities in a statistical framework where the system is always in a definite energy state.They treat |Snk(t)|2 as transition probabilities and describe superpositions as expressing possible transitions among quantum states.Source: §6.1.1 · p. 190
• This Born–Heisenberg account differs from present-day quantum mechanics, which normally does not regard a superposition as always occupying only one energy state.The distinction is explicitly identified as a difference between their interpretation and contemporary quantum mechanics.Source: §6.1.1 · p. 190
• Born and Heisenberg claim that interference probabilities remain ordinary probabilities and reduce to the non-interfering expression when state populations are established beforehand.They assert that the interfering expression is compatible with probability calculus and becomes the non-interfering form under the stated condition.Source: §6.1.1 · p. 191
• Applying modern state-vector collapse creates an inconsistency with immediate repeated energy measurements, which should reproduce the measured value with certainty.The inconsistency appears absent if transition probabilities, rather than the usual collapse rule, determine the subsequent result.Source: §6.1.1 · pp. 192–193
• Born and Heisenberg’s presented quantum mechanics appears to contain no wave-packet collapse or state-vector reduction.Instead, they interpret |S_nm|^2 as ordinary transition probabilities even when interference is present.Source: §6.1.1 · p. 193
• Measured energies yield a non-interfering result through phase randomisation and the probability calculus.The same result is obtained using transition probabilities |S_nm|^2 and population probabilities |c_m(0)|^2.Source: §6.1.1 · pp. 191–193
• Without energy measurements, phase relations remain intact and the interfering expression predicts interference, unlike the non-interfering expression.The passage describes the corresponding population probabilities as ill-defined, so the usual probability formula cannot simply be applied.Source: §6.1.1 · p. 193
• The discussion suggests an uncertainty or complementarity relation between population probabilities and phase relations.Heisenberg’s argument treats their simultaneous sharp definition as operationally impossible, paralleling the incompatibility of definite position and momentum.Source: §6.1.1 · p. 194
• Interference requires definite phase relationships, so measuring population probabilities makes phases indefinite and averaging over them washes out interference.In the pilot-wave analysis, ordinary probability calculus cannot be applied when phase information and population probabilities are treated as jointly definite.Source: §6.1.1 · p. 194

6.2 Ma ros opi sup erp osition: Born's dis ussion of the loud

• Born’s cloud-chamber discussion treats macroscopic superposition as a measurement problem, while asserting that wave-packet collapse is not required to describe the observed track.The environmental apparatus is treated classically, and the resulting trajectory becomes concentrated along straight lines after repeated localization associated with visible ionization.Source: §6.2 · pp. 195–196

6.2.1 Quantum me hani s without wave p a ket ol lapse?

• Born considers describing the alpha particle and cloud-chamber atoms as one quantum system in multidimensional configuration space, but says this does not advance the collapse problem.The proposed space has three times as many dimensions as there are particles present.Source: §6.2.1 · p. 196
• Born models the apparatus and particles together in configuration space, suggesting that reduction is unnecessary for the full system but relevant to subsystem descriptions.His two-atom model represents the total localized packet in three-dimensional configuration space, with branches corresponding to alternative collision arrangements.Source: §6.2.1 · p. 197
• Born appears to associate collisions with a choice of the system’s propagation direction rather than collapse of the wave packet itself.The wave packet continues propagating in both directions and determines probabilities for the possible motions, while the system follows one selected direction.Source: §6.2.1 · p. 198
• Born’s treatment appears to replace wave-packet collapse with an update of probabilities, leaving the complete wave packet unchanged.The wave packet propagates in both directions, while the system’s motion is selected and the probability distribution undergoes an ordinary reduction.Source: §6.2.1 · pp. 197–198
• The account suggests stochastic electron trajectories whose possible paths are assigned probabilities by the wave function ψ.The passages leave unclear whether ψ is a physical field attached to individual systems or a quantity referring only to an ensemble.Source: §6.2.1 · p. 199

6.2.1 Quantum mechanics without wave packet collapse?

• Born’s later description of ψ as a “guiding wave” suggests lingering influence from ideas that inspired him in 1926, while the text cautions that his earlier intentions remain unclear.The discussion also notes that, in a multidimensional treatment, ψ does not really collapse, making its interpretation as merely an ensemble probability distribution difficult to determine.Source: §6.2.1 · p. 199
• The analysis argues that Born and Heisenberg’s account can retain all components of a superposition after measurement, with phase randomization replacing ordinary collapse.This interpretation is explicitly noted for energy measurements and appears possible in Born’s loud-chamber example, although the relative phases are unspecified.Source: §6.2.1 · p. 200
• Delayed-choice interference exposes the measurement problem: inserting a plate can imply past interfering trajectories, whereas omitting it can imply passage through one slit, even when the choice is made late.The passages connect this ambiguity to when definite outcomes emerge and to the boundary between quantum and classical domains.Source: §6.2.1 · pp. 200–201
• In de Broglie–Bohm theory, detections at P or P′ arise from particles traversing the bottom or top slit, respectively, contrary to the standard inference that no definite path exists.The passage attributes this result to Bell 1980; Bell 1987, chap. 14.Source: §6.2.1 · p. 201
• Dirac described outcomes as selected by nature when ψ decomposes into non-interfering states, but did not specify when such decompositions occur or whether they are unique.These omissions leave unresolved the mechanism and uniqueness of state reduction.Source: §6.2.1 · p. 202

6.2.1 Quantum me hani s without wave p a ket ol lapse?

• Dirac’s interpretation treats interference as evidence that nature may postpone choosing where the electron appears when later experimental arrangements make causal histories indistinguishable.Without mirrors, detection can suggest a direction was chosen at scattering; with mirrors, interference prevents tracing the causal chain to such an earlier choice.Source: §6.2.1 · p. 203
• Heisenberg’s objection shows that later-added mirrors can produce interference, making Dirac’s claim that nature chose a scattering direction at collision difficult to sustain.The issue is how a choice made at scattering could be undone by subsequently adding mirrors; the authors identify this as a deep difficulty at the heart of the measurement problem.Source: §6.2.1 · p. 205
• Heisenberg argues that interference can be obtained by placing two mirrors far from the scattering region and measuring only after a long delay.His arrangement removes the original detectors and lets separated parts of the scattered wave re-overlap and interfere later.Source: §6.2.1 · p. 205
• The example makes the choice between which-way information and interfering paths potentially occur long after the particle has completed most of its journey.The discussion relates this delayed choice to Wheeler’s delayed-choice experiment.Source: §6.2.1 · p. 205
• Heisenberg’s remarks contain the essence of Wheeler’s delayed-choice experiment: an observer’s measurement makes the choice physically real and destroys interference.The account implies that a definite outcome appears only when an experimenter observes, although this interpretation faces objections about treating detectors and observers as ordinary physical systems.Source: §6.2.1 · p. 206
• Delayed-choice reasoning correctly predicts experimental statistics but makes whether past paths were distinguishable depend on the experimenter’s present decision.The precise definition of subsequent distinguishability remains problematic because later experimental choices determine the relevant status of earlier paths.Source: §6.2.1 · p. 207
• De Broglie’s pilot-wave theory explains delayed-choice interference without present actions influencing the past: the particle follows one route while wave components traverse alternatives and later recombine.An empty wave can be reflected and affect the particle when it overlaps the wave carrying it in the interference region.Source: §6.2.1 · p. 207

6.4 F urther remarks on Born and Heisen b erg's quan tum

• Born and Heisenberg shift from probabilistic transitions between definite energy states to probabilistic transitions between measurements, leaving their treatment of definite values unclear.They state earlier that systems always occupy definite energy states, but later make unmeasured population probabilities unmeaningful, an apparent inconsistency.Source: §6.4 · p. 208
• Born and Heisenberg’s interference discussion appears to replace collapse to a single energy eigenstate with phase randomization that suppresses interference.The account links interference to uncertain phase relationships and notes that averaging over random phases destroys interference.Source: §6.4 · pp. 208–209
• The scope of this phase-randomization argument remains unclear: the sources provide no clear evidence that Born or Heisenberg generalized it beyond energy measurements.A possible position-measurement generalization would randomize relative phases and destroy positional interference, but the text says no clear evidence supports that extension.Source: §6.4 · p. 209
• Born and Heisenberg’s Solvay contributions do not clearly establish their precise view of the relationship between probabilities and the wave function.The discussion suggests they may not yet have reached a definitive position, partly because Born and Heisenberg brought different perspectives to the subject.Source: §6.4 · p. 210

7.1 Einstein's 1927 argumen t for in ompleteness

• A substantial literature later arose from Einstein, Podolsky, and Rosen’s 1935 paper, “Can quantum-mechanical description …,” on incompleteness.The supplied passage identifies the paper as the famous EPR paper and notes the extensive literature it generated.Source: §7.1 · p. 212

7.1 Einstein's 1927 argument for incompleteness

• Einstein’s conclusion was that quantum theory is incomplete, matching the later EPR conclusion despite the arguments’ different forms.The 1927 reasoning was presented as a simpler argument leading to the same conclusion as EPR in 1935.Source: §7.1 · p. 212
• Einstein argued that ψ cannot be a complete description of an individual electron if locality is assumed, because detection at one location would require action at a distance elsewhere.He contrasted ψ as an electron ensemble with ψ as an individual-electron description and rejected the latter as incompatible with locality.Source: §7.1 · pp. 212–213
• The argument analyzes a diffracted electron whose extended wave reaches multiple possible detection points, creating the appearance that detection at one point prevents detection elsewhere through action at a distance.The text distinguishes probabilities for a given particle from probabilities for this particle, a distinction that underpins the multiple-detection issue.Source: §7.1 · pp. 212–213
• The historical presentation is qualified by an incomplete surviving German source and possible translation ambiguity concerning Einstein’s distinction between probabilities for a given particle and for this particle.Only a fragment of Einstein’s contribution survives, and the French wording may not clearly convey the intended probability distinction.Source: §7.1 · p. 213
• To avoid the locality problem, Einstein endorsed describing the process with the Schrödinger wave while also localizing the particle during propagation, a direction he associated with de Broglie’s theory.He said that using only Schrödinger waves makes the interpretation of |ψ|2 conflict with relativity and that the wave function must therefore be incomplete.Source: §7.1 · p. 214

7.1 Einstein's 1927 argumen t for in ompleteness

• Bell (1964) showed that reproducing quantum correlations requires nonlocality, leaving quantum theory’s completeness or incompleteness unresolved.The argument assumes locality initially, but the resulting conflict is between locality and quantum correlations.Source: §7.1 · pp. 214–215
• Bell’s derivation assumes no common cause between hidden variables and measurement settings, no backward causation, and single-outcome measurements, excluding many-worlds.These assumptions constrain how broadly the Bell-inequality argument applies.Source: §7.1 · p. 215

7.2 A pre ursor: Einstein at Salzburg in 1909

• Einstein’s 1909 Salzburg argument anticipated his 1927 Solvay reasoning by applying a locality-based analysis to light quanta rather than electrons.Einstein presented light as having both particle and wave aspects and inferred a localized energy fragment alongside the spreading wave.Source: §7.2 · p. 216
• Einstein’s 1927 argument for localized electrons accompanying de Broglie–Schrödinger waves had the same form as his 1909 argument for localized light quanta.Both arguments use a source that emits a wave spreading across a detector, and both assume no action at a distance.Source: §7.2 · p. 218
• Einstein used wave–particle duality to argue that a wave function cannot completely describe an individual electron, because locality entails localized particles moving through space as well.The authors connect this critique to Einstein’s longstanding effort to reconcile localized energy quanta with radiation’s wave aspect.Source: §7.2 · p. 218
• Einstein’s locality argument was historically compelling but should not now establish localized photons, because Bell’s theorem appears to rule out its locality premise.The authors argue that the point deserved greater attention before Bell, rather than claiming it remains physically conclusive today.Source: §7.2 · p. 219

7.3 More on nonlo alit y and relativit y

• Einstein’s 1927 objection to configuration space added that a complete wave-function view is tied to a multidimensional representation lacking an evident basis for local action.The discussion places this objection alongside his concerns about the nonseparability of many-body wave mechanics.Source: §7.3 · p. 220
• Modern quantum theory retains no controllable nonlocal signals, yet its entanglement structure and configuration-space or Hilbert-space formulations make quantum nonlocality fundamental under most interpretations.In pilot-wave theory, the wave propagates in configuration space and generally connects the motions of spatially separated particles.Source: §7.3 · p. 220

7.3 More on nonloal ity and relativity

• De Broglie’s many-body pilot-wave dynamics lacked an easy relativistic wave Ψ, reflecting a broader conflict between configuration space and relativity.The difficulty was identified in 1927 and persisted in pilot-wave theory thereafter.Source: §7.3 · p. 221

8.1 Time in quan tum theory

• In Born and Heisenberg’s 1927 account, closed systems undergo no temporal changes, while the time-dependent Schrödinger equation describes subsystems effectively and approximately.The relationship of this view to Campbell’s proposal is explicitly left unclear.Source: §8.1 · p. 223
• Original matrix mechanics represented closed systems through stationary energy eigenstates and observables, without a general state or notion of an initial state.Its matrices collectively represented the energy eigenstates of a closed system with Hamiltonian H.Source: §8.1 · p. 223
• Born and Heisenberg supplemented the matrix formalism with a picture of subsystems occupying stationary states and making random, indeterministic quantum jumps.They introduced the time-dependent Schrödinger equation to avoid the matrix theory’s awkwardness in describing processes developing in time.Source: §8.1 · p. 224

8.1 Time in quantum theory

• The time-dependent theory is presented as emergent or approximate, leaving the time-dependent Schrödinger equation without fundamental status.This formalism nevertheless supports the statistical view by calculating time development from an initial wave function under external time-dependent perturbations.Source: §8.1 · p. 224
• For isolated systems, time may have no fundamental role, with ordinary macroscopic time emerging statistically from many transitions between stationary states.The resulting description is a sequence of discontinuous transitions without meaningful comparisons of intervals between them.Source: §8.1 · p. 225
• Schrödinger ultimately develops an alternative in which a general wave function ψ(q, t) describes continuous time evolution for a single isolated system.This approach treats continuous time evolution as fundamental even at the level of one atomic system, despite the difficulties of a discrete jump-based view.Source: §8.1 · p. 226
• Schrödinger’s and de Broglie’s interpretations did not gain widespread acceptance, but their view of time coincides with the generally accepted contemporary view.Modern quantum mechanics uses a continuous spacetime background and an evolution operator for arbitrary initial quantum states.Source: §8.1 · p. 226

8.1 Time in quan tum theory

• Canonical quantum gravity’s “problem of time” remains unresolved despite decades of effort and recent technical progress.Proposals face difficulties ensuring a well-behaved time parameter t with one physical state associated with each value of t.Source: §8.1 · p. 227

8.2 Determinism and probabilit y

• De Broglie’s guidance and Schrödinger equations are deterministic, determining trajectories during interference, diffraction, atomic transitions, and elastic scattering, while evolution maintains the distribution P = |Ψ|2.Source: §8.2 · pp. 227–228
• De Broglie theory permits nonequilibrium distributions P ≠ |Ψ|2, which would violate standard quantum constraints and enable more precise measurements and trajectory observations (Valentini 1991b, 1992; Valentini 2002b; Pearle and Valentini 2006).Such distributions could allow measurements more accurate than normally permitted by the uncertainty principle and reveal ordinarily invisible particle trajectories.Source: §8.2 · p. 228
• A general H-theorem and numerical simulations support efficient relaxation toward quantum equilibrium in purely deterministic de Broglie–Bohm theory (Valentini 1991a, 1992, 2001; Valentini and Westman 2005).The simulations show exponential decay of the coarse-grained H-function, making added stochastic fluctuations unnecessary.Source: §8.2 · p. 228
• Dirac argues that observation is theory-laden and that measurement disturbance alone cannot establish indeterminism, because experimenters may compensate for disturbances using interaction knowledge.Quantum-theoretical numbers describe controllable experimental operations, whereas other initial-state numbers are unobservable and absent from the treatment.Source: §8.2 · p. 229
• Born locates the quantum–classical difference in the laws governing probability-packet propagation, not in the use of wave packets or imperfect initial-state preparation.For Born, prediction precision depends on initial-position measurement accuracy in classical mechanics as well, whereas quantum and classical packet laws differ.Source: §8.2 · p. 230

8.3 Visualisabilit y and the spa etime framew ork

• Pilot-wave theory is grounded in configuration space rather than 3-space, so its interactions occur outside ordinary spacetime and create relativistic difficulties.Bell’s theorem indicates that complete single-world descriptions require nonlocality, while pilot-wave theory represents this through a field on configuration space.Source: §8.3 · p. 231
• De Broglie and Schrödinger sought ultimately visualizable three-dimensional processes, treating configuration-space dynamics as provisional or hoping for a deeper double-solution account.De Broglie presented the configuration-space guiding field as a makeshift effective theory, while Schrödinger hoped for physical content interpretable through processes in three-space.Source: §8.3 · p. 232
• Pauli argued that quantum mutual interactions cannot be described ordinarily in space and time, making multidimensional configuration space mathematically necessary.He regarded configuration space as technically mathematical, while denying that the resulting interactions could be represented in ordinary spacetime.Source: §8.3 · pp. 232–233
• For two widely separated ground-state hydrogen atoms, the ordinary spacetime description predicts no mutual interaction, whereas the multidimensional method gives a different result consistent with experiment.Pauli used this contrast to illustrate why interparticle interactions resist a simple spacetime account.Source: §8.3 · p. 233
• Quantum perturbation theory produces an attractive van der Waals interaction proportional to 1/R6, resolving a problem that classical explanations had failed to solve.The account traces the quantum solution to work beginning with Wang in 1927 and distinguishes these forces from exchange forces associated with the exclusion principle.Source: §8.3 · pp. 233–234
• Pauli attributed the correct van der Waals result to the entangled perturbed wave in multidimensional configuration space, which cannot be properly understood in three-dimensional space alone.A classical oscillating-dipole picture is only heuristic; the proper multidimensional wave-mechanical treatment was, in Pauli’s view, not intelligible within a spacetime framework.Source: §8.3 · p. 234
• Pilot-wave theory was challenged for failing to explain single-photon radiation pressure and discrete energy exchange in inelastic collisions.Kramers questioned the spacetime picture’s tenability, while Pauli attributed the difficulty to deterministic particle dynamics described entirely by ordinary spacetime kinematics.Source: §8.3 · p. 235
• Earlier guiding-field proposals predicted single-event energy-momentum nonconservation, conflicting with experiment until guiding fields were introduced in configuration space.These proposals formed part of the background to the Solvay discussions.Source: §8.3 · p. 236

9.1 Einstein's early attempts to form ulate a dynami al theory

• Einstein’s early theory-building treated light quanta as localized energy fragments accompanied by extended fields, with deterministic motion equations for the singular points.His 1909 conception was described as reminiscent of de Broglie’s pilot-wave theory and the later double-solution theory; large numbers of associated fields were intended to recover Maxwell’s electromagnetic field.Source: §9.1 · pp. 236–237
• Einstein’s 1909 proposal pictured quanta as singularities surrounded by vector fields whose collective behavior could explain interference, though its single- versus many-photon scope remained unclear (Einstein 1909).The exchange leaves ambiguous whether interference concerned individual photons or the collective behavior of many quanta.Source: §9.1 · p. 238
• Einstein’s reply was retrospectively reasonable because de Broglie’s pilot-wave theory straightforwardly explains particle interference, despite uncertainty about Einstein’s precise proposal (Einstein 1909).The passage notes that Einstein’s associated waves could guide quantum motion in a way accounting for interference.Source: §9.1 · p. 239

9.2 The failure of energy-momen tum onserv ation

• Einstein’s 1925 guiding-field idea assigned directions probabilistically to light quanta and electrons, so energy–momentum conservation would hold only statistically, conflicting with individual events (Pais 1982; Howard 1990).He therefore never published or fully accepted the idea, although his thinking remained reminiscent of de Broglie’s work.Source: §9.2 · p. 240
• The difficulty was later addressed by a single generally entangled configuration-space guiding field determining collective probabilities for all particles, enabling stronger correlations than separate particle fields.Separate guiding fields precluded entanglement and could not guarantee energy–momentum conservation for individual events.Source: §9.2 · p. 240
• Entangled quantum states conserve total momentum in every individual scattering outcome, unlike separable states that conserve it only on average.The separable model gives four equiprobable outcomes with final total momenta 0, +2p, −2p, and 0.Source: §9.2 · p. 241
• The BKS theory likewise allowed energy-momentum conservation only on average, because correlations between widely separated atoms were statistical rather than event-by-event.In BKS, a distant atom’s transition was influenced probabilistically by another atom’s virtual radiation field, without requiring the first atom’s corresponding energy decrease.Source: §9.2 · p. 242
• Bothe-Geiger and Compton-Simon experiments observed individual-event energy-momentum conservation, and the BKS theory was consequently widely judged wrong.The experiments tested simultaneous outgoing photon and recoil-electron events, which BKS did not predict.Source: §9.2 · pp. 242–243
• A purely statistical interpretation of the wave function nevertheless yields correct predictions consistent with conservation laws for elementary processes such as scattering.This contrasts with Einstein’s claim that an ensemble interpretation conflicted with elementary conservation, which the section presents as tied to his specific conception of that interpretation.Source: §9.2 · p. 243
• Configuration-space wave fields enabled strict energy–momentum conservation in individual emission and scattering events, but at the cost of fundamental nonseparability.Assigning separate guiding fields to particles cannot secure both probabilistic individual behavior and correlations sufficient for strict conservation; entangled configuration-space fields provide the needed correlations (Howard 1990, p. 73).Source: §9.2 · p. 244
• The chapter frames elastic and inelastic scattering discussions as closely related to quantum measurement, with questions raised by Born, Pauli, and Kramers and replies from Brillouin and de Broglie.De Broglie could not answer Kramers’s question about the recoil of a single photon on a mirror.Source: §9.2 · p. 245
• De Broglie and Brillouin correctly answered Born’s question about elastic scattering, whereas Pauli’s inelastic-scattering objection was subtler and distorted by a misleading optical analogy.The account argues that de Broglie’s reply contained the essential ingredients for treating inelastic scattering, although the proper pilot-wave treatment was not supplied until 1956.Source: §9.2 · p. 245

10.1 S attering in pilot-w a v e theory

• For fixed-potential elastic scattering, a wave packet evolves as ψ = ψinc + ψsc, and the scattering amplitude determines the differential cross section dσ/dΩ = |f(θ, φ)|2.The textbook current-density calculation is recast in pilot-wave terms, where the relevant current is interpreted as the actual velocity field of the particle.Source: §10.1 · pp. 246–247
• Finite incident wave packets were central to the Solvay discussions because they allow outgoing scattering packets to separate spatially.A literally infinite incident plane wave would overlap the scattered wave everywhere, unlike a physically finite packet.Source: §10.1 · p. 247
• After channel separation, pilot-wave dynamics guides the actual particles by one non-overlapping branch, producing an effective and practically irreversible collapse to φ_iψ_i.The occupied branch determines the scattered and atomic-electron velocities, while later interference becomes practically unlikely after interaction with additional degrees of freedom.Source: §10.1 · p. 248
• The probability of ending in the i-th outgoing packet, and thus leaving the atom in φ_i, agrees with the usual quantum result.The outgoing packet ψ_n is associated with an asymptotic radius r_n = (ℏk_n/m)t, with k_n fixed by energy conservation.Source: §10.1 · pp. 247–248
• Treating the incident wave as an actually infinite plane wave makes outgoing channels overlap everywhere, so trajectories fail to settle into definite asymptotic branches or atomic states.The text identifies this as a mathematical limit rather than a realistic scattering situation; with spatially limited packets, only one outgoing component guides the scattered trajectory outside the incident beam.Source: §10.1 · pp. 248–249
• Pauli’s objection to de Broglie’s proposal used the plane-wave assumption to argue that inelastic scattering would not approach stationary energies, contrary to experiment.The objection concerned inelastic scattering by a rigid rotator and was presented as conceptually equivalent to scattering by a hydrogen atom.Source: §10.1 · p. 249
• Bohm attributed Pauli’s objection to the excessively abstract infinite-plane-wave model, but Pauli and de Broglie regarded de Broglie’s interpretation as untenable, after which de Broglie abandoned it.In Bohm’s account, incoming and outgoing wave fields are bounded packets that separate into classically describable alternatives, with the outgoing particle entering one packet.Source: §10.1 · pp. 249–250
• Pauli’s objection was not based on overlooking finite initial wave packets; instead, it relied on a misleading optics analogy about their necessary extent.The objection involved conditions thought to prevent separation into non-overlapping components, whereas finite packets play a crucial role in pilot-wave theory.Source: §10.1 · p. 250

10.2 Elasti and inelasti s attering: Born and Brillouin,

• Brillouin’s mirror example describes a photon guided by a finite packet: it approaches uniformly, undergoes interference and scattering near the mirror, then departs in a finite outgoing packet.The trajectory bends in the interference zone, moves approximately parallel to the mirror, avoids dark fringes, and eventually follows the reflected beam.Source: §10.2 · p. 251
• Plane waves served only as a calculational convenience: Brillouin’s approximation gives v = c sin θ parallel to the mirror, while at normal incidence the photon is stationary in the overlap region before departing.The realistic incident packet is limited longitudinally as well as laterally; uniform incoming and outgoing motion holds strictly only when guiding packets become sufficiently large.Source: §10.2 · pp. 251–252

10.2 Elasti and inelasti s attering: Born and Brillouin

• Maxima and minima in the predicted inelastic-scattering angular distribution had been observed experimentally by Dymond, though only qualitatively.De Broglie related these results to Born’s predictions and the new mechanics.Source: §10.2 · p. 253
• Pauli accepted agreement for elastic collisions but argued that de Broglie’s theory does not work for inelastic collisions, particularly scattering by a rotator.His concern was whether trajectories could account for the discrete energy exchange observed in individual electron–atom collisions.Source: §10.2 · pp. 253–254
• De Broglie’s theory agrees with Born’s predictions for elastic-scattering statistics and predicts selective electron scattering in directions of amplitude maxima.The maxima were associated with crystal regularity and the role of the pilot wave Ψ.Source: §10.2 · pp. 252–253
• In the rotator example, de Broglie’s prescription therefore gives a nonconstant final angular velocity, contrary to the expected stationary-state behavior.The analysis identifies Fermi’s optical analogy as having only very limited validity, while a finite incident packet would avoid the corresponding elastic-scattering problem.Source: §10.2 · pp. 254–255
• For an infinitely extended incident wave, the superposed outgoing energy states guide a final configuration whose velocity is not constant.This follows because the different spectral orders remain superposed throughout configuration space.Source: §10.2 · p. 255
• The apparent pilot-wave difficulty comes from treating an unlimited wave as analogous to scattering from an infinite grating, where symmetry forbids beams beyond zeroth order.Beam separation requires lateral localization, which the Fermi analogy excludes; this creates the misleading conclusion that the theory is untenable.Source: §10.2 · pp. 256–257
• Including the measuring apparatus in the quantum description could also produce non-overlapping branches and definite outcomes, but this was not standard in 1927.Modern pilot-wave scattering instead generally obtains definite outcomes through separation of the scattered-particle packets.Source: §10.2 · p. 257
• Localized incident waves separate into non-overlapping outgoing packets, so the pilot-wave theory produces a definite scattering outcome and directly answers Pauli’s objection.The total configuration occupies one outgoing branch, with constant electron speed and uniform rotator motion after separation.Source: §10.2 · pp. 257–258
• Fermi’s analogy fails because its effective optical “speed of light” depends on energy, so it applies only to a single energy rather than realistic finite incident packets.For incident waves containing different momenta, the rotator-scattering problem cannot be reduced to light scattering from a grating in a homogeneous medium.Source: §10.2 · pp. 258–259

10.2 Elasti and inelasti s attering: Born and Brillouin,

• De Broglie’s later analysis gave the first proper pilot-wave treatment of Pauli’s example, with outgoing packets moving at different speeds.The different speeds correspond to the electron’s possible post-collision kinetic energies, unlike the optical analogy’s same-speed prediction.Source: §10.2 · pp. 259–260
• De Broglie’s 1927 reply recognized the importance of lateral limitation but was misled by Fermi’s generally false optical analogy.The analogy has limited validity because the effective light speed is frequency-dependent and cannot describe a finite incident wave straightforwardly.Source: §10.2 · p. 260
• Finite incident waves separate into non-overlapping configuration-space packets, providing the mechanism required for definite scattering outcomes.After separation, the system velocity becomes constant and corresponds to a stationary state of the rotator.Source: §10.2 · p. 261
• Born had already described analogous non-overlapping wave-function branches in configuration space for cloud-chamber tracks, making de Broglie’s reply familiar to the audience.The example involved an initially localized packet separating into two non-overlapping branches.Source: §10.2 · p. 262

10.3 Quan tum measuremen t in pilot-w a v e theory

• The general theory of quantum measurement in pilot-wave theory was first developed by Bohm in his second paper on the theory.The supplied passage introduces this historical point but does not provide the full bibliographic citation.Source: §10.3 · p. 262
• Pilot-wave measurement includes system and apparatus coordinates in one guided trajectory, with the wave function separating into non-overlapping configuration-space branches.The particle configuration follows the branch selected by the actual apparatus coordinates.Source: §10.3 · pp. 262–263
• At measurement’s end, the system is guided by a reduced wave function φ_i(x), while the pointer plays the same role as a scattered particle.The active branch factorizes in system and pointer coordinates, so the system velocity is determined by φ_i(x) alone.Source: §10.3 · p. 263
• The branch probability is |c_i|^2, matching the standard Born rule, and effective collapse to the selected state is practically irreversible.Subsequent interactions with many apparatus degrees of freedom make later interference between distinct branches very unlikely.Source: §10.3 · p. 263

10.4 Re oil of a single photon: Kramers and de Broglie

• Pilot-wave analysis must include macroscopic apparatus coordinates in the total wave function so measurement dynamics can produce non-overlapping branches.The text presents this as an insight already present in Born’s loud-chamber discussion, though de Broglie did not carry it through generally in 1927 and Bohm did in 1952.Source: §10.4 · p. 264
• De Broglie and Brillouin could not explain the sudden mirror-momentum change from a single reflected photon; de Broglie said the theory then covered only mean pressure from photon ensembles.The missing ingredient was treating the mirror position through pilot-wave dynamics and including it in the wave function.Source: §10.4 · pp. 264–265

10.4 Recoil of a single photon: Kramers and de Broglie

• De Broglie could not answer Kramers's question about a single photon's mirror recoil because the pilot-wave description needed a joint wave function including the macroscopic mirror.The proceedings state that no current theory then gave an answer to Kramers's question, while ordinary quantum theory likewise had to treat the mirror as part of the quantum system.Source: §10.4 · pp. 265–266
• De Broglie presented a nonrelativistic many-body pilot-wave dynamics and correctly explained definite scattering outcomes through wave-packet separation in configuration space.His 1927 work included applications to interference, diffraction, and atomic transitions.Source: §10.4 · p. 266
• Bohm revived de Broglie's theory in 1952 and supplied its measurement account by treating macroscopic equipment as part of the quantum system.Bohm provided a detailed derivation of quantum phenomenology from de Broglie's 1927 dynamics, with different dynamical equations.Source: §10.4 · p. 266
• Bell treated de Broglie–Bohm theory as empirically equivalent to quantum theory and used it to investigate whether nonlocality was specific to that model or common to hidden-variable theories.Bell's conclusion was that viable theories reproducing EPR-type correlations must be nonlocal; Bell-inequality tests targeted local theories, not de Broglie–Bohm theory.Source: §10.4 · p. 267
• Pilot-wave theory became widely accepted as an internally consistent alternative only after about three-quarters of a century, despite de Broglie's substantial 1920s contributions.The proceedings argue that careful examination of the Fifth Solvay Conference changes perceptions of the theory both physically and historically.Source: §10.4 · pp. 267–268

11.1 Histori al mis on eptions

• De Broglie presented pilot-wave theory for a many-body system in configuration space at the 1927 Solvay conference, not merely for a one-body system.The many-body theory with a guiding wave in configuration space was originally due to de Broglie, while Bohm later supplied a many-body interpretation.Source: §11.1 · pp. 268–269
• The claim that de Broglie abandoned the theory because of Pauli’s criticism is false; his 1927 reply already contained the essential ideas needed for a proper response.The section states that de Broglie’s 1927 scattering theory contained the essentials later associated with Bohm’s treatment.Source: §11.1 · p. 269
• Both de Broglie’s 1927 theory and Bohm’s 1952 theory were regarded as provisional, although Bohm also contemplated relaxing the initial momentum constraint to permit modifications of quantum theory.De Broglie treated the configuration-space pilot wave as provisional, while Bohm suggested that relaxing the constraint could lead to corrections to quantum theory.Source: §11.1 · pp. 269–270
• The cited passages contrast de Broglie’s 1927 presentation, where the guidance equation was the basic law of motion, with Bohm’s 1952 Newtonian, acceleration-based dynamics involving a quantum potential.Source: §11.1 · p. 269
• De Broglie’s theory was extensively discussed after his lecture and during the general discussion, with support expressed by Brillouin and Einstein.The proceedings include varied comments on multiple aspects of the theory, including Einstein’s query about photon speed.Source: §11.1 · pp. 270–271
• The proceedings show that de Broglie’s pilot-wave theory received substantial discussion, contrary to later claims that it attracted hardly any attention or that the conference centered mainly on Bohr and Einstein.Such claims rely partly on de Broglie’s much later recollections and selectively summarized discussions.Source: §11.1 · pp. 271–272
• The account rejects the claim that Pauli’s criticism primarily caused de Broglie to abandon pilot-wave theory, noting that de Broglie’s book does not mention Pauli’s criticism.The energy-measurement problem was posed by de Broglie himself, although his response contained the essence of a correct reply to Pauli’s query.Source: §11.1 · pp. 272–273
• De Broglie’s 1930 account gave three reasons for rejecting pilot-wave theory: unresolved wave-packet collapse, problematic energy measurements, and superluminal photon trajectories.He considered the wave a symbolic representation of probability and found superluminal trajectories unacceptable.Source: §11.1 · p. 272
• By late 1928, de Broglie accepted quantum theory’s formalism and statistical wave-function interpretation but still viewed wave–particle duality’s physical meaning as unresolved.He regarded the Heisenberg–Bohr perspective as containing much truth, while maintaining that the physical interpretation remained extremely difficult.Source: §11.1 · p. 273
• De Broglie’s later reflections continued to question whether a corpuscle could retain a well-defined position in a wave, reinforcing the unresolved interpretive status of the theory.His 1929 Nobel Lecture still presented this as a difficult question rather than a settled conclusion.Source: §11.1 · p. 273
• Later recollections by de Broglie should not be treated as a reliable guide to what happened at the 1927 conference.The section presents this as a historical misconception alongside the tendency to portray de Broglie–Bohm theory as merely completing quantum theory.Source: §11.1 · pp. 273–274
• De Broglie’s pilot-wave theory was not an add-on to pre-existing quantum mechanics: its guiding waves and velocity formula preceded matrix and wave mechanics, and Bohm later reinstated elements present from the beginning.The theory culminated de Broglie’s independent work from 1923, with a major input from Schrödinger in 1926; its historical priority was subsequently downplayed, including in Pauli’s 1927 remarks.Source: §11.1 · pp. 274–275
• The Schrödinger equation has a natural place in original de Broglian pilot-wave theory because it arose from de Broglie’s wave program and its optical-mechanical analogy.This differs from Bohm’s 1952 Newtonian reformulation, where the wave equation is appropriated to generate a quantum force and lacks an internal derivation within that formulation.Source: §11.1 · pp. 275–276

11.2 Wh y w as de Broglie's theory reje ted?

• The section asks why de Broglie’s theory failed to gain widespread support soon after 1927 and identifies Bonk (1994) as applying Bayesian reasoning to relevant conference discussions.The supplied passage introduces this question and approach but does not provide the reasons or conclusions.Source: §11.2 · p. 276
• De Broglie answered Pauli’s criticism with an essentially correct rebuttal but failed to address Kramers’s difficulty, requiring renewed evaluation of why his theory was rejected.The authors identify these as two overlooked historical points relevant to judging whether de Broglie made a convincing case at the fifth Solvay Conference.Source: §11.2 · p. 276
• The free-particle Schrödinger equation follows from a plane de Broglie wave with the nonrelativistic dispersion relation, making this derivation as natural in pilot-wave theory as in standard quantum theory.This point challenges the assumption that standard quantum theory had an exclusive conceptual route to the Schrödinger equation.Source: §11.2 · p. 276
• Standard quantum theory had a practical advantage in 1927 because a simple collapse postulate could predict measurement outcomes without quantum-mechanically analyzing the apparatus.Pilot-wave theory instead required analyzing the measurement process, including the apparatus, within its own dynamics; ordinary collapse rules were pragmatic and imprecisely bounded but often yielded experimentally testable predictions.Source: §11.2 · p. 277

11.3 Einstein's alternativ e pilot-w a v e theory (Ma y 1927)

• Einstein’s unpublished May 1927 proposal described an alternative pilot-wave theory in which many-body wave functions determined particle trajectories differently from de Broglie’s theory.Einstein presented the proposal on 5 May 1927, withdrew it before publication, and its contents are known from a manuscript version (Howard 1990, pp. 89–90; Belousek 1996).Source: §11.3 · pp. 277–278
• Einstein’s proposal was connected to his argument that locality required both Schrödinger-wave propagation and simultaneous localization of the particle.Einstein regarded de Broglie’s search in this direction as correct, while his own version was relevant to his incompleteness argument for quantum theory.Source: §11.3 · p. 278
• Einstein’s proposed velocity law determines system velocities, up to signs, from the wave function and its derivatives.The construction matches kinetic-energy expressions term by term and uses a non-Euclidean metric determined by kinetic energy.Source: §11.3 · pp. 279–280
• Einstein regarded his theory as physically unacceptable because independent subsystems with product wave functions could have velocities depending on one another’s instantaneous coordinates.Grommer’s proposed Ψ → ln Ψ replacement was intended to avoid this problem.Source: §11.3 · p. 280
• Holland (2005) showed that replacing Ψ with ln Ψ removes Einstein’s separability difficulty, but identified further problems that undermine the scheme.These include restricted quantum states and configuration domains, plus failure of the continuity equation to preserve an initial Born-rule distribution and reproduce standard quantum predictions.Source: §11.3 · pp. 280–281
• De Broglie’s phase-gradient velocity field was simpler than Einstein’s and immediately ensured separable particle motions for product states.The relevant velocity is proportional to ∇S, paralleling the Hamilton–Jacobi velocity formula.Source: §11.3 · p. 281
• In the electron–grating example, Heisenberg predicted discrete scattering regardless of an obstacle, but the analysis is characterized as inappropriate reasoning borrowed from classical physics.The discussion notes that consistency with observation can be obtained when particle motion is strictly tied to the waves, as in de Broglie’s theory.Source: §11.3 · pp. 282–283
• Heisenberg argued that pilot-wave trajectories would require particle–wave motion tied together in a way that sacrifices quantum theory’s simplicity and intelligibility.He also objected to unobservable causal connections and viewed Einstein as willing to sacrifice simplicity to preserve causality.Source: §11.3 · pp. 282–283

11.4 Ob je tions: in 1927 and to da y

• Many objections commonly directed at pilot-wave theory today were already voiced in 1927, including doubts about the reality and observability of de Broglie’s trajectories.The dispute contrasts those who value hidden entities for their explanatory role with those who reject theoretical roles for unobservable details.Source: §11.4 · pp. 283–284
• Pilot-wave theory itself treats trajectory unobservability as an accident of Born-rule initial conditions, so quantum nonequilibrium could violate the uncertainty principle and permit trajectory observation.Such nonequilibrium would depart from quantum theory’s statistical predictions; alternative velocity laws could likewise produce trajectories differing from de Broglie’s.Source: §11.4 · p. 284

11.4 Obje tions: in 1927 and to day

• The central objection is whether subquantum dynamics should conform to classical measurement intuitions, given that it supplies the underlying measurement dynamics.Some trajectory features have been called “surreal” (Englert et al. 1992, Aharonov and Vaidman 1996), while others regard them as understandable within pilot-wave theory (Valentini 1992, Dewdney, Hardy and Squires 1993, Dürr et al. 1993).Source: §11.4 · p. 285
• Pilot-wave trajectories are expected to depart from classical expectations because de Broglie’s velocity-based dynamics is a genuinely distinct form of dynamics.This reframes apparently strange trajectories as consequences of the theory rather than defects measured against classical mechanics.Source: §11.4 · p. 285
• De Broglie’s original pilot-wave dynamics was a radical departure from classical dynamics, whereas Bohm’s later pseudo-Newtonian formulation fostered the mistaken view that it restored classical physics.Recent work again recognizes de Broglie’s original dynamics as a new form in its own right (Dürr, Goldstein and Zanghì 1992, Valentini 1992).Source: §11.4 · p. 285

Bey ond the Bohr-Einstein debate

• The famous Bohr–Einstein exchanges consisted mainly of informal, semi-private discussions rather than an official conference debate.They occurred chiefly over breakfast and dinner and were overheard by only a few participants.Source: Bey ond the Bohr-Einstein debate · p. 286
• The standard historical account is skewed because later recollections of private discussions overshadow the published proceedings and the rest of the conference.The available evidence suggests that much more memorable and important material occurred beyond the celebrated clash.Source: Bey ond the Bohr-Einstein debate · p. 286, §12.1 · p. 287
• Einstein’s main target was not the uncertainty relations but what he saw as quantum theory’s nonseparability.The chapter argues that this concern was misunderstood by Bohr, Heisenberg, and Ehrenfest.Source: Bey ond the Bohr-Einstein debate · p. 286

12.1 The standard histori al a oun t

• Because the interpretation of quantum theory remains an open question, the conference should be reassessed through its published proceedings rather than mainly through later recollections by two participants.There is little independent evidence from the time about what Bohr and Einstein said privately.Source: §12.1 · pp. 287–288
• The standard account portrays the 1927 conference as centered on Bohr’s private debate with Einstein, overlooking de Broglie, Schrödinger, Lorentz, Dirac, and other recorded discussions.Heisenberg’s account emphasizes uncertainty-relations paradoxes and their clarification, while omitting alternative theories and extensive proceedings discussions.Source: §12.1 · p. 289

12.2 T o w ards a histori al revision

• Bohr’s later recollection did not capture Einstein’s true concern: the nonseparability of quantum theory.The revision identifies Einstein’s objection as the inability to treat spatially separated systems independently, rather than primarily an attempted refutation of uncertainty relations.Source: §12.1 · p. 290, §12.2 · p. 291
• The cited discussion presents Einstein’s concerns about quantum theory’s failure of separability as predating the 1935 EPR paper and links configuration-space wave mechanics to violated separability.Source: §12.2 · p. 290
• The 1927 general discussion concerned locality and completeness, like the later EPR argument, not the uncertainty relations.Bohr’s reply indicates that he did not understand Einstein’s precise point, suggesting the issue was not properly recognized at the time.Source: §12.2 · p. 291
• Bohr’s account portrayed the photon-box proposal as an attempted violation of the energy-time uncertainty relation that failed because of gravitational time dilation.Ehrenfest’s 1931 report instead indicates that Einstein accepted the uncertainty relations and intended the experiment to show that distant incompatible quantities could be predicted through alternative measurements.Source: §12.2 · pp. 291–292
• Einstein’s photon-box and two-slit arguments targeted quantum nonseparability and incompleteness rather than the uncertainty relations.The photon-box experiment has been reinterpreted as an early form of the later EPR argument, while the two-slit case emphasized nonseparability between the particle and screen.Source: §12.2 · pp. 291–292
• Einstein’s EPR-style reasoning held that locality and separability require distant systems to possess definite properties, making quantum theory incomplete.Different measurements on one wing associate different wave functions with the same distant real state; Howard’s analysis details this argument.Source: §12.2 · p. 292
• The authors advocate re-evaluating the Solvay conference because no definitive interpretation now commands wide acceptance and earlier opposing views may have been misunderstood or undervalued.They specifically call for renewed consideration of de Broglie’s pilot-wave theory, Schrödinger’s ideas, and the roles of time and wave-function collapse in Born and Heisenberg’s quantum mechanics.Source: §12.2 · p. 293

H. A. Loren tz †

• The proceedings section begins with a memorial acknowledgment of H. A. Lorentz, who died suddenly in February 1928 after serving as Solvay’s chairman and guiding spirit.The tribute presents Lorentz as an illustrious teacher and physicist whose leadership shaped the meetings.Source: H. A. Loren tz † · p. 297
• H. A. Lorentz was portrayed as an authoritative theorist, teacher, international scientific organizer, and exceptionally effective conference chairman.His broad knowledge, fair judgment, multilingual ability, and skill in guiding discussion helped participants express opinions and reach conclusions.Source: H. A. Loren tz † · pp. 297–298
• Lorentz retained a devotion to classical theory while following the disruptive development of quantum theory and the new mechanics.The account presents this intellectual flexibility as an important part of his legacy to the Solvay Institute.Source: H. A. Loren tz † · p. 298
• The Fifth Physics Conference of the International Institute of Physics convened in Brussels from 24 to 29 October 1927.The institute was founded by Ernest Solvay under statutes providing for these physics conferences.Source: H. A. Loren tz † · p. 299
• The conference brought together leading physicists including Curie, Bohr, Born, de Broglie, Dirac, Einstein, Heisenberg, Planck, Schrödinger, and Wilson, with Lorentz as chairman.J.-E. Verschaffelt served as secretary; additional scholars attended as guests, while some invited participants were excused or absent.Source: H. A. Loren tz † · p. 299
• The institute’s scientific committee included Lorentz, Curie, Einstein, Guye, Langevin, Richardson, and others, with Cabrera replacing resigning member W. H. Bragg.The committee also selected a replacement for its late chairman, H. Kamerlingh Onnes.Source: H. A. Loren tz † · p. 300

• The paper presents X-ray diffraction spectroscopy as strong evidence for classical laws, with crystal refraction offering no exception to them.Source: By Mr W. L. BRA GG · p. 301
• Diffraction intensities reflect crystal structure and can be used qualitatively to infer atomic positions.Source: By Mr W. L. BRA GG · p. 301
• Quantitative intensity analysis is difficult because experimental results are often uncertain, leading investigators to rely on qualitative methods.Source: By Mr W. L. BRA GG · p. 302
• The paper critically surveys the subject’s development as X-ray analysis becomes sufficiently reliable for more complicated crystalline structures.Source: By Mr W. L. BRA GG · p. 302
• Unlike microscopy, X-ray examination measures scattered-beam intensities rather than phase relations or a direct image of the crystalline unit.Source: By Mr W. L. BRA GG · p. 302
• The method is constrained by limited scattering directions, near-theoretical resolving power, and inconvenient available wavelengths.Source: By Mr W. L. BRA GG · p. 302
• The analysis proceeds from measuring diffracted intensities, through theoretical reduction to scattered-wave amplitudes, to reconstructing the structural unit.Source: By Mr W. L. BRA GG · p. 303

Theoretical foundations

• Laue [1], Darwin [2], and Ewald [3] established mathematical treatments of X-ray reflection, including distinct formulas for ideally imperfect and ideally perfect crystals.The perfect-crystal formula predicted results of a different order of magnitude and accounted for extinction when crystal-plane interactions mattered.Source: §2 · pp. 303–304

Early experimental tests

• Experiments showed rock-salt reflection intensities matched the imperfect-crystal formula, while rapid high-order decline additionally reflected the spatial distribution of atomic scattering matter.The transmitted beam was extinguished more rapidly than ordinary absorption predicted in sufficiently perfect crystals.Source: §2 · p. 304

Quantitative measurement

• Bragg’s quantitative measurements established smooth angle-dependent reflection curves and an integrated reflection constant independent of experimental arrangement for a given mosaic-crystal reflection.The measurements used a crystal moved at constant angular velocity while total reflected radiation was recorded.Source: §2 · p. 305

Experimental confirmations

• Bragg demonstrated Darwin’s extinction effect in diamond and observed Debye’s predicted temperature-related reduction in reflection intensity at the expected order.Measurements also suggested that, for a very perfect crystal, intensity was proportional to atomic-plane scattering power rather than its square.Source: §2 · p. 305

Compton’s measurements

• Compton [7] derived the mosaic-crystal formula differently, linked atomic electron distributions to Bohr’s model, and found absolute reflection intensities consistent with the imperfect-crystal formula for rock-salt but strongly extinguished for calcite.Debye and Compton also examined how electron distribution affects reflection intensity.Source: §2 · pp. 305–306

Measurement refinement

• Measurements with James and Bosanquet improved integrated-reflection accuracy across many planes and corrected secondary, but not primary, extinction [9].The work combined absolute measurements of the strongest reflections with comparisons to weaker ones.Source: §2 · p. 306

• Intensity measurements agreed very closely with earlier measurements, reinforcing confidence in the measurement method and in the selected crystal.The crystal was fortunate because primary extinction was very small.Source: §2 · p. 306
• The analysis used scattering intensities to infer atomic structure, including electron distributions in sodium and chlorine.The authors published curves showing these distributions.Source: §2 · p. 306
• The theory’s predictions were verified, although the authors acknowledged failing to credit Compton’s 1917 absolute measurements.The omission resulted from their lack of awareness of that earlier work.Source: §2 · p. 306
• Comparing incident and reflected radiation can yield accurate quantitative estimates of scattering matter when crystal conditions permit.This ideal is more readily achieved with imperfect or mosaic crystals, whereas strong reflections and near-perfect crystals make extinction and interpretation more difficult.Source: §2 · p. 307

• Measured reflection intensities yield F values that represent the crystal unit’s scattering power and encode its crystalline or atomic structure.Source: §3 · pp. 307–308
• The resulting F values are characteristic of a given crystal across wavelengths, while shorter wavelengths increase resolving power by sampling more coefficients.Source: §3 · p. 308
• Thermal agitation must be accounted for because atomic motions and changing relative mean positions complicate structural deductions, especially in complex crystals.Source: §3 · p. 308
• The quantitative formulas produce results indicating that the theory is correct across simple and highly complex crystals.Source: §3 · p. 308
• For simple crystals, the analysis tests electron number and atomic size, with F tending toward the atom’s electron number at small sin θ.Source: §3 · p. 308
• Observed F curves generally agree with Thomas-model calculations and reach maxima near the atom’s electron number, supporting reconstruction of atomic structure.Source: §3 · pp. 308–309
• The agreement between observed and calculated F curves indicates that the measurements outline a picture of the atom and extends across other comparable atomic models.Source: §3 · p. 309

Interpretation of measurements of F

• Direct inversion of observed F curves produced images and relative scattering-matter distributions in crystals, including results by Duane, Havighurst, and Compton.Duane’s generalized Fourier method represented crystal density as a triple Fourier series, which Havighurst applied to rock salt and other crystals; Compton extended the formulas to absolute electron-density distributions under classical-electron scattering assumptions.Source: §4 · pp. 310–311
• Fourier reconstruction can display atomic structure as either a sheet distribution parallel to a plane or a radial distribution around the nucleus.The method uses measured F values over a sufficiently wide range to build either representation, but radial distributions converge more slowly and are more affected by missing higher Fourier terms.Source: §4 · p. 311
• Working right at the resolving limit, ignorance of relevant values can produce false detail in an X-ray image, while atomic structure smaller than half this value cannot be resolved.Source: §4 · p. 312
• Testing a proposed atomic model by calculating its F curve and comparing it with experiment is presented as the most satisfactory model test because it requires no assumptions about higher coefficients.The comparison nevertheless tests model details only on scales comparable with the X-ray wavelength; reflection from atomic planes can provide roughly twice the resolving power of the best microscope for a given wavelength.Source: §4 · p. 312
• An earlier shell-based analysis of sodium and chlorine fitted scattering over the observed range and found a consistent distribution across shell choices, with sodium’s electron distribution limited at about 1.1 Å.The procedure avoided extrapolating the F curve beyond experimental coverage and solved for unknown shell populations using simultaneous linear equations.Source: §4 · p. 312

4. Interpret a tion of measurements of F

• The distribution clearly indicates 1.8 Å in chlorine.Source: §4 · p. 313
• In general outline, the distribution corresponds to the more direct Fourier analysis.Source: §4 · p. 313
• Examples in paragraph 7 are cited to show the correspondence with Fourier analysis.Source: §4 · p. 313

• Fourier analysis condenses information from all reflection orders into one curve, offering a direct but data-dependent alternative for interpreting X-ray examinations.Its usefulness depends on the reliability of experimental observations, while its directness and significance mark an advance in analytical technique.Source: §5 · pp. 313–314
• Havighurst’s Fourier analysis of sodium chloride produced density peaks at the atomic positions, illustrating the method’s first application to accurate data.The calculation used measured structure factors and three-dimensional Fourier series along selected crystal directions.Source: §5 · p. 314
• For beryl, Fourier curves corresponded closely with independently determined oxygen-group positions, and the older crystal-analysis methods agreed with the Fourier interpretation.The Fourier coefficients required prior analysis by older methods, which supplied the needed structural information.Source: §5 · p. 315
• Fourier series for related alums showed scattering density varying at metal-atom positions while remaining constant elsewhere, as expected across substituted crystals.The examples involved substitutions among potassium, ammonium, rubidium, caesium, and thallium, with other atomic positions expected to remain similar.Source: §5 · p. 316
• Waller’s heat-motion formula closely matched rock-salt measurements up to 500°C, while higher temperatures caused an even faster intensity decline and diffuse NaCl peaks.The Fourier interpretation shows sharply defined sodium and chlorine peaks at low temperatures becoming diffuse as heat motion increases.Source: §5 · pp. 317–319

Examples of analysis

• Sodium atoms have greater average vibrational amplitudes than chlorine atoms because sodium is more loosely bound, whereas chlorine is more firmly pinned by its neighboring atoms.The difference in heat motion must be analyzed before observations can be reliably reduced to absolute zero.Source: §5 · pp. 319–320
• Mean vibrational amplitude is 0.21 Å at room temperature and about 0.58 Å at 900° abs., according to both heat-factor and Fourier analyses.The attainable accuracy may allow direct testing of whether zero-point energy exists, provided a reliable atomic model is available.Source: §5 · pp. 320–321
• Observed atomic distributions closely resemble the Thomas model through most of the crystal, while small departures supply evidence for individual atomic features omitted by its smooth electron distribution.The comparison also illustrates how difficult it is to recover fine atomic structure from observations.Source: §5 · p. 321

The mechanism of X-ray scattering

• Observed X-ray curves indicate the correct number of electrons, but Compton scattering means they provide a consistently similar yet untrue picture of atomic structure.Including the Compton effect in recalculated curves produced a better fit to experiment.Source: §5 · p. 322

Examples of analysis

• Including the Compton effect smooths the calculated chlorine-scattering hump at sin θ = 0.4, producing curves much more like those observed; the Thomas distribution is closest.The hump arises from irregular outer-electron contributions absent from the observed curve, while Compton scattering makes those contributions very small at large angles.Source: §5 · pp. 323–324
• X-ray diffraction measurements can be trusted to measure the Schrödinger continuous distribution of electricity in the atom.The coherent radiation is calculated from the Schrödinger density distribution, with incoherent radiation accounting for wavelength-changing scattering.Source: §5 · p. 325
• Mark and Szilard’s [31] experiment found forbidden RbBr reflections absent with soft CuK radiation but appreciable with SrK radiation, consistent with phase reversal by K electrons.The reflections are weak because rubidium and bromine have nearly equal atomic numbers, and the phase reversal occurs in one radiation case but not the other.Source: §5 · pp. 325–326

The analysis of atomic structure by X-ray intensity

• X-ray intensity measurements assign the correct number of electrons to atoms in crystals and indicate their spatial extension, but cannot determine their detailed electron arrangements.Analyses treat atoms as spherically symmetric and determine scattering matter between radii r and r + dr; different methods yield distributions of the same general type.Source: §5 · p. 326

5. Examples of analysis

• Fine distribution details remain uncertain because Fourier convergence is slow, extrapolation is sensitive, and earlier error estimates may have been too large.Smoothing shell-derived humps could still match observations within experimental error, so detailed features require caution.Source: §5 · pp. 326–327
• Experimental fluorine data agree very satisfactorily, while alternative Fourier interpretations reproduce the same overall distributions but place peaks differently.The comparison uses James and Randall’s results [17] and Claassen’s analysis [16].Source: §5 · pp. 328–329
• Fourier analyses broadly agree with reasonable atomic models and observed radial distributions, supporting the method’s fundamental correctness despite incomplete theory–experiment agreement.The authors describe the results as encouraging further investigation, while noting that satisfactory agreement has not yet been reached.Source: §5 · p. 330
• More accurate data from shorter wavelengths and low temperatures could provide direct pictures of atomic structure through Fourier analysis.The proposed conditions permit a larger range of F values and more accurate radial distributions.Source: §5 · p. 330

8. The refraction of X-rays

• X-ray refraction measures coherent radiation scattered forward, and experiments detected small deviations from simple reflection laws, including for harder rays.Darwin [2] predicted an index differing from unity by about one part in a million; related treatments were given by Ewald [34], Stenström [35], Duane and Patterson [36], and Siegbahn and Hjalmar [37].Source: §5 · p. 331

Examples of analysis

• X-ray refractive indices measured by total reflection, crystal reflection, and prism refraction agreed with the high-frequency dispersion formula within experimental error.Prism experiments produced a measurable deflection and an X-ray dispersion spectrum; total reflection occurred at sufficiently glancing incidence.Source: §5 · pp. 331–332
• 1–5%: experiments using wavelengths between 0.5 and 2 Å determined 1 − µ with this accuracy and accurately numbered electrons in scattering units.The agreement held while atomic critical frequencies were not approached.Source: §5 · p. 332
• Near atomic critical frequencies, the simple dispersion formula still matched experiment except when the critical frequency was very closely approached.This regime opened a way to investigate atomic responses near characteristic frequencies.Source: §5 · p. 333
• Measurements on iron pyrites reproduced the refractive indices using constants corresponding to two K electrons near the K absorption edge, indicating their existence.The data supported the Drude–Lorentz theory even in some regions approaching electron natural frequencies.Source: §5 · p. 333
• Mark and Szilard’s experiment showed that scattering power changes as scattered-radiation frequency passes through a critical value, another aspect of anomalous dispersion.The passage presents refractive-index measurements as a potentially fruitful probe of atomic response near characteristic frequencies.Source: §5 · p. 333

Discussion of Mr Bragg's report

• Assuming zero-point energy yields scattering curves agreeing very well with Schrödinger-based Hartree calculations for sodium and chlorine, whereas omitting it produces deviations exceeding experimental error.The passage states that confirmation by new experiments would provide direct and convincing proof of energy at absolute zero.Source: §5 · p. 336
• Hartree’s iterative method obtains atomic fields and good X-ray and visible-light energy levels, but Heisenberg characterizes it as useful rather than exact.Fowler adds that excluding each electron’s own field makes the method exact for hydrogen and extremely close to exact for helium.Source: §5 · pp. 336–337
• Pauli argues that atomic calculations should use an N-particle density |ψ(x1, ..., zN)|2 rather than a three-dimensional density, which cannot be rigorously exact.For sufficiently short waves, coherent scattered-radiation intensity is expressed through integration over all particle coordinates.Source: §5 · pp. 337–338
• Under a constant electron charge-to-mass ratio, classical crystal-reflection calculations give the same result for localized particles or irregularly distributed atomic charge.Reflection intensity depends on the average electron-charge density in different atomic regions.Source: §5 · p. 338

• Scattering by many nuclei in a crystal remained an unattempted problem, extending Thomas’s atomic model beyond a single centre of attraction.Bragg stated that no one had yet attacked the calculation of general electron density around many nuclei in a crystal.Source: §5 · pp. 338–339
• Scattering curves were independent of phase relations because the experiments measured average scattering in each direction from very many atoms.Source: §5 · p. 339
• An exact atomic description requires the total electron configuration in 3n-dimensional space; any three-dimensional charge model is only a rough approximation.Source: §5 · p. 339
• Compton scattering changes the curve so the same scattering curve can be assigned regardless of atomic type or wavelength, while total scattering still agrees with J. J. Thomson’s formula.Waller’s new-mechanics treatment separates coherent and modified radiation; the coherent component follows Hartree-like curves, and the total matches Thomson’s formula within stated limits.Source: §5 · pp. 339–340
• Prins observed a strong decrease in the effective total-reflection angle near a metal’s absorption limit, explainable through absorption using a complex refractive index, though the predicted anomaly remained unconfirmed.The Fresnel treatment removes the sharp total-reflection limit and predicts a smaller effective angle that decreases with absorption; the experiments had not yet demonstrated the related real-index effect.Source: §5 · pp. 340–341

Analytic dispersion analysis

• Dispersion formulas from Lorentz’s classical theory and modern quantum mechanics are equivalent, while the analytic formulation excludes singular behavior for exponentially growing waves.The same formula applies to atoms with continuous absorption regions and is equivalent to formulas proposed by de Laer Kronig, Mark, and Kallmann.Source: §5 · p. 342

Experimental comparison

• Measurements of X-ray refractive indices by Doan agree better with the Drude-Lorentz formula than with Kronig’s quantum-theory expression.Source: §5 · p. 342

X-ray wavelength measurements

• Thibaud and Soltan’s tangent-grating wavelengths differ systematically from Dauvillier’s crystal-diffraction values, with the discrepancy increasing with wavelength.For a boron line, Thibaud and Soltan found 68 Å versus Dauvillier’s 73.5 Å, a difference of 5.5 Å.Source: §5 · pp. 342–343

Refractive-index correction

• The wavelength discrepancy is attributed to refractive-index effects in crystal diffraction, with measurements near 70 Å agreeing with δ = Kλ2 and with the Kallmann–Mark formula.The modified formula gives very good agreement with experiment and helps explain why long-wavelength fatty-acid-grating lines are broad.Source: §5 · p. 343

Limits of the explanation

• Explaining detailed Laue phenomena and deviations from Bragg’s law remains difficult because crystal-layer vibrations generate series that cannot be replaced by integrals.Near an absorption edge, Röntgen rays show phenomena resembling classical absorption-band optics, but the absorption edge does not correspond to a frequency physically present in the particles.Source: §5 · pp. 343–344

Editorial discrepancies

• The French edition differs from the typescript through omitted quotation marks, words, references, names, captions, and bibliographic details, while also adding or altering wording.Examples include omissions of “Beryl,” “& Compton,” references, and figure labels, alongside added phrases such as “Mlle” and “dans la pyrite.”Source: §5 · pp. 345–346

Textual and mathematical corrections

• The notes identify typographical and formula-level corrections, including a spurious denominator, altered notation, missing overbars, and a published statement that incorrectly described fractions as unchanged.They also distinguish typescript, French, and published forms for absorption-edge terminology, numerical values, and displayed formulas.Source: §5 · pp. 345–346

Discussion topic

• The section includes the discussion heading “Disagreements between experiment and the electromagnetic theory of radiation.”Source: §5 · p. 347

• Classical electromagnetic theory cannot account for several radiation–matter phenomena, including the photoelectric effect and high-frequency X-ray scattering with recoil electrons.Compton also identifies unresolved difficulties involving the ether, wave production, and individual quantum interactions.Source: By Mr Ar thur H. COMPTON · pp. 347–348
• Experiments on individual radiation–electron interactions indicate that radiation quanta of energy hν proceed in definite directions.Compton presents this as evidence against a complete classical wave account.Source: By Mr Ar thur H. COMPTON · p. 348
• Compton proposes that radiation may consist of units with energy hν and momentum hν/c, while allowing corpuscular and wave descriptions to be complementary.He suggests that a guiding wave might direct energy-carrying corpuscles.Source: By Mr Ar thur H. COMPTON · pp. 348–349
• He calls each radiation unit a “photon,” following G. N. Lewis, because the term avoids committing to a particular physical interpretation.The name is presented as shorter and less theory-laden than alternatives such as “radiation quantum” or “needle ray.”Source: By Mr Ar thur H. COMPTON · p. 349
• Compton also considers Bohr, Kramers, and Slater’s virtual-radiation theory, in which atomic energy exchanges are statistically related rather than directly paired.The theory describes virtual radiation and oscillators associated with transitions among stationary states.Source: By Mr Ar thur H. COMPTON · p. 349

Experimental difficulty

• Michelson–Morley-type evidence would strengthen the ether argument if it detected relative motion, but the passage says that no one has observed such motion apart from recent, doubtful experiments.The proposed ether would need perturbations to travel at a definite speed relative to each observer, regardless of motion.Source: By Mr Ar thur H. COMPTON · p. 350

Ether and wave theory

• The wave theory requires a medium, or ether, to transmit light, but this ether has become an uncomfortable burden because no relative motion through it has been detected.The text notes that the analogy between waves in such a medium and elastic waves is consequently very distant.Source: By Mr Ar thur H. COMPTON · p. 350

Relativistic alternative

• Under relativity, energy can move at the observer-relative limiting speed, making corpuscles simpler than waves if the ether is abandoned.The passage frames this as an alternative to retaining a medium whose properties differ substantially from the simple ether.Source: By Mr Ar thur H. COMPTON · p. 350

The emission of radiation

• Wave theory cannot reconcile atomic stationary-state frequencies with emitted frequencies, whereas photon emission follows an atom’s transition to lower energy.The photon carries the energy lost by the atom, with its energy, direction, and polarization specified without assigning a radiation frequency.Source: The emission of radia tion · p. 351
• X-ray intensity and average frequency are greater forward than backward, requiring emitting particles moving at roughly 25% of light speed.The Doppler asymmetry therefore points to rapidly moving free electrons rather than atoms as the emitters.Source: The emission of radia tion · p. 351
• The photon theory explains both the Doppler shift of moving radiators and the continuous X-ray cutoff at hν = eV.Continuous X-ray emission is thereby treated as the same kind of event as other radiation emission, rather than requiring a distinct source.Source: The emission of radia tion · p. 352
• Virtual oscillators and virtual radiation reconcile wave theory with sudden absorption while retaining stationary states.Photon absorption raises an atom to a higher stationary state, whereas wave-theory absorption would otherwise be continuous and conflict with the absence of intermediate states.Source: The emission of radia tion · p. 353

Energy and speed

• The photoelectric equation was confirmed from low-frequency light through γ-rays, with the photon theory accurately predicting emitted-electron speeds.The proportionality constant matched Planck’s h, and the equation remained precise even for the highest-speed β-rays.Source: The emission of radia tion · p. 354

Emission direction

• Photoelectron emission is concentrated near the incident wave’s electric-field direction, with an appreciable forward component and substantial angular variation.Auger’s measurements found that the average forward component increases with the X-ray-tube potential.Source: The emission of radia tion · p. 354

Polarization

• Polarized X-rays produce a strong excess of photoelectrons in or near the plane containing the incident rays’ electric vector.Bubb’s incompletely polarized X-ray measurements also showed some electrons emitted at right angles, plausibly because multiple scattering reduced the polarization.Source: The emission of radia tion · p. 355

Wave-theory test

• For fixed wavelength, increasing light intensity did not increase photoelectron speed: intense sunlight ejected electrons no faster than faint starlight.This contradicted the classical expectation that stronger electromagnetic waves should produce faster electrons.Source: The emission of radia tion · p. 356
• The electromagnetic-wave theory also failed energetically, because photoelectrons could emerge with nearly the incident electron’s energy even though an X-ray pulse’s energy was dispersed over a spherical wave.The observed ejection from a small bismuth particle was incompatible with that energy being supplied by a single wave pulse.Source: The emission of radia tion · pp. 356–357

The emission of radia tion

• The gradual-absorption hypothesis conflicts with evidence for discrete atomic stationary states and does not explain photoelectrons’ forward emission tendency.Radiation pressure transfers the forward impulse to the atom rather than leaving it with the absorbing electron.Source: The emission of radia tion · p. 357
• The photon theory explains photoelectron velocity through energy conservation and predicts nearly 90-degree emission for light, consistent with uncertain visible- and ultraviolet-light data.The prediction uses α = γ/λ, with γ = h/mc = 0.0242 Å; α is small except for very hard X-rays and γ-rays.Source: The emission of radia tion · pp. 357–358
• Calculated X-ray emission curves agree satisfactorily with the observed direction of maximum emission, supporting that photons transfer both energy and momentum to photoelectrons.The proposed photon vector property produces a forward impulse analogous to electromagnetic radiation pressure.Source: The emission of radia tion · p. 358
• Later experiments found the photoelectron’s forward component almost twice the theoretical prediction, indicating that photon–atom interaction is more complex than assumed.The forward momentum remains comparable in order of magnitude to the incident photon’s, suggesting photon momentum transfer plus an additional atomic impulse.Source: The emission of radia tion · p. 358
• The theory fails to account fully for broad angular emission: Bubb obtained the correct dispersion order, but predicted stronger dispersion for heavy than light atoms contrary to experiment.If photon action is short relative to the electron’s orbital period, the nucleus receives little momentum and the photoelectron acquires the incident photon’s forward momentum within experimental error.Source: The emission of radia tion · p. 359

The emission of radiation

• The X-ray wave vector indicates the most probable impulse direction, explaining why photoelectrons emerge over a wide angular range rather than one definite direction.This interpretation treats the wave vector as probabilistic guidance for electron impulses.Source: The emission of radia tion · p. 359
• In Bohr, Kramers, and Slater’s virtual-radiation theory, photoelectron properties remain consistent because energy and momentum are conserved only statistically, without predicting individual electron motion.The passage notes that the theory’s success instead comes from applying conservation principles to individual photon–electron interactions.Source: The emission of radia tion · p. 360
• X-ray scattering exposed failures of classical wave theory through wavelength changes, intensity behavior, and recoil electrons, while light-element scattering increased wavelength for both X-rays and γ-rays.The scattering phenomena were organized under three headings, and absorption experiments established wavelength increases for light elements.Source: The emission of radia tion · pp. 360–361
• Absorption measurements found no critical absorption limit for the proposed short-wavelength fluorescent radiation, and spectroscopy found no corresponding spectral lines, indicating a different softening process.The result contrasted the behavior of light elements with heavy elements, where characteristic fluorescent X-rays were present.Source: The emission of radia tion · pp. 360–361
• The scattered spectrum contained an unchanged line and a shifted line for each primary-ray line; the observed shift agreed with theory to within a small fraction of 1 percent.The shifted line was somewhat broadened, but its center matched the predicted displacement.Source: The emission of radia tion · p. 362

• Recoil electrons appeared as short tracks pointing along the primary X-ray beam, interspersed with longer photoelectron tracks.Source: The emission of radia tion · p. 362
• The observed ratio of short to long tracks agreed satisfactorily with the ratio of scattered to absorbed X-rays, indicating about one scattered energy quantum per short track.Source: The emission of radia tion · p. 363
• The experiments contradicted classical wave expectations because recoil-electron motion carried only 5 or 10 percent as much energy as the scattered X-rays.Source: The emission of radia tion · p. 363
• Recoil-electron energies, directions, and velocities closely matched photon-theory predictions, providing strong evidence for its fundamental hypotheses.Track ranges under different directions and primary X-ray wavelengths satisfactorily verified the predicted energies.Source: The emission of radia tion · p. 363
• Scattered-wavelength changes followed the Doppler-effect dependence expected for an electron moving along the primary beam, but virtual-radiation theory struggled to predict these phenomena while denying individual-action conservation principles.Source: The emission of radia tion · pp. 363–364
• The photon theory quantitatively predicted the scattered X-ray wavelength changes and recoil-electron characteristics, whereas the virtual-radiation theory remained problematic.Source: The emission of radia tion · pp. 364–365

The emission of radiation

• Unmodified rays are plausibly attributed to X-rays scattered by electrons firmly bound within atoms, but no quantitatively satisfactory theory of their origin has been published.The account explains their major characteristics, while further investigation may require assumptions about photon–electron interaction.Source: The emission of radia tion · p. 365
• Classical electromagnetic theory cannot explain the low intensity of scattered X-rays, while proposed solutions introduce no new concepts about radiation or scattering.The discussion therefore turns to experiments on individual photon–electron interactions.Source: The emission of radia tion · p. 365

Interactions between radiation and single electrons

• Bothe’s coincidence experiment found no unexplained simultaneous detections, supporting the photon hypothesis and contradicting virtual radiation’s prediction of frequent coincidences.Only coincidences attributable to high-speed β-particles crossing both counters were observed.Source: The emission of radia tion · p. 366
• In the composite photoelectric effect, heavy atoms sometimes eject multiple electrons simultaneously, yet their total energy does not exceed that of one incident radiation quantum.The photon theory reportedly interprets all observed electrons without difficulty.Source: The emission of radia tion · p. 367

Bothe and Geiger’s coincidence experiments

• Bothe and Geiger observed about ten recoil electrons for every recorded scattered photon, with 66 coincidences and only a 1-in-400,000 chance under virtual-radiation statistics.The result agrees with photon-theory predictions and contradicts the statistical scattering view.Source: The emission of radia tion · p. 368

Directional emission of scattered X-rays

• Directional-scattering measurements found 18 of 38 plates with both recoil and secondary tracks satisfying the photon-theory angle, while only about 1 in 250 would occur by chance.The result implies that scattered X-ray energy travels in directed quanta associated with individual recoil electrons, contrary to statistical virtual radiation.Source: The emission of radia tion · p. 370

Reliability of experimental evidence

• The evidence is difficult to reconcile with classical waves, but coincidence experiments remain vulnerable to sparks, radioactive impurities, and other extraneous sources.Bothe and Geiger’s photographic timing method allowed chance-coincidence probabilities to be estimated and auxiliary tests to identify spurious coincidences.Source: The emission of radia tion · p. 370

The emission of radia tion

• Photon theory predicted absent coincidences in fluorescence but present coincidences in scattering, making it difficult for both counter-experiments to have been seriously affected.Source: The emission of radia tion · p. 371
• Stray radiation tended to hide the sought effect rather than create a spurious one, while contamination and scattered X-rays could place β-particles elsewhere in the chamber.Only by minimizing stray radiation were the directional relations clearly observed in photographs.Source: The emission of radia tion · p. 371
• The apparent tendency of β-particle angles to fit the theoretical formula was unlikely to arise from measurement errors in judging those angles.The authors identified psychological misjudgment as the only consistent error capable of vitiating the result, but considered it insufficiently large.Source: The emission of radia tion · p. 371
• The two experiments independently reached conclusions opposed to the statistical theory of β-ray production.Their initial publications appeared simultaneously, indicating independent convergence on the same opposing conclusion.Source: The emission of radia tion · p. 371
• Given the experiments’ difficulty and the importance of their conclusions, repetition by physicists from other laboratories was considered highly desirable.Source: The emission of radia tion · p. 371

Competing theories

• Wave and virtual-oscillator theories retain major difficulties, including ether-related problems and failures to predict photoelectron and recoil-electron characteristics.The classical wave theory is also described as inconsistent with photoelectric experiments and unable to account for scattered-radiation wavelength changes or recoil electrons.Source: Summar y · pp. 371–372

Photon theory

• The photon theory explains photoelectric effects, scattering wavelength changes, recoil electrons, and individual radiation-quanta experiments more successfully than statistical theory.It concludes that radiation consists of directed energy quanta, or photons, with energy and momentum conserved in interactions with electrons or atoms.Source: Summar y · p. 372

Discussion of Mr Compton's report

• Weak-light diffraction experiments imply that diffraction cannot arise solely from photon interactions, requiring something that guides photons through the electromagnetic field.The argument notes that the average photon number in the relevant region can be below one, including intervals with no photon present.Source: Summar y · p. 373
• Experiments indicate that X-ray energy propagates as particles rather than extended waves, leaving even the electromagnetic ether unsatisfactory.The discussion distinguishes the wave-like propagation of electromagnetic quantities from the particle-like propagation of X-ray energy bundles.Source: Summar y · p. 374
• Williams finds that photoelectrons’ average forward momentum component is markedly larger than the light-quantum momentum hν.The measurements used monochromatic X-rays between 0.5 Å and 0.7 Å and inferred forward momentum from observed initial electron directions.Source: Summar y · p. 374

Momentum transfer

• The experiments find an average electron momentum of 1.8 hν, while the positive residue’s average velocity is nonzero, so its recoil term cannot be neglected.The wavelength range is too small to attach particular importance to the numerical value 1.8, but the observations suggest Compton’s simple law may fail.Source: Summar y · pp. 374–375

Theoretical accounts

• Classical analysis attributes the forward momentum mainly to the lighter charge, whereas quantum treatment predicts complete transfer of hν only at very short wavelengths and binding-force effects at long wavelengths.A reflecting-box example illustrates how diffuse reflection can produce surplus forward momentum.Source: Summar y · p. 376

Momentum transfer

• Photographs of photoelectron trajectories support Auger’s prediction that the electron receives, on average, the photon’s forward momentum, leaving no average forward momentum for the rest of the atom.This conclusion concerns averages over many electrons emitted in different directions.Source: Summar y · pp. 375–376

Waves and photons

• The Compton effect is difficult to reconcile with a classical picture because photon descriptions of incident radiation rely on measurements whose instruments are interpreted using wave theory.This creates an apparent logical contradiction between the photon account and the wave-theoretic description of detection.Source: Summar y · p. 377

Wave-field representation

• A finite spatial and temporal extension of wave fields, using four mutually meeting fields in scattering, removes incompatibilities with a space-time description.The representation includes incoming and outgoing electron and light-quantum fields, each finite in extension and overlapping in the same space-time region.Source: §5 · p. 326, Summar y · p. 378

Radiation symmetry

• Photoelectron emission under moderate polarized radiation is symmetric about the electric field and follows the probability law A cos2 ϕ.The same distribution remains valid in the wave plane, and superposing orthogonally polarized waves yields a fully symmetric distribution.Source: Summar y · p. 379
• These symmetry requirements challenge structural photon theories because natural light has no privileged direction in the wave plane.The argument treats the continuous wave description as necessary to reproduce the observed symmetry of emission distributions.Source: Summar y · p. 379

Radiation and quantum theory

• Radiation’s discontinuity appears through individual energy and momentum conservation, but detailed phenomena are interpreted more naturally from a continuous viewpoint.The discussion rejects abandoning photons and statistically weakening conservation laws because experiments support conservation for individual processes.Source: Summar y · pp. 377, 379

Photoelectron emission

• Wenzel’s wave-mechanical treatment predicts increased forward photoelectron emission for harder radiation but appears incomplete because it assumes an immobile nucleus.The discussion adds that photon momentum must be distributed between the nucleus and emitted electron.Source: Summar y · p. 380

Photon absorption

• Compton’s photon picture makes complete absorption leave no radiation field and restricts the possible trajectories of the emitted photoelectron through angular-momentum conservation.Lorentz likewise states that angular momentum transferred to the electron–nucleus system comes from the radiation field.Source: Summar y · p. 380

Absorption measurements

• For carbon, separating true absorption τ from scattering absorption σ is necessary, but the available absorption measurements are imprecise and cover limited wavelength ranges.The procedure uses σ at 0.71 Å and total absorption µ across wavelengths; because τ is small for carbon, the inferred σ cannot be very precise.Source: Summar y · pp. 380–381

Compton broadening

• 0.005 Å is the estimated broadening for a 0.7 Å ray scattered by carbon, but the underlying experiments are unsatisfactory, so the estimate is provisional.The modified ray is broader than the unmodified ray, yet the evidence supporting this figure is described as far from satisfactory.Source: Summar y · p. 381

Theoretical explanations

• Quantum mechanics gives two explanations for the broadening—electron velocity distributions causing Doppler broadening and wave-mechanical treatment—and they yield the same result when the electron can be treated as free.The free-electron condition is required for the procedure to be legitimate; Jauney’s Bohr-orbital velocity assumption instead predicts broader lines than experiment.Source: Summar y · pp. 381, 383

Wave-mechanical account

• The wave-mechanical account requires two infinitely extended monochromatic wave trains in the unperturbed solution, an assumption criticized as unsatisfactory and corrected only by Dirac’s quantum electrodynamics.For bound electrons, differing component speeds explain the lack of sharpness in the shifted scattered ray, which is another form of the velocity-distribution explanation.Source: Summar y · p. 382

Biological effects

• Collision electrons are presented as fundamental to biological effects from high-frequency radiation, with γ-rays producing ionisation indirectly through accompanying secondary β-rays.The biological effect is attributed, under the stated assumption, to ionisation in cells exposed to radiation rather than directly to the γ-rays.Source: Summar y · p. 383

Compton-effect energy deposition

• A calculated 2% conversion of primary γ-ray energy into electron energy per unit absorbing mass makes collision electrons a possible explanation for radiation’s biological effects.The estimate uses α = 1.2 and ρ = 0.02.Source: Summar y · p. 384
• Multiple Compton scattering can multiply collision electrons and reduce photon energies until photoelectron absorption becomes increasingly important.The passage describes a repeatable cascade in which scattered quanta generate additional collision electrons and smaller quanta.Source: Summar y · p. 384

X-ray therapy

• This Compton perspective supports using very-high-frequency, highly penetrating X-rays in therapy because their uniform irradiation need not imply negligible biological effect.The passage contrasts this conclusion with claims that high-voltage X-ray apparatus would be therapeutically ineffective.Source: Summar y · p. 385

Secondary γ-ray spectra

• Thibaud’s interpretation of secondary γ-ray spectral bands as Compton-produced photoelectrons is plausible but requires observing the corresponding collision-electron band.The proposed test concerns the band structure and the matching band from collision electrons associated with scattered γ-rays.Source: Summar y · p. 385

β-ray spectra

• Most discrete β-ray spectral lines are explained by internal conversion, whereas the continuous spectrum remains disputed between primary and Compton-based secondary origins.The passage describes internal Compton scattering as a possible mechanism whose electron-velocity distribution should follow Compton’s predictions.Source: Summar y · pp. 385–386

Experimental tests

• The predicted γ-ray diffraction broadening of 0.0485 Å has not been reported, although the relevant experiments are difficult and scarce.Each homogeneous γ-ray group should produce this broadening through associated scattered γ-rays.Source: Summar y · p. 386

• Overlapping photoelectron bands can make magnetic spectra difficult to analyze, especially for substances emitting several γ-ray groups with similar effectiveness.Each γ-ray group produces a band bounded by photoelectron velocity limits, but bands may partially overlap.Source: Summar y · p. 387
• Available experimental spectra were too complex for convincing conclusions, although RaD and mesothorium 2 showed features plausibly associated with monochromatic or primary γ-rays.RaD spectra displayed photoelectric lines forming upper band edges, while mesothorium 2 spectra showed a possible gap near 58 kilovolts.Source: Summar y · p. 387
• Bohr characterizes the simultaneous wave treatment as a symbolic analogy rather than a causal theory in the classical sense.The discussion connects this symbolic treatment with Klein’s work and describes it as characteristic of quantum theory’s methods.Source: Summar y · pp. 387–388
• Schrödinger’s wave-mechanical explanation of the Compton effect uses electron and photon waves before and after the encounter, but the later waves are not determined by the initial wave alone.The discussion states that an additional datum is required to make the problem well-defined, analogous to specifying an impact parameter in classical collisions.Source: Summar y · p. 388
• In wave mechanics, microscopic collision data are unavailable, so determining one particle’s post-collision motion requires prescribing the other particle’s motion after collision.Born argues that this requirement parallels classical mechanics, where additional collision information determines the outcome.Source: Summar y · p. 388

Principal points of view

• De Broglie models material corpuscles as periodic phenomena represented by stationary waves, extending the description from localized material points to wave quantities.The framework begins from the experimentally established existence of matter and radiation corpuscles and attributes periodicity to them.Source: Summar y · p. 391
• The theory makes a particle’s velocity equal to the group velocity of its associated wave, while its phase propagates faster than c as a relativistic consequence.This relation is established first for uniform motion and then recovered for motion in a potential through Hamilton’s equations.Source: Summar y · pp. 392–393
• Using the quantum relation between frequency and energy, de Broglie links wave quantities to mechanical energy, momentum, mass, and the Hamilton–Jacobi function.The wave phase is identified with the Jacobi function, making the principles of least action and Fermat identical for uniform rectilinear motion.Source: Summar y · pp. 392–393
• Quantum stability conditions follow from requiring single-valued phase around closed trajectories, yielding Planck and Sommerfeld conditions and an analogy between quantum stability and resonance.The associated whole-number conditions arise from phase consistency along closed or quasi-periodic motions.Source: Summar y · p. 393

• The wave conception of mechanics yields relations for light quanta that account for Doppler effects, radiation pressure, the photoelectric effect, and the Compton effect.The derivation treats light corpuscles by letting their proper mass tend to zero, so both relevant speeds approach c.Source: Summar y · p. 394
• It also unifies the kinetic theory of gases and blackbody radiation through a new statistical mechanics, coinciding with Bose’s proposal [2] and developed by Einstein [3] and later papers [4].Source: Summar y · p. 394
• Schrödinger definitively constructed the new doctrine [5] from wave equations whose phases recover the Jacobi equation in the geometrical-optics approximation.He used continuous-amplitude waves rather than singular wave amplitudes.Source: Summar y · pp. 394–395
• For a material point in a static potential field, Schrödinger’s wave equation agrees with the equation obtained from the dispersion law and Newtonian mechanics.Source: Summar y · p. 395
• Stationary-wave conditions define eigenfunctions representing stable atomic states, and the resulting quantisation method is described as correct, with contributions from Brillouin, Wentzel, and Kramers [6].The method replaces the Bohr–Sommerfeld approximation with stationary waves satisfying specified conditions.Source: Summar y · pp. 395–396
• Interpreting a material point as a localized wave group makes the electron spatially “smeared out,” eliminating definite position or velocity and creating absorption difficulties.The concern is that a light quantum occupying a wavelength-scale volume would be absorbed by an atom roughly a thousand times smaller.Source: Summar y · p. 396

Schrödinger’s wave mechanics

• Wave mechanics faces a conceptual difficulty because it constructs many-body configuration space from material-point coordinates while denying those points clearly defined positions.Wave propagation in abstract configuration space also lacks the straightforward physical meaning associated with waves propagating in ordinary space.Source: Summar y · p. 397
• Schrödinger showed that Heisenberg’s quantum mechanics could be translated into wave mechanics and used Ψ to derive an atom’s mean charge density.The translation connected Heisenberg’s matrix elements with wave-mechanical functions.Source: Summar y · p. 397
• Schrödinger equations are nonrelativistic, although later work supplied more general relativistic forms and an elegant higher-dimensional formulation without imaginary terms.The passage notes related developments by O. Klein [9] and the author [10], and a connection to Einsteinian gravity by De Donder [11].Source: Summar y · pp. 397–398

Born’s statistical interpretation

• Born rejected the wave packet as a direct localization of the particle and treated Ψ as a statistical representation rather than a causal description of individual phenomena.His interpretation combines quantum-mechanical determination with statistical prediction.Source: Summar y · p. 398
• In electron–atom collisions, Born’s final Ψ determines the probability that the collision produces one result or another.The treatment covers both elastic collisions, which leave the atom’s internal state unchanged, and inelastic collisions, which transfer energy as the atom changes stable state.Source: Summar y · pp. 398–399

Corpuscles and waves

• Wave mechanics therefore raises the central problem of reconciling pointlike energy elements with continuous waves and establishing a link between corpuscles and Ψ.The section frames these as the chief questions arising from the contemporary state of wave mechanics.Source: Summar y · p. 399

Core dynamics

• Each wave Ψ determines a class of motions, while an individual trajectory is fixed by the particle’s initial position and Ψ gives its presence probability when that position is unknown.The wave functions as both a pilot wave and a probability wave, while individual motion remains deterministic.Source: Summar y · p. 400

Scope and many-body systems

• The framework extends beyond the Newtonian regime, diverges from classical-Jacobi-based approaches outside old relativistic mechanics, and introduces a many-body-systems discussion.The supplied passage begins the many-body section but does not provide its substantive analysis.Source: Summar y · pp. 400, 402

The interpretation of interference

• The new theory predicts the same bright and dark interference fringes as the old theory, including for experiments averaging over space or time.This agreement is stated for short, intense irradiation and long, weak irradiation such as Taylor’s experiment.Source: Summar y · p. 402

The energy-momentum tensor of the waves Ψ

• The wave energy-momentum tensor separates into particle energy-momentum and wave stresses, with those stresses characterizing new constrained states predicted by wave mechanics.The stresses vanish for states consistent with old dynamics and help explain radiation pressure on a reflecting mirror despite interference preventing particles from striking its surface.Source: Summar y · p. 402

Summary

• Unlike the single-particle case, finding a wave Ψ that defines the system’s motion while incorporating relativity does not appear easy.This is identified as a remaining difficulty after the configuration-space interpretation addresses the meaning of the variables.Source: Summar y · pp. 403–404
• De Broglie represents an N-corpuscle system by a point in configuration space guided by a fictitious wave Ψ, with corpuscle positions always definite.The configuration-space wave plays the role of a pilot and probability wave for the representative point, analogous to the single-particle case.Source: Summar y · p. 403
• The many-body probability relation agrees with results obtained by Born for electron–atom collisions and by Fermi [17] for electron–rotator collisions.The text presents the revised relation as replacing the corresponding single-body relation for many-body systems.Source: Summar y · p. 404
• For atomic systems, de Broglie assigns corpuscles definite position and velocity, giving configuration-space variables a precise meaning and implying nonzero motion in most hydrogen stationary states.The hydrogen electron is motionless only when m = 0; otherwise the model yields uniform circular motion, contrasting with the inadmissible general conclusion of universal rest.Source: Summar y · pp. 404–405
• Applying the dynamics to hydrogen transitions permits a visual representation of electron motion, while ensemble averages recover charge and current densities matching Gordon, Schrödinger, and O. Klein [18].The transition velocity can be obtained when the initial position is specified, and the mean-atom formulas coincide with those authors’ results apart from notation.Source: Summar y · pp. 405–406

Correspondence and scattering

• Classical electromagnetism may retain only a statistical value, while mean-atom calculations can yield good predictions for radiation scattering.The method treats a mean atom in mean light and derives scattered radiation from the perturbed density ρ.Source: Summar y · p. 406

Corpuscles and singular waves

• A fuller theory should incorporate corpuscles into wave phenomena through singular wave solutions whose motions follow the propagation of Ψ.This correspondence is readily established without an external field but remains difficult in the general case.Source: Summar y · p. 407

Limits of wave mechanics

• Calculating electromagnetic quantities from Ψ does not by itself establish a deep connection between electromagnetism and wave mechanics.The proposed deeper link likely requires beginning with singular waves, and the magnetic-electron hypothesis remains outside wave mechanics.Source: Summar y · p. 407

Experimental prospects

• Electron wave phenomena should depend on wavelength, but precise experiments require electrons accelerated through at least a few volts, limiting λ to a few angstroms.The expected effects are more analogous to X-ray phenomena than ordinary-light diffraction.Source: Summar y · p. 408
• Crystals or grazing-incidence gratings offer practical routes to observe electron diffraction, including rings analogous to those found in X-ray experiments.Diffraction through crystalline or amorphous materials was proposed as an observable analogue of X-ray scattering.Source: Summar y · p. 408

Crystal diffraction

• For electron beams scattered by crystals, regular atomic structure should produce amplitude maxima in Laue-predicted directions and selective electron scattering.The exact theoretical prediction remained insufficiently advanced, and the crystal potentials were not known precisely.Source: Summar y · p. 409

Experiments by Mr E.G. Dymond

• Most scattered electrons had the same energy as the primary electrons, indicating elastic collision, while many lost about 20 to 55 volts, showing helium changed from normal 11S to excited 21S.A very pronounced maximum was observed for a low value of θ.Source: Summar y · pp. 409–410
• Dymond’s angular-distribution results were related to Mr Born’s predictions and the new mechanics, but could substantiate wave mechanics only in the most general way.The theoretical side was judged insufficiently advanced to provide detailed predictions of the expected phenomena.Source: Summar y · p. 410

Experiments by Messrs C. Davisson and L. H. Germer

• Earlier Davisson and Kunsman measurements [25] showed voltage- and surface-dependent angular-scattering maxima whose interpretation was assigned to the new mechanics.Above 200 volts scattering decreased steadily with deviation angle; at smaller voltages two maxima appeared, while magnesium-coated platinum produced a single small maximum below 150 volts.Source: Summar y · pp. 410–411
• Davisson’s experiments with Germer [26] confirmed the general predictions and even the formulas of wave mechanics for electron scattering from nickel crystals.The experiment used homogeneous electron beams incident normally on a nickel crystal and examined scattering along distinguished azimuths.Source: Summar y · p. 411
• Davisson and Germer observed distinct electron-scattering maxima at predicted directions, directly numerically confirming the new dynamics’ formulas, although the explanation remained incomplete.The maxima appeared only near particular accelerating voltages, a condition the account says still required explanation.Source: Summar y · pp. 411–412

Crystal-plane interference

• Selective scattering occurs only at certain electron velocities when interference from several parallel crystal planes satisfies an additional relation.Experiments show significant intensity only for particular values of V.Source: Summar y · p. 412

Davisson–Germer interpretation

• Patterson [27] showed that two superficial crystal planes can exactly predict the selective electron reflection variations observed near the Davisson–Germer peaks.The agreement supports explaining the phenomenon as wave diffraction in the outermost crystal layers.Source: Summar y · pp. 412–413

Implications and limitations

• A complete analysis of such diffraction experiments could provide information about actual crystal-surface conditions and investigate previously unobserved crystal regions.The discussion notes that the superficial planes may differ in spacing from deeper planes and that exact prediction requires a smaller near-surface plane separation than crystallographic measurements provide.Source: Summar y · pp. 412–413

Experiments by G. P. Thomson and A. Reid [28]

• Thomson and Reid observed three rings from electrons passing through cellulose, appearing near 2500 volts and remaining visible up to 16,500 volts.Ring radii decreased with increasing electron energy, approximately inversely with speed and therefore with wavelength λ.Source: Summar y · p. 413

Discussion of Mr de Broglie's report

• De Broglie extended phase quantisation to non-closed trajectories through pseudo-periods, claiming this recovered Sommerfeld’s conditions.He used quasi-periods that return the trajectory arbitrarily close to its initial position and require the wave to finish in phase.Source: Summar y · p. 416
• Born challenged de Broglie’s trajectory definition for elastic electron–atom collisions, while later analysis judged the quasi-period derivation faulty.The cited criticism states that phase closure after any quasi-period does not imply the separate conditions listed in the derivation.Source: Summar y · p. 416
• Brillouin argued that no serious objection had been made to de Broglie’s viewpoint because its trajectories were not proven nonexistent and its formulas agreed with Gordon’s accepted results.Born’s objection concerned observability and existence, but Brillouin emphasized that it did not establish a contradiction with other authors’ views.Source: Summar y · p. 417
• Pauli and others questioned whether a conserved-current construction supplied a satisfactory particle velocity and energy–momentum tensor compatible with Maxwell’s theory.The discussion compared current-density and momentum-density analogies and noted that independent moving particles do not yield the required energy–momentum tensor.Source: Summar y · pp. 417–418
• For hydrogen stationary states, de Broglie’s velocity formula produced circular rather than the old theory’s elliptical orbits, and the electron speed could be zero.Schrödinger confirmed the circular-orbit result for separated polar-coordinate solutions while leaving open whether other coordinate choices yield different trajectories.Source: Summar y · pp. 418–419

Atomic orbits

• For degenerate cases, the physically relevant state is an arbitrary constant-coefficient linear combination of all particular solutions sharing an eigenvalue, not any uniquely selected solution.Because the solutions cannot be distinguished in principle, more complicated orbit types may arise, and circular orbits can reflect an unjustified choice of particular solutions.Source: Summar y · p. 419

Pilot-wave examples

• In a reflected limited light beam, photons travel rectilinearly, bend at the interference-zone edge, follow bright fringes parallel to the mirror, and avoid dark fringes before retreating with the reflected beam.No photon strikes the mirror, yet special stresses in the interference zone supplement photon momentum flux to reproduce the classical Maxwell tensor.Source: Summar y · p. 421
• De Broglie’s hypotheses preserve classical radiation-pressure values while avoiding the paradox that straight-line photon paths would concentrate impacts and torque a mirror.The example contrasts the contradictory requirements of rectilinear photon paths and localization only in bright interference fringes; de Broglie’s system avoids the resulting torque paradox.Source: Summar y · pp. 421–422
• The pilot-wave account predicts non-total reflection when a second prism is placed within roughly a wavelength of the first, producing transmitted light and attraction between the prisms.The attraction is reported as unavailable from corpuscular motion, which produces only positive pressure in the kinetic-gas analogy.Source: Summar y · p. 423

Quantum mechanics

• Quantum mechanics seeks a self-consistent and complete account of conceivable micromechanical experiments by beginning with observations rather than assuming a fixed boundary between observable and unobservable quantities.Once a conceptual scheme exists, observations can support inferences about facts that are not directly observable.Source: Summar y · pp. 425–426
• The approach is motivated by atomic discontinuities that classical mechanics introduces only through artificial auxiliary assumptions, including corpuscles and discrete stationary states.The passage identifies particles and discrete stationary states as two characteristic discontinuities of atomic physics.Source: Summar y · p. 426
• Quantum mechanics treats atomic discontinuities and statistical transition probabilities as fundamental, yet preserves exact conservation of energy and momentum and offers no empirical argument against fundamental microphysical indeterminism.The theory is described as empirically testable despite its statistical character, while accounting for apparently causal processes.Source: Summar y · p. 426

The mathematical methods of quantum mechanics

• The mathematical formalism was developed initially to describe spontaneous radiation and began from empirical spectral laws after classical mechanical models failed to explain line spectra.Ritz’s combination principle represents each spectral frequency as ν_ik = T_i − T_k, motivating a quadratic array containing frequency, intensity, and phase information.Source: Summar y · p. 427

• Matrix multiplication applied to atomic oscillations introduces noncommutativity while preserving the classical frequency structure, since Ritz’s formula yields no new frequencies [2, 3].Real quantities are represented by Hermitian matrices, and repeated addition and multiplication define arbitrary matrix functions.Source: Summar y · p. 428
• The Born–Heisenberg–Jordan construction [4] transfers canonical equations to matrices, producing noncommuting conjugate variables and commuting noncorresponding coordinates and momenta.The Hamiltonian is treated as the same function of matrix p and q as in classical mechanics, despite ordering ambiguities such as p^2q versus pqp.Source: Summar y · p. 428
• For non-degenerate systems, matrix Hamiltonian dynamics conserves energy by becoming diagonal, with diagonal elements representing hT_n; the resulting requirement replaces canonical equations and remains unambiguous under degeneracy.The formalism was tested on harmonic and anharmonic oscillators, and the conservation theorem identifies the diagonal Hamiltonian entries with Bohr’s frequency condition.Source: Summar y · pp. 428–429
• Matrix mechanics handles rotating systems through Cartesian components and yields angular-momentum magnitudes proportional to h√(j(j + 1)), along with the correct transition selection rules and intensity formulas.These results agree with empirical rules associated with Landé’s Zeeman-effect analysis and earlier correspondence arguments.Source: Summar y · p. 429
• Pauli [6] used matrix mechanics to obtain hydrogen energy values and some intensity features, while the search for general valid coordinates extended canonical variables and transformations.The hydrogen result avoided angular variables and covered the energy spectrum plus selected intensity aspects.Source: Summar y · p. 429
• Matrix mechanics reduces the fundamental problem to diagonalizing a quadratic form in infinite-dimensional space, with invariant eigenvalues W_n representing the system’s energy values.The eigensolutions form the transformation matrix S, and the eigenvalues remain invariant under the corresponding canonical substitutions.Source: Summar y · pp. 431–432
• The general theory shows that a form cannot always be decomposed into a sum of squares and requires invariant integral components associated with a continuous distribution ψ(W).Continuous spectra therefore enter naturally, but retain quantum discontinuities: spontaneous transitions remain jumps between spectral points W′ and W′′.Source: Summar y · p. 432
• Matrix mechanics is cumbersome for non-periodic quantities, including angular variables and coordinates that become infinitely large, motivating the operator-calculus and q-number approaches.The passage identifies Born and Wiener’s operator calculus and Dirac’s q-number theory as two essentially different routes for overcoming this difficulty.Source: Summar y · p. 432
• Dirac’s q-number formalism uses abstract noncommutative quantities while preserving commutation relations, and it produced independent solutions for the hydrogen atom and Compton-effect radiation intensities [7] [12].Its drawbacks include tiresome noncommutative algebra and replacing q-numbers with ordinary numbers such as J = hn to obtain experimentally comparable results, reintroducing special quantum conditions.Source: Summar y · p. 433
• Operator calculus instead employs concrete operators or functions over infinitely many variables that obey the same noncommutative laws, developing from Schrödinger’s wave mechanics through later work.Unlike q-number theory, it does not introduce abstract hypercomplex numbers.Source: Summar y · p. 433
• The formalism shows that continuous-coordinate operator representations and discrete matrix representations are equivalent descriptions related by changes of orthogonal coordinates.Equation (17) is explicitly stated to be equivalent to equation (9).Source: Summar y · pp. 434, 436
• Quantum-mechanical quantities are represented by linear operators in Hilbert space, whose eigenvalues give the values they can take.The operator framework generalizes matrix representations and permits discrete or continuous spectra.Source: Summar y · pp. 434–435
• Schrödinger’s wave equation appears as a special case of the general operator theory.Constructing a Hamiltonian from conjugate operators converts the eigenvalue equation into a differential equation for ϕ(q).Source: Summar y · p. 435
• The wave-equation formulation replaces earlier quantum conditions with the requirement that eigenfunctions remain finite, generating discrete spectra automatically when applicable.The eigenfunctions also function as transformation matrices connecting wave and operator representations.Source: Summar y · pp. 435–436
• Matrix mechanics initially describes possible states and processes rather than actual phenomena, giving no account of when a state is present or when change occurs.Its focus on closed periodic systems explains why it can calculate stationary states and transition-related oscillations without describing genuine temporal processes.Source: Summar y · p. 437

Transition probabilities

• The squares of S-matrix elements determine transition probabilities for the corresponding state changes.An individual term |S_n1n2,m1m2|^2 contributes to the transition probability associated with jumps in both subsystems.Source: Summar y · p. 438

Interpretation

• Continuous beating and discrete quantum jumps do not conflict: for observable mean values, both descriptions produce the same result.The equivalence follows from a mathematical theorem relating time-averaged changes to weighted transition probabilities.Source: Summar y · pp. 437–438

Indeterminism

• Quantum mechanics yields correct mean values but cannot predict when an individual quantum event occurs, making indeterminism a matter of principle.The passage presents this limitation as deeply rooted in the possibilities of physical knowledge rather than as a gap removable by further probing.Source: Summar y · p. 438

Statistical framework

• Quantum mechanics merges mechanics with statistics by eliminating unobservable microscopic coordinates from empirically testable formulas.Atomic-physics observations are described as predominantly statistical, such as counts of atoms in a given state.Source: Summar y · p. 439

Time dependence

• Explicit time dependence can be introduced by treating the system as coupled to another system whose reaction is neglected, extending the statistical viewpoint.This formalism is described as convenient and applicable to explicitly time-dependent perturbations added to a time-independent energy function.Source: Summar y · pp. 439–440

Statistical interpretation

• Perturbations transform an initially definite quantum state into a superposition, which Bohr’s principles interpret statistically rather than as simultaneous occupation of several states.The transition probability is associated with the relevant coefficient magnitude, while Ehrenfest’s adiabatic theorem implies that infinitely slow changes suppress jumps.Source: Summar y · pp. 440–441
• The coefficients |c_n(t)|2 are identified as state probabilities, but initial state probabilities alone generally do not determine the perturbation’s subsequent course.The phases of the complex coefficients also matter, creating an interference effect in the final probability distribution.Source: Summar y · pp. 441–442

Matter waves

• Matter requires both corpuscles and waves: continuously propagating probability amplitudes govern the frequencies of discontinuous particles.The passage rejects identifying particles solely with wave packets and instead treats the wave description as probabilistic.Source: Summar y · pp. 442–443
• Collision phenomena provide the strongest support for interpreting matter waves as probability waves, with diffraction evidence linking particle-number distributions to wave intensity.The cited discussion refers to electron reflection by crystals and related diffraction experiments.Source: Summar y · p. 443

• Wave mechanics derives Rutherford’s law for α-particle scattering by heavy nuclei [31, 32].The scattering calculation uses a solution with an incident plane wave and an outgoing spherical wave, whose intensity determines directional detection probabilities.Source: Summar y · p. 443
• For many-particle processes, quantum-mechanical waves propagate in multidimensional configuration space rather than ordinary three-dimensional space.This distinction shows that quantum-mechanical waves differ fundamentally from classical light waves.Source: Summar y · p. 443
• The squared amplitude |ϕ|2 is interpreted as a probability density, while ϕ itself is a “probability amplitude.”A current vector constructed from the wave solution measures particle flow and supports this probabilistic interpretation.Source: Summar y · pp. 443–444
• Jordan’s transformation framework assigns |ϕ(q′, Q′)|2 as the probability distribution between two observables’ principal-axis representations [39].The framework extends conjugate-quantity reasoning to discrete spectra and expresses amplitudes as rotation-matrix elements.Source: Summar y · pp. 444–445
• Transition probabilities are given by |S(q′, q′′)|2, while transition amplitudes combine according to the interference rule.The transformation depends on an external parameter such as time or a weakly coupled system.Source: Summar y · p. 445
• Jordan [39] formulates an axiomatic system intended to connect observations directly to the theory’s mathematical principles.The proposed axioms are to be critically checked against all observations conceivable by contemporary means.Source: Summar y · p. 446

Axiomatic formulation

• The axioms make noncommutative operator multiplication a consequence of the formalism, removing much of its apparent strangeness.Dirac’s method [38] is described as completely equivalent to Jordan’s formulation, apart from its non-axiomatic organization.Source: Summar y · pp. 446–447

Limitations

• Quantum mechanics is summarized as a system centered on calculable probabilities, but its amplitudes are not directly measurable and their phases are only indirectly determined.The formalism also relies on improper functions such as the Dirac δ for continuous-variable representations.Source: Summar y · p. 447
• The amplitude-based probability definition does not express invariance under orthogonal transformations of Hilbert space, leaving a mathematical gap in the theory.The passage presents this as a defect alongside the use of improper functions and indirect phase observability.Source: Summar y · p. 447

Mathematical completion

• von Neumann [41,42] fills these gaps with invariant definitions of arbitrary operators’ eigenvalue spectra and relative probabilities, without assuming eigenfunctions or improper functions.Although not fully elaborated in every direction, the result establishes that quantum mechanics can have a mathematically irreproachable grounding.Source: Summar y · p. 447

Empirical interpretation

• The theory raises the problem of how statistically determined individual processes can preserve the deterministic order observed in composite macroscopic phenomena.Testing it requires determining where classical concepts such as position, velocity, momentum, and energy remain applicable.Source: Summar y · p. 447

Uncertainty principle

• For noncommuting quantities, simultaneous measurements have a characteristic indeterminacy: narrowing one variable’s range necessarily widens the other’s.The slit example connects wave diffraction with the corresponding momentum distribution, while Planck’s constant h measures this universal indeterminacy and supports quantum mechanics’ mixture of mechanical and statistical principles.Source: Summar y · pp. 447–449

• Future experimental results remain generally indeterminate, with quantum laws constraining only their statistical expectations.Measurements replace previously valid probability functions with new ones corresponding to observed outcomes.Source: Summar y · p. 449
• For simple systems, wave-packet propagation agrees with classical trajectories under initially limited precision, but generally differs in diffraction and interference [18,19].The transition from microscopic to macroscopic mechanics follows from the smallness of Planck’s constant h.Source: Summar y · p. 450

Applications of quantum mechanics

• The matrix treatment of electron spin and anomalous Zeeman effects is straightforward, whereas eigenoscillation methods require Dirac–Jordan theory [11].Two three-dimensional wave functions are associated with each electron, motivating an analogy with polarized light.Source: Summar y · p. 450
• For many-body systems, resonance decomposes similar particles into non-combining subsystems, with symmetric and antisymmetric states governing their behavior [28,29,40].Pauli’s principle permits only the antisymmetric subsystem for multi-electron atoms, enabling systematic line spectra and electron grouping.Source: Summar y · p. 451
• Bose–Einstein statistics applies to light quanta, while antisymmetric states yield Fermi–Dirac statistics with state weights restricted to 0 or 1 [56,16,57,37].Symmetric states assign weight 1 to permutation-equivalent configurations, whereas antisymmetric eigenfunctions vanish when quantum numbers coincide.Source: Summar y · pp. 451–452

Summary

• Fermi–Dirac statistics hold for electrons and likely protons, while Bose–Einstein statistics follow for neutral structures such as molecules.The claims are supported by spectral observations [28,43] and hydrogen’s low-temperature specific heat [55].Source: Summar y · p. 452
• Quantising cavity-field oscillators reproduces light quanta obeying Bose–Einstein statistics [4], while quantising Schrödinger-function harmonics yields Fermi–Dirac statistics with occupation numbers restricted to 0 and 1.This establishes a formal analogy between light quanta and electrons, although interacting material particles may create deep difficulties.Source: Summar y · p. 452
• Dirac’s investigations [51,52] rigorously derive Einstein’s spontaneous-emission probabilities and provide a dispersion formula with damping that includes Wien’s luminescence-decay experiments.The treatment couples atoms to a quantised electromagnetic field and clarifies atomic absorption and light scattering.Source: Summar y · p. 453
• Light scattering requires momentum variables and a matrix-operator equation rather than Born’s wave-mechanical collision theory [30], because light quanta have energy and momentum but no readily assigned spatial position.The remaining radiation difficulties concern relativistically invariant, closed electromagnetic formulations rather than the light-quantum–wave dualism itself.Source: Summar y · pp. 453–454
• Quantum mechanics is presented as a closed theory whose fundamental assumptions cannot be modified, with indeterminism agreeing with currently accessible experience.The discussion treats the corpuscle–wave dualism as compatible with this closed theory.Source: Summar y · pp. 453–454

Bibliography

• The bibliography lists works [5]–[59] spanning quantum mechanics, wave mechanics, quantum statistics, scattering, and related developments.The entries include contributions by Bohr, Pauli, Dirac, Heisenberg, Schrödinger, Born, Jordan, and others.Source: Summar y · pp. 455, 456, 458

Editorial policy

• The bibliography’s style was modernized and uniformized, with missing details and amendments supplied in square brackets, mostly without commentary.Corrections to typographical errors in the French edition were generally adopted, while mistakes found only there were recorded in endnotes.Source: Summar y · p. 455

Bibliography

• A subsection headed “On statisti s also” lists references by N. S. Bose [56] and E. F ermi [57].Source: Summar y · p. 458

Discussion of Messrs Born and Heisenberg's reports

• Matrix mechanics was presented as corresponding to classical mechanics’ second method by treating all possible initial conditions simultaneously, though it groups states with finite differences in the action variables.Its experimentally comparable results require substituting numerical action-variable values, creating difficulties because general results are matrix elements associated with two action-variable sets; only diagonal elements have direct physical meaning.Source: Summar y · p. 459
• Lorentz argued that representing coordinates and potential energy by matrices marked a major abstraction because these quantities had lost their original meanings.He also expressed surprise that the matrices satisfy equations of motion and called their construction mysterious, while noting their connection to observable atomic radiation frequencies.Source: Summar y · p. 459
• Lorentz inferred from constrained phase changes that emitted radiation might be derivative of a simpler underlying wave substrate, analogous to Schrödinger’s theory.The argument was that matrix elements can undergo phase changes while preserving the equations of motion, yet not all phase changes are arbitrary if emitted radiation is fundamental.Source: Summar y · p. 460
• Born compared matrices with hypercomplex numbers, Dirac related their arbitrary phases to classical action-angle variables, and Bohr stressed that phase is essential in calculations but scarcely enters observational interpretation.Bohr regarded the meaning of arbitrary phases as important for the consistency of quantum-theory methods, while Born rejected the idea that anything mysterious was hidden behind them.Source: Summar y · pp. 460–461

Translation and wording

• The notes record translation and wording differences, including omitted words, changed terminology, rearranged sentences, and uncertain readings in the source material.The French edition sometimes omits words or clauses, translates “Übergang” as “transformation,” and may reflect a mistranscription of “Überstand” as “Überelstand.”Source: Summar y · pp. 462–463

Editorial corrections

• The editors identify numerous discrepancies in equations, symbols, reference numbers, and notation between the typescript and French edition, correcting or standardizing them where necessary.Examples include missing equation numbers, altered summation indices, omitted absolute values, inconsistent H notation, and a probable incorrect reference that should be (17) or (22a).Source: Summar y · pp. 462–463

Context and bibliography

• The apparatus supplies contextual and bibliographic clarification, including Jordan’s terminology, the uncertain status of a purported Bohr work, and corrections to titles, dates, page numbers, and references.It notes that no extant Bohr work matches the cited title, while related archival material and Bohr (1985, p. 478) are identified.Source: Summar y · pp. 463–464

• Schrödinger presents a strictly relativistic theory based on waves in three-dimensional space, termed four-dimensional wave mechanics.Source: By Mr E. SCHR ÖDINGER · p. 465
• A second theory describes a wave-like process in the q-space of an arbitrary mechanical system and is termed multidimensional wave mechanics.Source: By Mr E. SCHR ÖDINGER · p. 465
• The complete unification of the two conceptions has not yet been achieved.Source: By Mr E. SCHR ÖDINGER · p. 465
• Beyond single-electron motion, the theory is treated only in the multidimensional version, which mathematically solves problems posed by Heisenberg–Born matrix mechanics.Source: By Mr E. SCHR ÖDINGER · p. 465
• The translation follows Schrödinger’s German typescript, with discrepancies from the French edition noted and Schrödinger’s French summary translated in footnotes.Source: By Mr E. SCHR ÖDINGER · p. 465

I.  Mul ti-dimensional theor y

• Wave mechanics represents an n-coordinate system by one function ψ(q1, q2, …, qn), rather than n coordinate functions qk(t).The qk become independent variables in a partial differential equation, whose variational formulation corresponds to classical equations of motion.Source: §I · pp. 466, 468
• The variational equation generally has admissible, finite, continuous solutions only for discrete eigenvalues that coincide in special cases with Bohr’s stationary energy levels.The recalculated values are reported to explain experimental facts better than the old values when discrepancies occur.Source: §I · p. 467
• Wave-mechanical eigenfunctions allow calculation of the matrix elements appearing in Heisenberg–Born theory, including the matrix element associated with q.The stationary-state framework yields transition-related quantities and reproduces matrix quantities from the wave-mechanical eigenfunctions.Source: §I · p. 468
• The formalism can treat interacting subsystems and material–radiation interactions by representing them as one larger conservative system with suitable potential terms.Subsystem coordinates enter the combined potential, while radiation can be modeled through additional oscillators and interaction terms.Source: §I · p. 468
• For an isolated system, the time variable may play no role, so the system’s internal passage of time would not be perceived.This possibility is explicitly associated with restricting attention to an isolated system.Source: §I · p. 468

I.  Mul ti-dimensional theor y

• The eigenvalues of the multidimensional equation are precisely Bohr’s stationary energy levels, and the corresponding functions form a complete orthogonal system.Source: §I · p. 468
• For non-conservative systems, the same equation is proposed to apply, although its solution no longer retains the simple conservative form and leads toward dispersion theory.Source: §I · p. 470
• The treatment can recover transition probabilities and the matrix elements qi(k, k′) of the same mechanical problem in Heisenberg-Born theory.Source: §I · p. 470
• The formulation contains no explicit time, allowing events to be viewed as sudden transitions between quantum states of the total system.Time would then be defined statistically by counting quantum jumps rather than by assigning ordinary durations to individual transitions.Source: §I · p. 470
• The section identifies a central interpretive question: how the ψ-function describes a system in three-dimensional space rather than merely an abstract multidimensional configuration space.One proposed picture treats the real system as a superposition of classical configurations weighted by ψψ∗.Source: §I · p. 471

I. 16 Multi-dimensional theory

• The charge-distribution picture yields experimentally successful radiation predictions, including Bohr’s frequency condition, spectral polarisation, and selection rules.Calculations reproduce observed radiation frequencies, intensities, and polarisations, including zero intensity where an allowed frequency-condition line is absent.Source: §I · p. 472
• The electrodynamic model is not complete: radiation reaction is absent from the wave equation, and treating one atom’s spread-out charges as fields acting on another is wrong in principle.The provisional classical coupling may be useful, but continuously radiating wave solutions could persist unabated, and interatomic interactions cannot consistently be assigned by directly evaluating one atom’s smeared charge at the other’s position.Source: §I · p. 473
• ψ represents a continuous electric-charge distribution whose density is calculated from ψψ*, replacing the classical picture of charged point particles.The quantum system is described as a superposition of classical configurations in q-space, with ψ serving as a weighting function.Source: §I · p. 473
• The fluctuating charges are governed by the wave equation rather than Coulomb or Biot–Savart forces, while acting as sources of the atom’s electric and magnetic fields and radiation.Because ψ generally depends on time, the charge distribution fluctuates; its field-source role is distinct from assuming that the charges directly act on one another.Source: §I · p. 473

I. Mul ti-dimensional theor y

• A unified wave-mechanical description would represent the emitting and receiving systems with one wave equation and distance-dependent coupling terms.This approach would avoid explicitly describing processes in the radiation field, but the correct coupling remains unresolved.Source: §I · p. 474
• At separations of several wavelengths, the usual Coulomb potentials cannot provide the required coupling terms.Retarded potentials are suggested instead, but they are more complicated than ordinary functions in common q-space.Source: §I · p. 474
• Without important amendments, the theory can in reality be applied only to very small systems.This limitation marks a provisional boundary of the theory rather than a completed general treatment.Source: §I · p. 474
• The available procedure is therefore an approximate treatment that nevertheless appears very useful.Its practical usefulness is retained despite the theory’s unresolved coupling problem and provisional limits.Source: §I · p. 474

Four-dimensional theory

• The multidimensional wave equation becomes the ordinary three-dimensional equation for de Broglie’s phase waves when relativistic effects are neglected.For slow electron motion, the derivation reproduces the classical, nonrelativistic approximation of de Broglie’s phase-wave equation.Source: I I.  F our-dimensional theor y · p. 475
• Lorentz invariance generalizes the equation to arbitrary electromagnetic fields, placing time and the three spatial coordinates on an equal footing.A Lorentz transformation of a static electric field generates a magnetic field, motivating the fully electromagnetic form.Source: I I.  F our-dimensional theor y · p. 476
• A four-dimensional variational principle produces the wave equation together with four retarded-potential equations.Adding Maxwell’s field Lagrangian yields Euler equations for both ψ and the electromagnetic potential.Source: I I.  F our-dimensional theor y · pp. 476–477
• The resulting charge and current densities are quadratic forms in ψ and agree with the multidimensional rule for extracting true charge distributions.The associated stress-energy-momentum tensor combines with the Maxwell tensor to satisfy energy and momentum conservation.Source: I I.  F our-dimensional theor y · p. 477
• The formulation is internally inconsistent because the same potentials modify the wave equation while being determined by charges derived from that equation.The wave equation determines ψ, ψ determines charge and current densities, and those densities act as sources for the retarded potentials.Source: I I.  F our-dimensional theor y · p. 477

Four-dimensional formulation

• The relativistic wave equation makes time enter symmetrically with spatial coordinates and follows from a four-dimensional variational principle that also yields electromagnetic laws.The resulting stress-energy-momentum tensor, combined with Maxwell’s tensor, satisfies conservation laws.Source: I I.  F our-dimensional theor y · p. 477

Hydrogen spectrum

• For hydrogen, the relativistic equation produces Sommerfeld’s fine-structure formula but disagrees with experience because both radial and azimuthal quantum numbers become half-integers.The result matches what the Bohr-Sommerfeld theory would give under that half-integer assumption.Source: I I.  F our-dimensional theor y · pp. 478–479

Limits of field theory

• The pure field theory is inadequate because adding fluctuating charge potentials to the wave equation would produce completely wrong results.The proposed supplement individualises extended electron charge densities while excluding each particle’s self-field from its single-particle wave equation.Source: I I.  F our-dimensional theor y · p. 478

Spin

• Wave mechanics had not yet established how electron spin is represented, although replacing the scalar ψ with a vector was identified as promising.The hydrogen result also indicates that spin-related modifications may be needed to complete the theory.Source: I I.  F our-dimensional theor y · p. 479

The many-electron problem

• For many-electron atoms, approximation methods reduce the multidimensional problem to three-dimensional hydrogen eigenfunctions and charge distributions whose interaction potentials determine the atom’s behaviour.This suggests that deeper understanding might ultimately restore intelligibility in three dimensions.Source: I I.  F our-dimensional theor y · p. 479

• The many-electron problem is formulated in 3n-dimensional configuration space using products of hydrogen eigenfunctions as a complete orthogonal basis.With electron–electron interactions initially neglected, the eigenvalues are sums of the corresponding hydrogen eigenvalues.Source: I I.  F our-dimensional theor y · p. 481
• The expansion coefficients become functions of time, and their governing equations are determined by Coulomb-energy integrals that reduce to sextuple integrals.Orthogonality eliminates many terms, leaving couplings associated with the electrostatic energies of charge distributions.Source: I I.  F our-dimensional theor y · pp. 481–482
• The resulting system contains infinitely many differential equations and cannot be solved directly, so the immediate practical gain appears limited.This is a limitation of implementation rather than a loss of formal completeness.Source: I I.  F our-dimensional theor y · p. 483
• Formally, the coefficient equations determine an exact solution of the many-electron problem and contain no remaining multidimensional dependence.Their coefficients are simple Coulomb energies of charge distributions already encountered in the hydrogen problem.Source: I I.  F our-dimensional theor y · pp. 483, 487
• The many-electron solutions are represented as combinations of products of hydrogen eigenfunctions, while nonzero pairwise products generate hydrogen-like charge distributions.This construction is presented as analogous to constructing higher atoms from hydrogen trajectories in Bohr’s theory.Source: I I.  F our-dimensional theor y · p. 483

Four-dimensional theory

• Assuming electron interactions and coupling coefficients are small, the first approximation reduces the problem to a denumerable set of finite principal-axes problems.The approximation treats the interaction contribution as small relative to the nuclear action and eigenvalue differences, neglecting rapidly fluctuating short-period terms.Source: I I.  F our-dimensional theor y · pp. 484–485
• The approximation procedure can continue indefinitely but becomes increasingly cumbersome, while the untruncated equations can also be transformed into infinitely many ordinary linear equations.When the coefficients are small, the energy E lies close to one of the eigenvalues E_l, yielding the principal-axes reduction.Source: I I.  F our-dimensional theor y · p. 484
• The multidimensional problem is solved in principle by an infinite system of equations whose coefficients have simple three-dimensional meanings, although direct solution is unavailable.The resulting charge distribution is a superposition of the hydrogen-problem distributions ρ_kk.Source: I I.  F our-dimensional theor y · p. 484

Discussion of Schrödinger's report

• Schrödinger explains the wave-mechanical picture by distributing many systems according to ψψ* dτ and taking the limiting superposed charge distribution as N = ∞.This construction treats ψψ* dτ as the probability of finding a system in a configuration-space volume element.Source: I I.  F our-dimensional theor y · p. 486
• The discussion exposes unresolved physical difficulties: radiation calculated from superposed solutions scales with c², while quadrupole moments complicate interpreting ψψ* as a charge density.Schrödinger says formal algebraic results do not yet provide a physical answer, and Born identifies quadrupole contributions as relevant both theoretically and experimentally.Source: I I.  F our-dimensional theor y · pp. 486–487

Four-dimensional theory

• The quadrupole-radiation question remained unresolved: Schrödinger defended the dipole calculations, but acknowledged that he had not examined whether wave and matrix mechanics agree for quadrupole radiation.No observations were available to decide a possible disagreement between the two approaches.Source: I I.  F our-dimensional theor y · p. 487
• Wave mechanics treats many-particle systems through multidimensional equations, with interaction problems becoming substantially more complex and simplified treatments recovering Schrödinger’s quantisation method as a special case.The discussion considers external nuclear action, electron-cloud interactions, relativistic and electromagnetic effects, and an infinitely massive nucleus as a first approximation.Source: I I.  F our-dimensional theor y · p. 488
• The helium ground-term calculation agreed with the empirical value, but its precision remained below measurement and below Kellner’s Ritz-based calculation.Born’s perturbative calculation used terms through second order; for helium, convergence was described as very slow.Source: I I.  F our-dimensional theor y · pp. 488–489
• Heisenberg argued that Schrödinger’s approximation replaces n-dimensional differential equations with infinitely many linear equations whose coefficients can be interpreted in three-dimensional space, but this does not establish a classical three-dimensional understanding.Schrödinger’s method makes calculations possible through multidimensional wave equations while seeking a clearer physical meaning rather than a simple three-dimensional partial differential equation.Source: I I.  F our-dimensional theor y · p. 489
• Schrödinger maintained that the central unresolved issue was the physical interpretation of the theory, whose existing formulations reproduce classical action at a distance and fail when light-crossing times become relevant.He described the mathematical criticism as unexceptionable but insisted that further theoretical development required physical interpretation.Source: I I.  F our-dimensional theor y · p. 490

Born’s many-electron problem

• Born argues that an n-electron atom has 3n quantum numbers, making an ordinary three-dimensional eigenvalue problem with the required multiplicity doubtful.The number of quantum numbers increases by three with each additional electron, and the relevant eigenvalue would need a multiplicity of ∞^3n.Source: I I.  F our-dimensional theor y · p. 490
• Born suggests obtaining the necessary range of quantum numbers by treating the eigenfunction as a Dirac q-number and quantizing its amplitude in time.An n-quanta oscillation of this amplitude, together with three spatial quantum numbers, supplies the required multiplicity; the electron number itself then appears as a quantum number.Source: I I.  F our-dimensional theor y · p. 490

Schrödinger’s wave-mechanical account

• Schrödinger states that the periodic system is already encoded in the three-dimensional hydrogen problem, whose degeneracies multiplied by spin yield the periodic numbers.The degeneracies 1, 4, 9, 16, and so on are multiplied by 2, with the factor 2 attributed to spin.Source: I I.  F our-dimensional theor y · p. 491
• Schrödinger interprets Pauli exclusion as the absence of a third eigenfunction with principal quantum number 1 in the three-dimensional model, whereas the multidimensional equation has too many eigenfunctions.The exclusion principle is therefore needed to eliminate the excess solutions of the multidimensional equation.Source: I I.  F our-dimensional theor y · p. 491

Causality, determinism, probability

• Lorentz sought to retain distinct space-time images of phenomena, including localized electrons with trajectories, even if a future theory allowed electrons to transform or dissolve into clouds.He was willing to accept new theories only if they could be re-expressed through clear, distinct images.Source: Ca usality, determinism, pr obability · pp. 494–495
• Lorentz described wave mechanics as determining the wave function ψ before and after collisions, while experiments also required accounting for individual electrons and photons.His collision example treats the wave function as satisfying a differential equation in a high-dimensional space, but invokes corpuscles alongside waves.Source: Ca usality, determinism, pr obability · p. 495
• Lorentz concluded that probabilities for electrons and photons in a given volume could suffice to explain the experiments, including examples presented by Heisenberg.He regarded this probabilistic description as achieving everything experiment permits, while treating indeterminacy as a conclusion of theoretical considerations rather than an a priori axiom.Source: Ca usality, determinism, pr obability · pp. 495–496
• Lorentz maintained faith in determinism for fundamental phenomena and questioned whether indeterminism must be elevated to a principle.He left open the possibility that a deeper mind could know the motions of electrons, even if current experiments supported indeterminacy.Source: Ca usality, determinism, pr obability · p. 496
• The surviving notes on Bohr’s remarks at this point are particularly fragmentary and incomplete, limiting reconstruction of the discussion.The editorial account notes that archival notes were too incomplete for reproduction in Bohr’s Collected Works, despite preserving topics such as wave-corpuscle aspects and momentum measurement.Source: Ca usality, determinism, pr obability · p. 496

Bohr’s complementarity and phase indeterminacy

• Bohr connects complementarity to the loss of phase information in stationary-state observations, illustrated by Stern–Gerlach distinguishability and the time–energy uncertainty relation.The distinguishability condition is ε > α, where diffraction spreading must be smaller than the beam’s angular separation.Source: Ca usality, determinism, pr obability · p. 498

Born’s wave–corpuscle problem

• Born frames the wave–corpuscle problem through radioactive α-particle tracks: spherical emission waves must account for tracks that appear as nearly straight lines.He states that explaining this requires appealing to Heisenberg’s “reduction of the probability packet.”Source: Ca usality, determinism, pr obability · p. 500

Probability-packet reduction

• Observing ionisation triggers local wave-packet reduction, producing a ray-shaped packet that corresponds to the phenomenon’s corpuscular character.The spherical-wave description remains valid only until ionisation is observed through cloud droplets.Source: Ca usality, determinism, pr obability · p. 500

Multidimensional treatment

• Born notes that the process can instead be represented in a multidimensional space with three times as many dimensions as particles, but this does not advance the fundamental questions.The treatment is presented as an intuitive example, with a simplified case involving two atoms that may be hit.Source: Ca usality, determinism, pr obability · p. 500

Probability and wave-packet representation

• The two-particle examples assign probabilities according to geometry, including certain hits when points lie on the same side and complementary outcomes when they lie on opposite sides.The examples give hit distributions of 0:1 and 0:0, 1:1 for the two configurations, respectively.Source: Ca usality, determinism, pr obability · p. 502
• A three-dimensional wave-packet construction represents these alternatives through branching propagation, trajectory kinks, and reduction to one propagation direction after a hit.The construction places the two points at x0 = 0, x1 = a, x2 = ±b and uses the packet’s branches to determine which point can be hit.Source: Ca usality, determinism, pr obability · pp. 502–503
• A complete description of processes in a system of several molecules requires a space with several dimensions.The example generalizes the wave-packet construction beyond a one-dimensional graphical simplification.Source: Ca usality, determinism, pr obability · p. 503

Limitations and objections

• The presentation acknowledges that its author has not entered deeply enough into quantum mechanics and explicitly raises objections to Conception II.The supplied passage records the limitation and the transition to objections but does not specify those objections here.Source: Ca usality, determinism, pr obability · pp. 503, 505

Competing conceptions of quantum theory

• Conception I treats de Broglie–Schrödinger waves as describing an electron cloud and ensembles, whereas Conception II describes individual particles through localized wave packets.Under Conception I, |ψ|2 gives ensemble probabilities; Conception II claims to describe each law-governed individual process as completely as possible.Source: Ca usality, determinism, pr obability · p. 504
• Conception II derives conservation laws for elementary processes, explains the Geiger–Bothe experiment, and accounts for nearly continuous alpha-particle tracks in Wilson chambers.The passage states that these consequences follow only from Conception II, while Conception I does not provide them.Source: Ca usality, determinism, pr obability · p. 505

Causality, determinism, probability

• A purely Schrödinger-wave account of |ψ|2 appears to require a peculiar action at a distance, motivating de Broglie’s proposal to localise particles during propagation.The objection concerns how a spatially distributed wave avoids producing simultaneous effects at multiple screen locations.Source: Ca usality, determinism, pr obability · p. 505
• The multidimensional configuration-space representation is criticized because identical-particle permutations become distinct points and short-range forces appear less natural there.These objections target the conceptual suitability of configuration space, not merely its mathematical consistency.Source: Ca usality, determinism, pr obability · p. 505
• Bohr argues that three-dimensional and 3n-dimensional descriptions are fundamentally equivalent, although reducing the latter to particle coordinates may be neither natural nor simple.He characterizes this issue as concerning the theory’s form rather than its substance.Source: Ca usality, determinism, pr obability · pp. 506–507
• Pauli presents a function of ordinary-space material-wave amplitudes that avoids the identical-particle defect and, under suitable assumptions, reproduces Schrödinger’s multidimensional results.The construction assigns probabilities to particle-number configurations with specified kinematical properties.Source: Ca usality, determinism, pr obability · p. 507
• Bohr’s complementarity is presented as more general than any particular mathematical representation, implying that multiparticle interactions cannot be described in the ordinary manner in space and time.The point separates the conceptual claim about complementary physical descriptions from the technical choice of representation.Source: Ca usality, determinism, pr obability · p. 507
• Multidimensional wave mechanics predicts an interaction between distant neutral atoms that ordinary spatial-temporal description would miss, with neglected polarization effects dominating the calculated contribution.The result is presented as an attractive force analogous to coupled classical oscillators, and the neglected polarization contribution outweighs the effects calculated by Heitler and London.Source: Ca usality, determinism, pr obability · p. 508
• For Bothe’s experiment, many-dimensional wave mechanics assigns probability to a light quantum appearing in either counting chamber but zero probability for two simultaneous appearances.The three-dimensional wave picture cannot distinguish one quantum with probability p from two with probability 2p and no one-quantum outcome, whereas the many-dimensional wave function can.Source: Ca usality, determinism, pr obability · p. 509
• The general many-dimensional wave-function theory necessarily abandons relativity, and proposed three-dimensional wave quantization would not reproduce Bothe’s explanatory power.The passage presents this as a current limitation rather than a settled resolution.Source: Ca usality, determinism, pr obability · p. 509
• Matrix theory derives Schrödinger’s electron-density expression as a diagonal element, but the true quantum density is the whole matrix and its diagonal gives only the average density.The diagonal form is therefore invalid when the density must be multiplied by a dynamical variable represented by a matrix.Source: Ca usality, determinism, pr obability · pp. 509–510

Determinism and observability

• Classical theory deduces a complete final state from completely specified initial numbers, whereas quantum theory distinguishes observable from inherently unobservable initial quantities.Experimental disturbances are under the experimenter’s control, while other numbers describing the system cannot be directly observed.Source: Ca usality, determinism, pr obability · p. 510

Experimental records

• Experimental results are permanent records describing a causally connected chain of events and the world’s state at subsequent times.The Wilson cloud-chamber example links ion formation, water-drop formation, light scattering, and a photographic record.Source: Ca usality, determinism, pr obability · p. 511

Quantum probability

• Quantum mechanics normally evolves the wave function causally, but nature chooses among noninterfering alternatives with probability |c_n|2.An experimental result identifies an irrevocable choice that affects the future course of events, although interference can prevent tracing the causal chain back to the collision.Source: Ca usality, determinism, pr obability · pp. 511–512

Probability calculus

• Born relates Dirac’s considerations to von Neumann’s forthcoming work, which builds quantum mechanics from ordinary probability calculus using a small number of formal hypotheses.In that construction, probability amplitudes and the law of their composition do not play a substantive role.Source: Ca usality, determinism, pr obability · p. 513

Causality, determinism, probability

• Dirac’s interpretation makes the diagonal matrix terms represent the probability that an electron lies within a specified volume V, linking this result to ψ̄ψ from Schrödinger’s wave equation.The construction uses a function equal to one inside V and zero outside, transformed into the relevant matrix representation.Source: Ca usality, determinism, pr obability · p. 514
• Heisenberg rejects the claim that nature makes the choice, arguing that interference can be produced even when the alternatives are resolved only after a long time.He instead says the observer makes the choice when the observation occurs, at which point the phase relationship and interference power are destroyed.Source: Ca usality, determinism, pr obability · p. 514
• Lorentz argues that treating nature’s choice as fundamental indeterminism means future events cannot be predicted in advance, contrary to the previously assumed possibility of prediction.The exchange therefore marks a fundamental disagreement about the meaning of measurement choices.Source: Ca usality, determinism, pr obability · p. 514

Photons

• De Broglie’s dualist corpuscle-and-wave representation predicts only mean mirror pressure for photon clouds, not local pressures during single-photon reflection.The discussion explicitly leaves the single-photon pressure problem unresolved and questions whether an additional mechanism is needed.Source: Photons · p. 515
• In de Broglie’s theory, photons have definite positions and velocities, but their speed falls below c inside interference zones and reaches c only during free propagation.For reflection, the parallel velocity component remains constant while the normal component disappears, yielding stationary photons at normal incidence.Source: Photons · pp. 515–516
• At the limit of normal incidence, increasing incidence slows photons until they become stationary in the interference zone.This follows from maintaining the velocity component parallel to the mirror while the normal component disappears.Source: Photons · p. 516
• De Donder’s correspondence principle identifies gravitational and electromagnetic field equations with wave-mechanical equations, yielding quantum current and tensor structures.The discussion states that these relations preserve charge continuity and support energy-momentum conservation through generalized quantum equations.Source: Photons · pp. 516–517
• De Donder’s quantum stresses contain Maxwell stresses as a special case, and his calculations recover the stresses associated with de Broglie’s model.This inclusion follows from deriving the correspondence principle partly from Maxwell’s equations and assigning quantum stresses an analogous formal role.Source: Photons · p. 517

Statistics and representation

• Bose–Einstein statistics emerge when constituents share motion states, while an alternative representation yields Fermi–Dirac statistics under a single representative point per cell.The new representation is presented as more appropriate for photons and particles treated as identical in nature but distinct in their states of motion.Source: Photons · pp. 518, 539
• Quantum mechanics provides no reason to prefer one statistics over another, and different statistics can be regarded as complete solutions independent of the interaction question.The participants nevertheless suggest Einstein–Bose statistics for light quanta and Fermi–Dirac statistics for positive and negative electrons, while noting that linking statistics to radiation–matter differences remains difficult.Source: Photons · p. 518

Photon quantization

• Quantizing cavity blackbody radiation with Boltzmann statistics produces Planck’s formula, equivalent to Bose–Einstein statistics for photons; a modified Dirac method similarly yields a Fermi distribution.The Fermi construction is described as artificial, relying on an unusual quantization method, and Jordan’s work is said to contain unresolved mathematical errors.Source: Photons · p. 519
• Both old and new cavity theories reproduce Planck’s radiation formula, but only the new theory gives exact local radiation fluctuations.The contrast is presented as a distinction between agreement on mean radiation density and disagreement over fluctuation values.Source: Photons · p. 519

Experimental evidence

• Experimental evidence associates protons and electrons with angular momentum and Fermi–Dirac statistics, while hydrogen specific heat and helium bands provide incomplete or competing evidence for composite-system statistics.Hydrogen specific-heat data were said to favor Bose–Einstein statistics, whereas helium bands indicate a nonspinning helium nucleus but cannot experimentally decide between Bose–Einstein and Fermi–Dirac statistics.Source: Photons · pp. 519–520

Electron exchange and exclusion

• Heisenberg argues that two electrons must have different energies, while weak interaction makes energy exchange too slow to resolve their symmetry state experimentally.The relevant exchange time is long, whereas identifying the symmetric or antisymmetric state requires at least approximately 1/ν.Source: Photons · p. 521

Spectroscopic evidence for nuclear spin

• Hydrogen-molecule spectra show odd-numbered lines intrinsically three times more intense than even-numbered lines, with about 5% accuracy across multiple vibrational states.The 3:1 ratio is reported for at least five vibrational states of a three-electron excitation and additional states, supporting persistent nuclear spin across excitation states.Source: Photons · p. 522

Photon–electron analogy

• The discussion asks whether the analogy between light waves and photons and between de Broglie waves and electrons can generate equally fertile experimental predictions.It specifically raises possible photon counterparts to electron charge and variable speed, while noting the wave theory of electrons explains Davisson–Germer results.Source: Photons · pp. 522–523

Photon polarization

• If each photon carries elliptic polarization, relativistic invariance would imply distinct photon species for different ellipticities, unlike the spinning-electron analogy.Retaining the electron analogy also raises how linearly polarized light should be represented and how photon motion at light speed affects spin descriptions.Source: Photons · p. 523
• Zeeman’s experiment found equal absorption in parallel and antiparallel magnetic fields, as predicted by continuity considerations.This result addresses the proposed photon description of linearly polarized light.Source: Photons · p. 523

• The discussion further questions whether photon properties such as angular momentum and wave-particle motion can be defined consistently across reference frames and compared directly with polarization.Photon frequency ratios remain invariant only when their worldlines are exactly parallel, and the spin-polarization analogy is explicitly rejected as simple.Source: Photons · p. 524
• The photon analogy with electron spin fails because photon polarization does not commute with position, unlike electron spin variables.A photon’s position implies a localized electromagnetic disturbance that cannot have a definite polarization direction.Source: Photons · p. 525
• Photon descriptions also face a mismatch with classical radiation: representing energy density, energy flow, and the Poynting vector requires altered speeds, superposed currents, or smeared photons.Assigning every photon speed c can require superposing several photon currents or excluding some energy-momentum-tensor components.Source: Photons · p. 525
• The physical nature of the photon’s ψ-wave remains unknown, including whether it can be identified with the electromagnetic wave or requires a second wave type.De Broglie provisionally develops a photon theory by associating photons with ψ-waves, while Lorentz questions whether these waves differ from light waves.Source: Photons · p. 526
• De Broglie’s theory assigns freely moving photons speed c, but allows deviations in interfering fields through an effective proper mass M0 determined by the ψ-wave amplitude.For a free photon associated with an ordinary plane wave, M0 is zero and finite energy requires speed c.Source: Photons · p. 526

Photons

• De Broglie’s new mechanics curves photon trajectories at diffracting edges through a force that appears when the proper mass varies during interference.For a photon cloud, the same equations recover internal stresses and the radiation pressure for reflected light, although the relation between ψ and the electromagnetic wave remains unclear.Source: Photons · p. 527
• De Broglie’s conception agrees with Born’s theory for elastic scattering but not when inelastic collisions are included.This marks a limitation of the agreement between the two approaches in scattering experiments.Source: Photons · p. 527
• In the rotator-collision problem, configuration-space wave mechanics treats spectral orders as distinct ways of transferring energy between the incident particle and rotator.Each spectral order corresponds to a stationary state of the rotator after collision.Source: Photons · pp. 528–529
• With a rotator initially stationary, superposed diffraction orders make de Broglie’s post-collision angular velocity nonconstant, conflicting with stationary-state requirements.The argument depends on an incident wave unlimited along the configuration-space axis, so all diffraction orders are superposed at each point.Source: Photons · p. 529
• Pauli’s difficulty is presented as arising from de Broglie’s demand that individual collisions be fully determined in spacetime while avoiding probabilities.Born’s theory instead applies the separate configuration-space partial waves statistically; the discussion also notes that the incident wave should be laterally limited in configuration space.Source: Photons · p. 529

De Broglie’s wave model

• De Broglie argues that a laterally limited wave yields a constant representative-point velocity and a stationary rotator state after the incident beam separates.He challenges the artificiality of configuration space when periodic identification along the ϕ-axis makes separated points represent the same system state.Source: Photons · p. 530

Photon models

• The quantum potential is presented as producing the difference between physical and geometrical quantisation.The formulas compare de Broglie’s and Rosenfeld’s correspondence models, with the former using a modified-Jacobi framework and the latter retaining the classical one.Source: Photons · p. 532
• Rosenfeld’s photon model is favored because it permits the photon’s speed to remain that of light, unlike de Broglie’s model, which requires variable speed.Both models assign zero charge density; de Broglie’s replaces the zero proper mass with a modified mass, whereas Rosenfeld retains m_0 = 0.Source: Photons · p. 533

Atomic localization

• Lorentz objects that an atom is spatially limited while its wavefunction extends to infinity, whereas de Broglie’s variable proper mass removes the strict confinement of an electron within a fixed sphere.The discussion also notes that quantum-mechanical eigenfunctions can connect separated equilibrium regions, allowing transitions between them.Source: Photons · pp. 533–534

Measurement and description

• Born and Bohr maintain that speaking of an electron’s location and motion remains meaningful when an experiment can be specified to measure them.This position is stated against Schrödinger’s view that such talk is nonsensical, though the supplied passage ends before the experimental qualification is fully developed.Source: Photons · p. 534

• Atomic wave packets cannot be formed because available eigenfrequencies are too widely separated to provide the slightly different wavelengths needed for localization.The same spreading difficulty affects free electrons, whose localized packets generally do not retain their shape.Source: Photons · pp. 535–536
• Localized wave packets spread, making the wave-packet picture of an electron unsatisfactory beyond perhaps a short interval.Bohr’s procedure repeatedly re-localizes the packet after observation, but the packet then spreads again; Lorentz wants a description valid for unlimited time.Source: Photons · p. 536
• Wave mechanics treats the impossibility of constructing classical-like atomic wave packets as evidence that classical mechanics fails at atomic dimensions.Schrödinger describes the moving entity as a finite excitation domain rather than a point electron; when trajectories shrink to a few wavelengths, old mechanics loses meaning.Source: Photons · p. 536
• Schrödinger’s model recovers classical motion only when an excitation domain travels along a trajectory whose dimensions and curvature radii greatly exceed the domain’s size.For small-quantum-number orbits, the trajectory becomes comparable to a few wavelengths, so propagation along a trajectory is no longer meaningful.Source: Photons · p. 536
• Wave-packet spreading after collisions conflicts with evidence for both diffraction-like behavior and continued well-defined trajectories, leaving wave–particle combinations provisional.Davisson–Germer interference supports spreading, whereas Wilson chamber experiments indicate something continues to describe a definite trajectory after collision.Source: Photons · pp. 536–537

Electron–photon analogy

• Langmuir asks whether the parallel between electrons and photons—de Broglie versus electromagnetic waves—can be extended experimentally.He compares electron charge with possible photon velocity and asks what experiments might clarify the analogy and its differences.Source: Photons · p. 539

Wave mechanics

• Wave-packet construction fails for atoms and free electrons because the required slightly different wavelengths are unavailable among sharply distinct eigenfunctions.The discussion states that wave packets cannot be formed from the available atomic eigenfunctions and that such packets ultimately dissolve.Source: Photons · p. 540

Localization

• Localized wave packets spread and do not maintain themselves, making the wave-mechanical picture unsatisfying for describing localization over indefinite time.Bohr’s post-observation relocalization is mentioned, but the speaker requests a picture of the system throughout an indefinite period.Source: Photons · pp. 536, 540

Photons

• The bibliography includes foundational discussions of quantum theory, hidden variables, Bohmian mechanics, and quantum gravity.Entries include Bell, Bohm, Barbour, and related works on interpretation and dynamics.Source: Photons · pp. 541–542
• It also lists works by Bohr, Born, Heisenberg, Jordan, and others on quantum mechanics, atomic physics, and radiation theory.The entries cover quantum mechanics, quantum laws, atomic dynamics, and the quantum theory of radiation.Source: Photons · pp. 542–543
• Several references document de Broglie’s early work on light quanta, waves, diffraction, interference, and related quantum principles.The listed publications span 1922–1924 and include proposals concerning light quanta and waves.Source: Photons · p. 543

Related works

• The references include de Broglie’s works on quantum theory, wave mechanics, light quanta, interference, and the wave nature of electrons (1924b, 1924c, 1924d, 1924e, 1925, 1926, 1927a, 1927b, 1999).They also include later books and collected writings on wave mechanics and microphysics (1930, 1955, 1956, 1974).Source: Photons · pp. 543–544
• The bibliography covers historical and conceptual studies of quantum theory, including wave-particle duality, physical reality, information-theoretic constraints, and the Copenhagen and causal quantum-theory programs (2006, 1988, 2003, 1994, 1996).These entries include works by Camilleri, Chevalley, Clifton, Bub, Halvorson, Cushing, and Cushing, Fine, and Goldstein.Source: Photons · pp. 544–545
• The references also include foundational and technical contributions on quantum mechanics, radiation, scattering, electron diffraction, and quantum dynamics by Compton, Dirac, Davisson and Germer, and others.The listed works span 1925–1933 and include studies of scattered X-rays, Compton scattering, emission and absorption, dispersion, and electron scattering.Source: Photons · pp. 544–545

Bibliographic references

• The entries include multiple works by Einstein from 1924–1949 on quantum theory, ideal gases, physical reality, and correspondence.Source: Photons · p. 546
• Additional entries cover interpretations, quantum foundations, quantum information, nonlocality, consistent histories, and related physics.Source: Photons · pp. 546–547
• The bibliography lists Heisenberg’s works from 1925–1984 on quantum mechanics, kinematics, interpretation, and the development of quantum theory.Source: Photons · pp. 547–548

Related work

• The passages are bibliography fragments rather than substantive discussion, listing works by authors including Hilbert, von Neumann, Holland, Jordan, Mehra, Rechenberg, and Newton.The entries cover mathematical foundations, quantum-mechanical history, interpretive debates, and earlier physical theory.Source: Photons · pp. 548–550

Related literature

• Additional entries cover quantum mechanics, quantum gravity, scattering, radiation, interference, and historical studies of physicists and their work.The listed publications include books, journal articles, edited correspondence, and historical analyses.Source: Photons · pp. 550–552

Quantum foundations

• The references include works addressing quantum interpretations, measurement, probability, decoherence, and foundational debates.Examples include publications by Pearle, Saunders, Shimony, Unruh and Wald, and Valentini.Source: Photons · pp. 550–552

Schrödinger references

• The bibliography lists numerous Schrödinger publications on wave mechanics, quantization, radiation, and quantum-mechanical interpretation.These entries span works published from 1922 through 1995.Source: Photons · pp. 551–552

• The references include Valentini’s works on pilot-wave theory, signal locality, subquantum information, and quantum probabilities.Source: Photons · pp. 552–553
• They also cite Wallace on Everettian structure and rationality, alongside Wheeler and Zurek on quantum measurement and delayed-choice experiments.Source: Photons · p. 553
• Additional references cover photon-related work by Wentzel, Wigner’s reflections on quantum theory, complementarity, and Zurek’s account of decoherence.Source: Photons · p. 553