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The coming of age of Erwin Schrödinger: His quantum statistics of ideal gases

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References

  1. E. Schrödinger, “Quantisierung als Eigenwertproblem,”Ann. d. Phys.,79, pp. 361–376, 489–527;80, pp. 437–491;81, pp. 109–140 (January–June, 1926). Reprinted in English with other papers on wave mechanics inCollected Papers on Wave Mechanics (London: Blackie & Son, 1928).

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  2. W. Nernst, “Über die Anwendung des neuen Wärmesatzes auf Gase,”Zeitschrift für Elektrochemie 20 (1914), pp. 357–360.Nernst actually derived the formula for the expected degeneracy temperature in a later paper,W. Nernst, “Über einen Versuch, von quantentheoretischen Betrachtungen zur Annahme stetiger Energieänderungen zurückzukehren,”Verhandlungen d. Deutschen Phys. Gesellschaft 18, (1916), pp. 83–116.

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  3. A. Einstein, “Quantentheorie des einatomigen idealen Gases,”Berliner Berichte (1924), pp. 261–267, and (1925), pp. 3–14. The first part of this paper was read to the Berlin Academy on September 20, 1924, The second part was read on January 8, 1925.

  4. It must be added thatSchrödinger also contributed to the twelve-year neglect ofEinstein's prediction of a non-classical condensation of particles at low temperatures. See the discussion ofSchrödinger's “On the Einstein Gas Theory”, below. But, asStephen Brush has noted, that neglect may have arisen more fromUhlenbeck's criticisms of 1927. (Brush, draft of paper “Statistical Mechanics and the Philosophy of Science: Some Historical Notes”, presented to Philosophy of Science Association Meeting, Chicago, October 1976.) SeeG.E. Uhlenbeck,Over Statistische Methoden in der Theorie der Quanta ('s-Gravenhage: Nijhoff, 1927).

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  5. E. Schrödinger, “Zur Einsteinschen Gastheorie,”Phys. Z. 27 (1926), pp. 95–101.

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  6. Martin J. Klein, “Einstein and the Wave-Particle Duality,”The Natural Philosopher 3 (1964), p. 45.Klein's work has stimulated much of this study, and I owe him special thanks for his aid and criticism.

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  7. —, pp. 522.

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  8. This point was made twelve years ago byM. J. Klein.—, “Einstein and the Wave-Particle Duality,”The Natural Philosopher 3, (1964), See note 6, p. 45.

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  9. Schrödinger's interest inEinstein's work onBrownian motion and fluctuations is discussed inSchrödinger's research notes, which have been micro-filmed by the Sources for History of Quantum Physics Project. The Project has produced the Archive for History of Quantum Physics (hereafter “AHQP”), of which Microfilm 39 contains many ofSchrödinger's notes. I have transcribed one interesting notebook, “Smoluchowski's Last Works” (Microfilm 39, Section 5), which appears as an appendix in my doctoral dissertation,Erwin Schrödinger's Statistical Mechanics, 1912–1925 (New Haven, 1975). In the dissertation I also discussSchrödinger's reviews of the theories of specific heats of solids which extend the work ofEinstein.

  10. For a discussion of how and whySchrödinger might have had special reason to developde Broglie's idea into a full-fledged wave mechanics, seeV. V. Raman &Paul Forman, “Why was it Schrödinger Who Developed de Broglie's Ideas?”Historical Studies in the Physical Sciences, Volume1 (Philadelphia: Univ. of Penn., 1969), pp. 291–314.

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  11. —,Nernst, note 2, p. 359.

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  12. P. Ehrenfest &V. Trkal, “Deduction of the dissociation-equilibrium from the theory of quanta and a calculation of the chemical constant based on this,”Koninkl. Nederl. Akademie van Wetenschappen te Amsterdam, Proceedings 23 (1920), pp. 162–183.Schrödinger cited a re-publication which appeared in theAnn. d. Phys. 65 (1921), pp. 609 ff.

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  13. M.J. Klein, “Ehrenfest's Contributions to the Development of Quantum Statistics, I and II”,Koninkl. Nederl. Akademie van Wetenschappen te Amsterdam, Proceedings, Series B,62 (1959), pp. 41–62, especially pp. 51 ff.

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  14. The discussion ofN-dependence byEhrenfest & Trkal is cited repeatedly inSchrödinger, “Gasentartung und freie Weglänge”,Phys. Z. 25 (1924), pp. 41–45. See especially p. 41 after equation (4), p. 44 after equation (19); also the first and second paragraphs of column two, p. 44. Furthermore,Schrödinger published an additional note titled “Isotopie und Gibbssches Paradoxon” (Z. f. Phys. 5 (1921), pp. 163–166), which addressed theN! problem indirectly. There he discussed how a continuous homogeneous gravitational field could be used in principle to gain work in mixing two isotopes of a monatomic gas (RT ln 2 for one mole of each isotope), asGibbs andBoltzmann had also found for chemically different and for physically separable gases (p. 165). That gain of work meant that there was an entropy of mixing of the isotopes of magnitude 2R ln 2.Schrödinger did not mention that one could “understand” why there was no entropy of mixing for identical particles by correcting away the redundancies in the thermodynamic probability — that is, by dividing the thermodynamic probabilityW byN!. The paper may have interested him in the problem of identity of particles and introduced him to theN! correction, which he pondered further inEhrenfest & Trkal's paper of 1921 in theAnnalen der Physik.

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  15. Ehrenfest & Trkal, note 12 p. 177.

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  16. O. Sackur, “Die Anwendung der kinetischen Theorie der Gase auf chemische Probleme”,Ann. d. Phys. 36 (1911), pp. 958–980; alsoO. Sackur, “Die universelle Bedeutung des sog. elementaren Wirkungsquantums”,Ann. d. Phys. 40 (1912), pp. 67–86;H. Tetrode, “Die chemische Konstante der Gase und das elementare Wirkungsquantum”,Ann. d. Phys. 38 (1912), pp. 434–442;H. Tetrode, “Berichtigung ...”,Ann. d. Phys. 39 (1912), pp. 255–256.

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  17. Ehrenfest & Trkal, note 12, pp. 54–55.

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  18. Klein, note 13,, pp. 54–55.

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  19. Max Planck, “Absolute Entropie und Chemische Konstante”,Ann. d. Phys. 66 (1921), pp. 365–372; especially p. 371. This paper was received by the publisher October 31, 1921.Planck argued still more vehemently for the division byN! in “Zur Quantenstatistik des Bohrschen Atommodells”,Ann. d. Phys. 75 (1924), pp. 673–684. Here he argued against “several notable physicists” [among themSchrödinger, whose work, “Gasentartung ...”, note 14, he cited] who objected to his kind of calculation and considered the division byN! to be apost facto insertion. (p. 681.)

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  20. AHQP, Microfilm 40, Section 1. There are no dates in this notebook. However, since the heading of the first section is the same as the title ofPlanck's response toEhrenfest & Trkal, note 19, I estimate that this notebook dates between about January 1922 and Dcember 1923, the date of reception of the paper on gas degeneracy.

  21. Planck, note 19, “Absolute Entropie...”, p. 370.

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  22. AHQP, Microfilm 40, Sections 1, 2, and 3. Practically all ofSchrödinger's unpublished manuscripts of about 1924 in the Archive for the History of Quantum Physics refer to quantum statistics. Included are (1) a long outline of chapter headings for a proposed book on “Molecular Statistics”, (2) the previously discussed 34-page notebook on “Chemical Constants and Gas Degeneracy”, (3) about 50 pages of notes for lectures on quantum statistics, and (4) another 100 pages of rough research notes titled “Quantum Statistics”. The lecture notes suggest thatSchrödinger may have come upon the general problem of degeneracy through preparation for his lectures on “Gases”, a sub-section of item (3). These notes make no mention ofSchrödinger's work with entropy definitions or theEinstein gas theory, and the last topics of the rough research (or lecture) notes are osmotic pressure, theory of dissociation, and the “phenomenological statistics” ofEinstein (cf. Einstein,Ann. d. Phys. 33 (1910), pp. 1275–1298). The corpus of preserved notes on quantum statistics appears to be no more recent than early 1924, because the next topics covered after gas degeneracy are the theory of spectra (Microfilm 40, Section 4, dated late 1925–1926) and manipulations of the H-atom wave equation (Microfilm 40, Section 5, dated late 1925).

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  23. Fritz Reiche (The Quantum Theory (New York: E.P. Dutton, n.d.), translated byH. S. Hatfield & H. L. Brose, pp. 79–80) listed those important works appearing between 1911 and 1921 which considered the degeneration of gases. They wereSackur andTetrode (see note 16)O. Sackur, “Die Anwendung der kinetischen Theorie der Gase auf chemische Probleme”,Ann. d. Phys. 36 (1911), pp. 958–980,W. H. Keesom,Phys. Z. 15 (1914), pp. 695 ff.,Sommerfeld andLenz (see note 29)A. Sommerfeld,Vorträge über die kinetische Theorie der Materie und der Elektrizität (Leipzig: B. G. Teubner, 1914), pp. 125–146,Scherrer (see note 27)P. Scherrer, “Das ideale Gas als bedingt periodisches System im Sinne der Quantentheorie”,Nachrichten von der Kön. Gesellschaft d. Wiss. zu Göttingen (1916), pp. 154–159, andPlanck (see note 26).M. Planck, “Über die absolute Entropie einatomiger Körper”,Berliner Berichte (1916), pp. 653–667.

  24. W. Nernst note 2, “Über einen Versuch ...”,, pp. 100–101.

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  25. M. Planck, “Über die absolute Entropie einatomiger Körper”,Berliner Berichte (1916), pp. 653–667.

  26. P. Scherrer, “Das ideale Gas als bedingt periodisches System im Sinne der Quantentheorie”,Nachrichten von der Kön. Gesellschaft d. Wiss. zu Göttingen (1916), pp. 154–159.

  27. Schrödinger, note 14, “Gasentartung ...”, “, pp. 41–42.

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  28. A. Sommerfeld,Vorträge über die kinetische Theorie der Materie und der Elektrizität (Leipzig: B. G. Teubner, 1914), pp. 125–146.Sommerfeld presented some work which was originally done byW. Lenz.

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  29. Schrödinger note 14, “Gasentartung ...” “, p. 42.Martin Klein has pointed out to me that such a counting of modes should not be expected to yield a cogent theory since, at this time inSchrödinger's view, there was nothing to “carry” the oscillations (if one conceives of them as mechanical).

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  30. θ now became of order 10−4°K. This was well below experimentally observable values.

  31. Schrödinger, note 14, “Gasentartung ...”, “, pg. 44. Note, asSchrödinger expressedS, all of the thermodynamic behavior is in the first term; the chemical constant, dependant on quantum assumptions, is in the second term.

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  32. Schrödinger note 14, “Gasentartung ...”, “, p. 44.

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  33. Schrödinger credited this argument toEhrenfest & Trkal note 12, p. 626.

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  34. The idea of treating conduction electrons as an ideal gas dated to the turn of the century. SeeSommerfeld, note 38, for details.Sommerfeld points out that by 1905Lorentz had done a calculation which underlined the difficulties of the method: The necessity of arbitrary assumptions about the temperature-dependence of electron density in metals, the fact that the observed contribution to the specific heat of a metal was less than that predicted by the equipartition theorem, and the failing that the theory could not handle contact potentials. Thus in the twenty years before 1924 this treatment had fallen out of favor. SeeSommerfeld, note 38,A. Sommerfeld gives a detailed historical and critical discussion of the problem of the theory of electrons in metals, in the subsequent “Zur Elektronentheorie der Metalle”,Die Naturwissenschaften 15 (1927), pp. 825–832.

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  35. Schrödinger, note 14, “Gasentartung ...”, “, p. 45.

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  36. Einstein, note 3A. Einstein, “Quantentheorie des einatomigen idealen Gases,”Berliner Berichte (1924), especially Part II, p. 12.

  37. W. Pauli, “Über Gasentartung und Paramagnetismus”,Z. f. Phys. 41 (1927), pp. 81–102. Also,A. Sommerfeld gives a detailed historical and critical discussion of the problem of the theory of electrons in metals, in the subsequent “Zur Elektronentheorie der Metalle”,Die Naturwissenschaften 15 (1927), pp. 825–832.

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  38. W. T. Scott,Erwin Schrödinger An Introduction to His Writings, (Amherst: University of Massachusetts Press, 1967), p. 24.

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  39. Schrödinger, “Bemerkungen über die statistische Entropiedefinition beim idealen Gas”,Berliner Berichte (1925), pp. 434–441.

  40. S. Bose, “Plancks Gesetz und Lichtquantenhypothese”,Z. f. Phys. 26 (1924), pp. 178–181.

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  41. M. J. Klein, note 6., p. 27. This wasBose's only important contribution to physics, according toJ. Mehra's obituary “Satyendra Nath Bose”,Biographical Memoirs of Fellows of the Royal Society of London 21 (1975), pp. 116–154. See alsoWillaim A. Blanpied, “Satyendranath Bose: Co-Founder of Quantum Statistics,”Am. J. Phys. 40 (1972), pp. 1212–1220, for an interview and another evaluation ofBose's work.

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  42. Jammer (Conceptual Development of Quantum Mechanics (New York: McGraw-Hill, 1966) p. 26) says thatPlanck's reasoning in 1900 was inconsistent, asEinstein had shown in 1906 (Ann. d. Phys. 20, pp. 199–206), because “...in the electrodynamical part (1) [the] formula is based on Maxwell's theory ... and the assumption that the oscillator energy is a continuously variable quantity, whereas in the statistical part (2) this same energy is treated as a discrete quantity, capable of assuming only values which are multiples ofh v.” Subsequent attempts contained similar difficulties. See alsoKlein, note 6,Martin J. Klein, “Einstein and the Wave-Particle Duality”,The Natural Philosopher 3 (1964), p. 27.

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  43. Einstein, note 3,A. Einstein, “Quantentheorie des einatomigen idealen Gases,”Berliner Berichte (1924), pp. 261–267.

  44. Einstein, note 3.A. Einstein, “Quantentheorie des einatomigen idealen Gases,”Berliner Berichte (1924) pp. 261–267. In the classical limit he got theSackur-Tetrode equation which didnot satisfyNernst's theorem. But hisgeneral expression for entropy did go to zero asT goes to zero.

  45. Klein, note 6, pp. 33–34.

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  46. Einstein, note 3,A. Einstein, “Quantentheorie des einatomigen idealen Gases,”Berliner Berichte (1924) second communication, p. 6.

  47. A. Einstein, “Zum Gegenwärtigen Stand des Strahlungsproblems”,Phys. Z. 10 (1909), pp. 185–193.

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  48. Einstein, note 48, pp. 189; note 3A. Einstein, “Quantentheorie des einatomigen idealen Gases”,Berliner Berichte (1924), second communication, p. 8.

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  49. Einstein, note 3A. Einstein, “Quantentheorie des einatomigen idealen Gases,”Berliner Berichte (1924), second communication, p. 9.

  50. Schrödinger toEinstein, April 23, 1926, inLetters on Wave Mechanics (London: Vision Press, 1967), edited byK. Przibram and translated byM. J. Klein, p. 26.

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  51. A. Einstein, “Zur Quantentheorie des idealen Gases”,Berliner Berichte (1925), pp. 18–25; submitted January 29, 1925.

  52. Einstein, note 52,A. Einstein, “Zur Quantentheorie des idealen Gases”,Berliner Berichte (1925), pp. 23–25.

  53. Einstein, note 52,A. Einstein, “Zur Quantentheorie des idealen Gases”,Berliner Berichte (1925), p. 18.

  54. Martin Klein discussed this correspondence in a lecture course in the History of Quantum Physics, delivered in the fall of 1971 at Yale University. See alsoKlein, note 6,, p. 31. For a later criticism of condensation, seeUhlenbeck, note 4.G. E. Uhlenbeck,Over Statistische Methoden in der Theorie der Quanta ('s-Gravenhage: Nijhoff, 1927). The letters areAlbert Einstein toPaul Ehrenfest, 29 September 1924; andEinstein toEhrenfest, 2 December 1924,Einstein Archive, Princeton, New Jersey.

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  55. M. Planck, “Zur Frage der Quantelung einatomiger Gase,”Sitz d. Preuss. Akad. d. Wiss. (1925), pp. 49–57; dated Feb. 5, 1925.Pauli, also, suggested (note 38, p. 81) that controversy surroundedEinstein's theory when he noted that it had been “much discussed” since its appearance two-and-one-half years before.

  56. Planck, note 57M. Planck, “Zur Frage der Quantelung einatomiger Gase,”Sitz d. Preuss. Akad. d. Wiss. (1925), p. 50.

  57. Planck, note 57M. Planck, “Zur Frage des Quantelung einatomiger Gase”,Sitz d. Preuss. Akad. d. Wiss. (1925), p. 50.

  58. Planck, note 57,M. Planck, “Zur Frage der Quantelung einatomiger Gase,”Sitz d. Preuss. Akad. d. Wiss. (1925), p. 56.

  59. Planck, note 57,M. Planck, “Zur Frage der Quantelung einatomiger Gase,”Sitz d. Preuss. Akad. d. Wiss. (1925), p. 57.

  60. Schrödinger, note 40, “Bemerkungen über die statistische Entropiedefinition beim idealen Gas”,Berlinear Berichte (1925), pp. 434–441, andM. Planck “Über die statische, Entropiedefinition”,Berliner Berichte (1925), pp. 442–451; submitted July 23, 1925.

  61. It is a repeated disappointment that few letters dated before January 1926 remain, which might document exchanges betweenSchrödinger and other notable physicists. One letter betweenPlanck andSchrödinger is cited below (note 98)Schrödinger wrote toPlanck on February 26, 1926 (AHQP Microfilm 34, Section 2), congratulatingPlanck on his “clear, beautiful crowning” of the development of the expression for entropy.Schrödinger said that he was in complete agreement with it. However, he could not agree withPlanck's assertion that the definition was “entirely independent of any considerations of probability and of the arbitrariness associated with postulating equally probable states.” One must make additional, more specific assumptions. As soon as one states, for example, that the only rational ascription of weights of quantum states is “by the count of Zeeman components”, the arbitrariness has been removed no matter what definition of entropy has been used. In practice, such difficulties always cropped up, because “the system is not governed quantum-theoretically”.Schrödinger seemed to be saying thatPlanck's definitions were correct, but that in practice they were too general. One neeeed additional rules for weighing quantum states, rules which were prescribed byBose-Einstein counting methods and bySchrödinger's alternate method, which had not yet appeared in print.Planck commented no further, either in print or in correspondence that I have found. But for the most part letters have been of little help in this study.

  62. Einstein's theory appeared in print only six months before.

  63. Schrödinger, note 40, “Bemerkungen über die statistische Entropiedefinition beim idealen Gas”,Berliner Berichte (1925), pp. 434–437.

  64. Schrödinger, note 40, “Bemerkungen über die statistische Entropiedefinition beim idealen Gas”,Berliner Berichte (1925), p. 437.

  65. Schrödinger cited a work bySchidlof which appeared afterEinstein's but which did not include “such far-reaching conclusions for gas degeneracy”. SeeA. Schidlof,Archives de sciences physiques et naturelles (Genève)6 (1924), p. 281, 381.

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  66. Schrödinger, note 40, “Bemerkungen über die statistische Entropiedefinition beim idealen Gas”,Berliner Berichte (1925), p. 437.

  67. Schrödinger, note 40, “Bemerkungen über die statistische Entropiedefinition beim idealen Gas”,Berliner Berichte (1925), p. 440.

  68. Schrödinger, note 40, “Bemerkungen über die statistische Entropiedefinition beim idealen Gas”,Berliner Berichte (1925), p. 440–441.

  69. See, for example,F. Reif,Fundamentals of Statistical and Thermal Physics (New York, McGraw-Hill, 1965), pp. 244–245.

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  70. Planck, note 62,M. Planck, “Über die statistische Entropiedefinition”,Berliner Berichte (1925), p. 442.

  71. Planck, note 62,M. Planck, “Über die statistische Entropiedefinition”,Berliner Berichte (1925), pp. 442–443.

  72. Planck, note 62,M. Planck, “Über die statistische Entropiedefinition”,Berliner Berichte (1925), p. 451.

  73. Schrödinger published another note, which was sent after he submitted his “last” paper on quantum statistics of the ideal gas (“On the Einstein Gas Theory”,note 5). The note, “Die Energiestufen des idealen einatomigen Gasmodells”,Berliner Berichte (1926), pp. 23–26, developed a suggestion ofEinstein's, that one could quantize the entire gas usingPlanck's counting method.Schrödinger, followingEinstein, stated the major requirement of this treatment: “... the allowed energy levelsU n of the gas-body must follow one another in an interval which, measured in phase-space, is constant and equal toN!h 3N” (p. 23.) The paper was essentially stillborn, sinceSchrödinger realized that it developed a theory of “ideal degeneracy” — of dubious value because the effects of cohesion and self-volume of the molecules gain importance especially in the degenerate state. (p. 35.) Nevertheless,Pauli (note 38,W. Pauli, “Über Gasentartung und Paramagnetismus”,Z. f. Phys. 41 (1927), p. 84) foundSchrödinger's results of some interest because they suggested that one sees different degenerate behavior if one includes or excludes a zero-point energy.

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  74. Schrödinger, note 5, p. 95.

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  75. Schrödinger, note 5, p. 96.

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  76. A. Landé, “Lichtquanten und Kohärenz”,Z. f. Phys. 33 (1925), pp. 571–578; especially p. 571.

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  77. Landé, note 79,, p. 571.

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  78. Schrödinger had written toLandé on November 16, 1925. “On the Einstein Gas Theory” was submitted for publication a month later (15 December), while the mid-November date marks whenSchrödinger began to analyzede Broglie's thesis. SeeSchrödinger toLandé, AHQP, Microfilm 4, Section 20, for documentation ofSchrödinger's approval ofLandé's approach.

  79. Schrödinger, note 5, p. 95. Thede Broglie view was by no means universally accepted in December 1925 whenSchrödinger wrote this paper.Einstein had influencedSchrödinger to take it seriously, andSchrödinger acknowledged the debt in April 1926: “Besides, the whole thing [wave mechanics] would certainly not have originated yet, and perhaps never would have, (I mean not from me), if I had not had the importance of de Broglieś ideas really brought home to me by your second paper on gas degeneracy.” Reference note 51.Schrödinger toEinstein, April 23, 1926, inLetters on Wave Mechanics (London: Vision Press, 1967), edited byK. Przibram and translated byM. J. Klein, p. 26. For a short discussion of the unproven status ofde Broglie's thesis, seeKlein, note 6,Martin J. Klein, “Einstein and the Wave-Particle Duality”,The Natural Philosopher 3 (1964), pp. 31–33.

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  80. Schrödinger, note 5,, p. 96.

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  81. C. G. Darwin &R. H. Fowler, “On the Partition of Energy”,Philosophical Magazine 44 (1922), pp. 450–479 and 823–842; andR. H. Fowler, “Dissociation-equilibrium by the Method of Partitions”,Phil. Mag. 45 (1923), pp. 1–33.

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  82. See, for example,Schrödinger,Statistical Thermodynamics (Cambridge, England: Cambridge, 1946), second edition, pp. 45–46.

  83. SeeEinstein, note 3,A. Einstein, “Quantentheorie des einatomigen idealen Gases”,Berliner Berichte (1924), first communication, pp. 263–265. Compare toSchrödinger, note 5,E. Schrödinger, “Zur Einsteinschen Gastheorie,”Phys. Z. 27 (1926), p. 97. InSchrödinger's expression for Ψ (equation (7)) an additional term of lnr was negligible compared tonlnr.Schrödinger dropped it when he established that his result agreed withEinstein's. In ideal gases at high temperatures and low densities,A is the ratio of the cube of the “thermal wavelength” of a particle to the average volume in which it moves. ThusA is a rough measure of the interaction of particles. This is generally true in the low temperature regime also, though there the situation is more complicated. For details seeKerson Huang,Statistical Mechanics (N.Y.: Wiley, 1963), pp. 196–197, 200–201. ForEinstein in 1924,A was a constant whose physical significance lay in its measure of the onset of degeneracy as temperature goes to zero and density goes to infinity.

  84. Louis de Broglie, “Recherches sur la theorie des quanta,”Ann. de Phys. 3 [Series 10] (1925), pp. 22–128, esp. p. 34, 39.

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  85. Schrödinger, note 5,, p. 98.

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  86. Schrödinger, note 85, See, for example,Schrödinger,Statistical Thermodynamics (Cambridge, England: Cambridge, 1946), second edition, pp. 76–79.

  87. Schrödinger, note 5, p. 99.

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  88. Schrödinger, note 5,, p. 100, compared toEinstein, note 3, second communication, p. 8. It is easy to surmise whySchrödinger (followingPlanck, of course) wanted to use Ψ instead ofZ in the calculation. Since Ψ was a sum, the last two terms could be dropped in partial differentiation [cf. equation (7)], while differentiating theproducts inZ necessitated carrying constant factors throughout the entire calculation. In this case the effort saved is trivial, indeed non-existent, since he made an error; but dealing with a more complicated expression might justify the use of Ψ.

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  89. P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,”Ann. d. Phys. 30 (1909), pp. 755–776, andM. von Laue, “Die Freiheitsgrade von Strahlenbündeln”,Ann. d. Phys. 44 (1914), pp. 1197–1212.

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  90. Schrödinger, note 5,, p. 101.

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  91. Schrödinger, note 5,, p. 101.

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  92. Schrödinger, note 5,, p. 101.

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  93. Pauli, note 38,, pp. 81–82.Arnold Sommerfeld made a more general comment in recognition ofSchrödinger's work. He claimed thatEinstein and laterSchrödinger pointed out that “the new statistics correspond to the point of view of wave mechanics, in which each quantum state is replaced by an eigenfunction of the space considered.” (Sommerfeld, not 38,A. Sommerfeld gives a detailed historical and critical discussion of the problem of the theory of electrons in metals, in the subsequent. “Zur Elektronentheorie der Metalle”,Die Naturwissenschaften 15 (1927), pp. 825.)

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  94. M. Planck, “Eine neue statistische Definition der Entropie”,Z. f. Phys. 35 (1926), pp. 155–169. Submitted 30 October 1925.

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  95. Schrödinger wrote toPlanck on February 26, 1926 (AHQP Microfilm 34, Section 2), congratulatingPlanck on his “clear, beautiful crowning” of the development of the expression for entropy.Schrödinger said that he was in complete agreement with it. However, he could not agree withPlanck's assertion that the definition was “entirely independent of any considerations of probability and of the arbitrariness associated with postulating equally probable states.” One must make additional, more specific assumptions. As soon as one states, for example, that the only rational ascription of weights of quantum states is “by the count of Zeeman components”, the arbitrariness has been removed no matter what definition of entropy has been used. In practice, such difficulties always cropped up, because “the system is not governed quantum-theoretically”.Schrödinger seemed to be saying thatPlanck's definitions were correct, but that in practice they were too general. One needed additional rules for weighing quantum states, rules which were prescribed byBose-Einstein counting methods and bySchrödinger's alternate method, which had not yet appeared in print.Planck commented no further, either in print or in correspondence that I have found.

  96. Schrödinger, note 5,, p. 95.

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Hanle, P.A. The coming of age of Erwin Schrödinger: His quantum statistics of ideal gases. Arch. Hist. Exact Sci. 17, 165–192 (1977). https://doi.org/10.1007/BF02464979

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