Abstract
Einstein’s gas theory, and the short but infinitely far-seeing remarks that he suggested, form the link between De Broglie’s matter waves and Schrödinger’s wave mechanics.
Einstein’s gas theory, and the short but infinitely far-seeing remarks that he suggested, form the link between De Broglie’s matter waves and Schrödinger’s wave mechanics.
Martin J. Klein (1964), 46
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Notes
- 1.
In Seelig (2005), 316. Salaman would emigrate shortly afterwards to England, with a letter of recommendation from Einstein to introduce herself to Rutherford. Finally, she would end up devoting herself to the literature in that country.
- 2.
Among others, the manifesto was also signed by Russell, Gandhi and Tagore.
- 3.
Letter from S. Bose to A. Einstein, 4 June 1924. Reprinted in Kormos-Buchwald et al. (2015), 399.
- 4.
For more details about Bose’s academic and scientific life, as well as for the bibliography that justifies our statements in this section, see, for instance, Navarro (1996). In Spanish.
- 5.
Bose (1924 a), 181. This note is not included in the English translation cited in the bibliography.
- 6.
In France he was also in contact with the De Broglie brothers and in Germany, through Einstein, with Fritz Haber, Otto Hahn and Lise Meitner, among others.
- 7.
Interview quoted in Mehra; Rechenberg (1982 a, part B), 571.
- 8.
See, for example, Huang (1987), 278–283. Recall that in the grancanonical ensemble the variables characterizing the thermodynamic system are volume, temperature and chemical potential, which is zero in the case of a photon gas.
- 9.
See, for example, the letter from A. Einstein to M. Besso, 5 June 1925. In Nollar-James et al. (2018), 26–27.
- 10.
A typographical error: in Einstein’s original there is an extra logarithm before the summation.
- 11.
Among others, they had previously justified expressions similar to (4.35) for the entropy of the ideal monoatomic gas by the following: O. Sackur (1911), H. Tetrode (1912), W. H. Keesom (1914), W. Lenz (1915), A. Sommerfeld (1915), P. Scherrer (1916) and M. Planck (1916). See on this subject Mehra; Rechenberg (1982a, part B), 573–574.
- 12.
Einstein (1924). In Hentschel; James (2015), 282. Einstein does not justify the use of the term “degeneracy” to refer to a situation in which the description of the gas state departs from the classical one; and precisely the more so the higher the value of the parameter λ. Although it was a term sometimes used to refer to low temperatures, from now on we will associate it to situations in which the quantum effect is remarkable.
- 13.
Einstein (1924). In Hentschel; James (2015), 283. The situation described here is sometimes referred to as “Einstein’s paradox” and “Einstein’s mixture paradox”. This is how it appears for the first time in Ehrenfest; Uhlenbeck (1927).
- 14.
We would like to state for the record that the discussions with Enric Pérez Canals and Pere Seglar Comas, with their accurate comments on the true meaning of the paradoxical situation presented by Einstein, have been extremely useful to us when writing the following comments on the issue.
- 15.
Letter from A. Einstein to P. Ehrenfest, 2 December 1924. In Hentschel; Nollar-James (2015), 384.
- 16.
Depending on the definition adopted for “additivity” it may turn out that, for usual systems, the extensivity of a thermodynamic quantity implies its additivity (the reciprocal is not true); this contributes to the terminological confusion, which may also indicate conceptual confusion.
- 17.
For more details and bibliography on the context of the discovery of the Bose–Einstein condensation, see Pais (2005), 432–434.
- 18.
This correlation can be intuitively represented by the “statistical potential”, also sometimes called the “Uhlenbeck-Gropper potential”, introduced by both of them in 1932. See, e.g., Huang (1987), 200–203.
- 19.
Let us recall: a “selection rule” usually limits the possible transitions between quantum states. A “superselection rule” prohibits determined systems from occurring in certain pure quantum states, which implies limiting the accessible region of the corresponding Hilbert space.
- 20.
Einstein (1925 a). In Hentschel; James (2015), 380. Einstein specifically quotes Nernst for his explanation, in 1919, of the dependence of the viscosity coefficient on the temperature, on the basis of the incipient quantum ideas of that time.
- 21.
The theory Einstein refers to is the “free electrons model” in metals, devised by Drude around 1900 and refined by Lorentz about five years later. It was able to explain a good part of the experimental results known at that time about electrical and heat conductivity of metals.
- 22.
Boltzmann (1877), 385. It is precisely in this work that Boltzmann expounds his “combinatorial method” for the analysis of equilibrium. In his first section he introduces the discrete calculus, based on “energy elements”, and then moves on to the continuous to deduce the classical distribution of velocities.
- 23.
See Pérez; Sauer (2010). This is the only detailed and rigorous paper we know of, devoted to analysing the motivation, content, impact, and circumstances surrounding the publication of Einstein’s third paper on the quantum theory of ideal gases.
- 24.
In Masoliver; Ros (2010) an updated and clear version of the line followed by Schrödinger —to derive the equation that today bears his name— from the analogy between optics and mechanics, is presented.
- 25.
Letter from E. Schrödinger to A. Einstein, 23 April 1926. In Nollar-James et al. (2018), 277.
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Navarro Veguillas, L. (2023). The Last Collection (1924–1925): Formulation of The First Quantum Statistics. In: The Lesser-Known Albert Einstein. History of Physics. Springer, Cham. https://doi.org/10.1007/978-3-031-35568-4_4
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