Appendix II

References

Among general papers treating of the application of the theory of quanta to different parts of physics are:

1. A. Sommerfeld, Das Planck’sche Wirkungsquantum und seine allgemeine Bedeutung f¨ur die Molekularphysik, Phys. Zeitschr., 12, p. 1057. Report to the Versammlung Deutscher Naturforscher und Aerzte. Deals especially with applications to the theory of specific heats and to the photoelectric effect. Numerous references are quoted.

2. Meeting of the British Association, Sept., 1913. See Nature, 92, p. 305, Nov. 6, 1913, and Phys. Zeitschr., 14, p. 1297. Among the principal speakers were J. H. Jeans and H. A. Lorentz.

(Also American Phys. Soc., Chicago Meeting, 1913.⁶⁹)

3. R. A. Millikan, Atomic Theories of Radiation, Science, 37, p. 119, Jan. 24, 1913. A non-mathematical discussion.

4. W. Wien, Neuere Probleme der Theoretischen Physik, 1913. (Wien’s Columbia Lectures, in German.) This is perhaps the most complete review of the entire theory of quanta.

H. A. Lorentz, Alte und Neue Probleme der Physik, Phys. Zeitschr., 11, p. 1234. Address to the Versammlung Deutscher Naturforscher und Aerzte, K¨onigsberg, 1910, contains also some discussion of the theory of quanta.

Among the papers on radiation are:

E. Bauer, Sur la théorie du rayonnement, Comptes Rendus, 153, p. 1466. Adheres to the quantum theory in the original form, namely, that emission and absorption both take place in a discontinuous manner.

E. Buckingham, Calculation of $c_2$ in Planck’s equation, Bull. Bur. Stand. 7, p. 393.

E. Buckingham, On Wien’s Displacement Law, Bull. Bur. Stand. 8, p. 543. Contains a very simple and clear proof of the displacement law.

P. Ehrenfest, Strahlungshypothesen, Ann. d. Phys. 36, p. 91.

A. Joffé, Theorie der Strahlung, Ann. d. Phys. 36, p. 534.

Discussions of the method of derivation of the radiation formula are given in many papers on the subject. In addition to those quoted elsewhere may be mentioned:

C. Benedicks, Ueber die Herleitung von Planck’s Energieverteilungsgesetz Ann. d. Phys. 42, p. 133. Derives Planck’s law without the help of the quantum theory. The law of equipartition of energy is avoided by the assumption that solids are not always monatomic, but that, with decreasing temperature, the atoms form atomic complexes, thus changing the number of degrees of freedom. The equipartition principle applies only to the free atoms.

P. Debye, Planck’s Strahlungsformel, Ann. d. Phys. 33, p. 1427. This method is fully discussed by Wien (see 4, above). It somewhat resembles Jeans’ method (Sec. 169) since it avoids all reference to resonators of any particular kind and merely establishes the most probable energy distribution. It differs, however, from Jeans’ method by the assumption of discrete energy quanta $h\nu$. The physical nature of these units is not discussed at all and it is also left undecided whether it is a property of matter or of the ether or perhaps a property of the energy exchange between matter and the ether that causes their existence. (Compare also some remarks of Lorentz in 2.)

P. Frank, Zur Ableitung der Planckschen Strahlungsformel, Phys. Zeitschr., 13, p. 506.

L. Natanson, Statistische Theorie der Strahlung, Phys. Zeitschr., 12, p. 659.

W. Nernst, Zur Theorie der specifischen W¨arme und ¨uber die Anwendung der Lehre von den Energiequanten auf physikalisch-chemische Fragen ¨uberhaupt, Zeitschr. f. Elektrochemie, 17, p. 265.

The experimental facts on which the recent theories of specific heat (quantum theories) rely, were discovered by W. Nernst and his fellow workers. The results are published in a large number of papers that have appeared in different periodicals. See, e.g., W. Nernst, Der Energieinhalt fester Substanzen, Ann. d. Phys. 36, p. 395, where also numerous other papers are quoted. (See also references given in 1.) These experimental facts give very strong support to the heat theorem of Nernst (Sec. 120), according to which the entropy approaches a definite limit (perhaps the value zero, see Planck’s Thermodynamics, 3. ed., sec. 282, et seq.) at the absolute zero of temperature, and which is consistent with the quantum theory. This work is in close connection with the recent attempts to develop an equation of state applicable to the solid state of matter. In addition to the papers by Nernst and his school there may be mentioned:

K. Eisenmann, Canonische Zustandsgleichung einatomiger fester K¨orper und die Quantentheorie, Verhandlungen der Deutschen Physikalischen Gesellschaft, 14, p. 769.

W. H. Keesom, Entropy and the Equation of State, Konink. Akad. Wetensch. Amsterdam Proc., 15, p. 240.

L. Natanson, Energy Content of Bodies, Acad. Science Cracovie Bull. Ser. A, p. 95. In Einstein’s theory of specific heats (Sec. 140) the atoms of actual bodies in nature are apparently identified with the ideal resonators of Planck. In this paper it is pointed out that this is implying too special features for the atoms of real bodies, and also, that such far-reaching specializations do not seem necessary for deriving the laws of specific heat from the quantum theory.

L. S. Ornstein, Statistical Theory of the Solid State, Konink. Akad. Wetensch. Amsterdam Proc., 14, p. 983.

S. Ratnowsky, Die Zustandsgleichung einatomiger fester K¨orper und die Quantentheorie, Ann. d. Phys. 38, p. 637.

Among papers on the law of equipartition of energy (Sec. 169) are:

J. H. Jeans, Planck’s Radiation Theory and Non-Newtonian Mechanics, Phil. Mag., 20, p. 943.

S. B. McLaren, Partition of Energy between Matter and Radiation, Phil. Mag., 21, p. 15.

S. B. McLaren, Complete Radiation, Phil. Mag., 23, p. 513. This paper and the one of Jeans deal with the fact that from Newtonian Mechanics (Hamilton’s Principle) the equipartition principle necessarily follows, and that hence either Planck’s law or the fundamental principles of mechanics need a modification.

For the law of equipartition compare also the discussion at the meeting of the British Association (see 2).

In many of the papers cited so far deductions from the quantum theory are compared with experimental facts. This is also done by:

F. Haber, Absorptionsspectra fester K¨orper und die Quantentheorie, Verhandlungen der Deutschen Physikalischen Gesellschaft, 13, p. 1117.

J. Franck und G. Hertz, Quantumhypothese und Ionisation, Ibid., 13, p. 967.

Attempts of giving a concrete physical idea of Planck’s constant $h$ are made by:

A. Schidlof , Zur Aufkl¨arung der universellen electrodynamischen Bedeutung der Planckschen Strahlungsconstanten $h$, Ann. d. Phys. 35, p. 96.

D. A. Goldhammer, Ueber die Lichtquantenhypothese, Phys. Zeitschr., 13, p. 535.

J. J. Thomson, On the Structure of the Atom, Phil. Mag., 26, p. 792.

N. Bohr, On the Constitution of the Atom, Phil. Mag., 26, p. 1.

S. B. McLaren, The Magneton and Planck’s Universal Constant, Phil. Mag., 26, p. 800.

The line of reasoning may be briefly stated thus: Find some quantity of the same dimension as $h$, and then construct a model of an atom where this property plays an important part and can be made, by a simple hypothesis, to vary by finite amounts instead of continuously. The simplest of these is Bohr’s, where $h$ is interpreted as angular momentum.

The logical reason for the quantum theory is found in the fact that the Rayleigh-Jeans radiation formula does not agree with experiment. Formerly Jeans attempted to reconcile theory and experiment by the assumption that the equilibrium of radiation and a black body observed and agreeing with Planck’s law rather than his own, was only apparent, and that the true state of equilibrium which really corresponds to his law and the equipartition of energy among all variables, is so slowly reached that it is never actually observed. This standpoint, which was strongly objected to by authorities on the experimental side of the question (see, e.g., E. Pringsheim in 2), he has recently abandoned. H. Poincaré, in a profound mathematical investigation (H. Poincaré, Sur la Théorie des Quanta, Journal de Physique (5), 2, p. 1, 1912) reached the conclusion that whatever the law of radiation may be, it must always, if the total radiation is assumed as finite, lead to a function presenting similar discontinuities as the one obtained from the hypothesis of quanta.

While most authorities have accepted the quantum theory for good (see J. H. Jeans and H. A. Lorentz in 2), a few still entertain doubts as to the general validity of Poincaré’s conclusion (see above C. Benedicks and R. A. Millikan 3). Others still reject the quantum theory on account of the fact that the experimental evidence in favor of Planck’s law is not absolutely conclusive (see R. A. Millikan 3); among these is A. E. H. Love (2), who suggests that Korn’s (A. Korn, Neue Mechanische Vorstellungen ¨uber die schwarze Strahlung und eine sich aus denselben ergebende Modification des Planckschen Verteilungsgesetzes, Phys. Zeitschr., 14, p. 632) radiation formula fits the facts as well as that of Planck.

H. A. Callendar, Note on Radiation and Specific Heat, Phil. Mag., 26, p. 787, has also suggested a radiation formula that fits the data well. Both Korn’s and Callendar’s formulæ conform to Wien’s displacement law and degenerate for large values of $\lambda T$ into the Rayleigh-Jeans, and for small values of $\lambda T$ into Wien’s radiation law. Whether Planck’s law or one of these is the correct law, and whether, if either of the others should prove to be right, it would eliminate the necessity of the adoption of the quantum theory, are questions as yet undecided. Both Korn and Callendar have promised in their papers to follow them by further ones.