[ 500 ] 



XL VIII. Adiabatic Invariants and the Theory of Quanta. 

 By P. Ehrenfest *. 



Contents : — (1) Definition of a reversible adiabatic transformation of a 

 mechanical system. Systems which are adiabatically related to 

 each other. — (2) Formulation of an hypothesis on adiabatic trans- 

 formations for systems with periodic or quasi-periodic motions. — 

 (3) Adiabatic Invariants and their use. — (4) The adiabatic invariant 



2T e 



— for periodic motions, and - especially for harmonic motions. — 



(5) Geometrical interpretation of the adiabatic invariant _L in the 



V 



phase-space. — (6) Connexion with the formulae of the Theory of 

 Quanta, as proposed by Planck, Debye and others for systems of one 

 degree of freedom. — (7) Connexion with Sommerfeld's formulae 

 for systems of more degrees of freedom. — (8) Connexion with 

 the statistical roots of the Second Law of Thermodynamics. — 

 (9) Difficulties which arise by a passage through singular motions. 

 Aperiodical motions. — (10) Conclusion. 



Introduction. 



In the treatment of a continually increasing number of 

 physical problems, use is at the same time made o£ the 

 principles of classical mechanics and electrodynamics, and 

 of the hypothesis of the quanta, which is in conflict with 

 them. Through the study of these problems it is hoped to 

 arrive at some general point of view which may trace the 

 boundary between the "classical region " and the "region 

 of the quanta." 



One fundamental law stands amidst the theory of quanta, 

 which is wholly derived from classical foundations : the 

 Displacement Laiv of W. Wien on the change of the distri- 

 bution of energy over the spectrum involved by a reversible 

 adiabatic compression of radiation. This fact deserves our 

 attention. It might be possible that also in more general 

 cases, when we do not restrict ourselves to harmonic motions, 

 the reversible adiabatic transformations should be treated in 

 a classical way, whereas in the calculation of other pro- 

 cesses (e. g. an isothermal addition of heat) the quanta 

 come into play. 



From this point of view I started in some papers in which 

 on the one hand I studied Planck's hypothesis of energy 

 elements f, and on the other tried to extend this hypothesis 



* Communicated by the Author. Abridged translation of a paper 

 published in the Proc. Acad, of Amsterdam, xxv. (1916) p. 412. 



f P. Ehrenfest, Ann. d. Phys. vol. xxxvi. (1911) pp. 91-118 (quoted as 

 paper A). 



