﻿452 Messrs. C. Gr. Darwin and R. H. Fowler on 



probable. It will suffice to recall that for assemblies obeying 

 the laws o£ classical mechanics the theorem of Liouville 

 shows that the elements of equal probability may be taken 

 to be equal elements of volume in Gibbs' " phase space." 

 It follows out of this, for example, in the case of an assembly 

 of a number of identical systems — say simple free mole- 

 cules — that the elements of equal probability can be simpli- 

 fied down into 6-dimensional cells dq l dq 2 dq z dp l dp 2 dp z of 

 equal extension, where q l9 q 2 , q 3 are the coordinates, and 

 Pi> Pi, ps the conjugated momenta, of a single molecule. 

 We shall describe this by saying that the weight of every 

 equal element dq A ... dp z is the same, and by a slight 

 generalization, that the weights of unequal cells are pro- 

 portional to their 6-dimensional extension. The word weight 

 is here used in exactly the sense of the term a priori 

 probability, as used by Bohr and others. 



But when we come to the quantum theory, mechanical 

 principles cease to hold, and we require a new basis for 

 assigning the equally probable elements. Such a basis is 

 provided by Ehrenfest' s * Adiabatic Hypothesis and Bohr's f 

 Correspondence Principle. These show how the theorem of 

 Lionville is to be extended, and allow us to assign a weight 

 for each quantized state of a system. It is found that we 

 must assign an equal weight to every permissible state in 

 each quantized degree of freedom. At first sight this is a 

 little surprising, for it would seem natural to suppose that a 

 vibrator which could only take energy in large units would 

 be less likely to have a unit than one which could take it in 

 small ; but this is to confuse the two stages of the problem. 

 It is only by the supposition of equal weights that we can 

 obtain consistency with classical mechanics by the Corre- 

 spondence Principle. It is customary % in assigning a definite 

 weight to every quantized state to give it the value h, so as to 

 bring the result to the same dimensions as those of the 

 element dq dp in the classical case. But there is considerable 

 advantage in reversing this, and taking the quantized weights 

 as unity and the weight of the element in the phase space as 

 dq dp/h ; for if this is done, the arguments about entropy are 

 simplified by the absence of logarithms of dimensional 

 quantities. We shall adopt this convention here, though in 



* Ehrenfest, Proc. Acad. Amst. xvi. p. 591 ; Phil. Maof. xxxiii. p. 500 

 (1917), etc. 



t Bohr, " The Quantum Theory of Line Spectra," Dan. Acad. iv. p. 1 

 (1918). 



% Ehrenfest & Trkal, Proc. Amst. Acad. Sc. xxiii. p. 162. See in par- 

 ticular p. 165 and Additional Notes, No. 1. 



