14 



position and shape of this element and siniplj' proportional to its volume, defined 

 in the usual way by 



dW =\dq, ... dqsdp^ ... dps. (11) 



In Ihe quantum theory, however, these considerations cannot be directly applied, 

 since the point representing the state of a system cannot be displaced continuously 

 in the 2s-dimensional phase-space, but can lie only on certain surfaces of lower 

 dimensions in this space. For systems of one degree of freedom the phase-space is 

 a two-dimensional surface, and the points representing the states of some system 

 given by (10) will be situated on closed curves on this surface. Now, in general, the 

 motion will differ considerably for any two states corresponding to successive entire 

 values of n in (10), and a simple general connection between the quantum theory 

 and ordinary statistical mechanics is therefore out of question. In the limit, how- 

 ever, where n is large, the motions in successive states will only differ very little 

 from each other, and it would therefore make little difference whether the points 

 representing the sj'stems are distributed continuously on the phase-surface or situated 

 only on the curves corresponding to (10). provided the number of systems which 

 in the first case are situated between two such curves is equal to the number which 

 in the second case lies on one of these curves. But it will be seen that this condi- 

 tion is just fulfilled in consequence of the above hypothesis of equal a-priori pro- 

 babilitj' of the different stationary states, because the element of phase-surface 

 limited by two successive curves corresponding to (10) is equal to 



âW =^dpdq = \pdq - \pdq] = I^ — h-x = h, (12) 



so that on ordinary statistical mechanics the probabilities for the point to lie within 

 any two such elements is the same. We see consequently that the hypothesis of 

 equal probability of the different states given by (10) gives the same result as 

 ordinary statistical mechanics in all such applications in which the states of the 

 great majority of the systems correspond to large values of n. Considerations of 

 this kind have led Debye^) to point out that condition (10) might have a general 

 validity for systems of one degree of freedom, already before Ehrenfest, on the 

 basis of his theory of the mechanical transformability of the stationary states, 

 had shown that this condition forms the only rational generalisation of Planxk's 

 condition (9). 



We shall now discuss the relation between the theory of spectra of atomic 

 systems of one degree of freedom, based on (1) and (10), and the ordinary 

 theory of radiation, and we shall see that this relation in several respects shows a 

 close analogy to the relation, just considered, between the statistical applications of 

 (10) and considerations based on ordinarv statistical mechanics. Since the values 



') P. Debye, Wolfskehl-Vortrag. Göttingen 1913\ 



