384 BELL SYSTEM TECHNICAL JOURNAL 



This procedure is right in the old statistics, becomes wrong in the new. I 

 suppose that this is what some expositors mean when they say that in the 

 new statistics there is a correlation between positions and momenta, or 

 words to that effect. I say that the old statistics apparently furnished a 

 term equal to the fourth of (66) except for the power to which e is raised. 

 Actually the old statistics gives an additive term Nk In [{2Trmkef'^/Q\ and 

 the new statistics gives an additive term A''^ In [{2Trnik)^''e''^ /Q], but Q in the 

 former case is the volume of the region and in the latter case is the volume 

 of the cell. Giving the same value h^ to Q in the two cases is positively not 

 doing the same thing. However by doing this notwithstanding, and by 

 "tampering" with the old statistics in a certain way which I described at the 

 end of the previous article, it is possible to produce an expression exactly 

 like (66). 



Size of the Elementary Cell 



We have reached the final step, which consists in assigning a value to the 

 size of the elementary cell in /z-space. For this I have used the symbol h^, 

 implying (as everyone has guessed already) that it is taken to be the cube of 

 Planck's constant h so promiscuously found in Nature. What arguments 

 can be advanced to justify this choice? 



It may be remarked very simply, that since the volume of the elementary 

 cell has the "dimensions" of the cube of the product of length by momentum, 

 and since these are also the dimensions of h , and since both Ji and that 

 volume are very fundamental things, what could be more natural than to 

 identify them the one with the other? This was the argument used when 

 formula (66) was first derived from the old statistics with the aid of judicious 

 tampering. 



An argument more precise of aspect may be adduced from wave-me- 

 chanics. Imagine the box containing the gas to be a cube, its edges — these 

 being of length L so that Lz = V — being along the coordinate-axes x, y, z. 

 The doctrine of wave-mechanics avers that the momentum-components 

 px, py, pz of any atom are perforce integer multiples of h/2L; for this is the 

 condition that the waves which are associated with the atom shall form a 

 stationary wave-pattern with nodes at the walls of the cube, and upon this 

 condition wave-mechanics is insistent. Now let us reenter the momentum- 

 space, and place a dot at every point for which px, py and pz are integer 

 multiples of h/2L. The dots form a cubic lattice, and it would seem very 



^ The wave-length of the waves associated with a particle moving parallel to the x-axis 

 is h/px, and there must be an integer number of half-wave-lengths between the walls of 

 the cube which are perpendicular to the axis of x and face one another at a distance L. 

 The same may be said, mtitatis mutandis, of a particle moving parallel to the axis of y or z, 

 with momentum py or pz] while if an atom is moving obhquely so that two or all three of 

 its momentum-components differ from zero, each of these components is to be treated as 

 if it alone existed. 



