662 BELL SYSTEM TECHNICAL JOURNAL 



of the wave-motion partially but only partially described by (13), 

 and complete it by the assertion — not an inevitable nor a self-evident 

 assumption, but an original and daring hypothesis — that it is indeed a 

 wave-motion endowed with a frequency, and this the frequency 



V = E/h. (19) 



This manner of introducing into every mechanical system a 

 vibration-frequency linked with its energy by the vital quantum- 

 relation (19) was the invention of Louis de Broglie. 



The wave-equation to which this hypothesis leads us then is: 



V2^ + ^ (E - V)^ = 0. (20) 



This is a particular form of the wave-equation of de Broglie and 

 Schroedinger. It is the form which I will use throughout this article, 

 for it is adequate to the first steps in the processes of atom-design — 

 adequate, for instance, to supply a theory of the major features of the 

 spectrum of atomic hydrogen, though not of its fine-structure; ade- 

 quate also to interpret the data of the experiment of Davisson and 

 Germer, and sufiicient for an introduction to the ways of thinking 

 which constitute wave-mechanics. Nevertheless it is certainly not 

 the general wave-equation, for it is subject to at least two limitations. 



The first of these is, that equation (20) is based upon Newtonian, 

 not upon relativistic mechanics. We should therefore expect it to be 

 valid only for slow-moving particles, to be the limiting form of a 

 relativistic wave-equation appropriate to all velocities. Such an 

 equation, indeed, was the first propounded by de Broglie. The past 

 history of atomic theory suggests that we should need it when em- 

 barking upon the enterprise of explaining the fine-structure of the 

 hydrogen spectrum. The latest developments in that history, how- 

 ever, indicate that the mere replacement of equation (20) by its 

 relativistic analogue would not suffice for that enterprise; due allow- 

 ance must be made in addition for the "spin " of the electron.^ Wave- 

 mechanics being yet too young to have furnished an answer to this 

 twofold problem, the relativistic equation still wants what may in 

 the end turn out to be its main experimental support. Yet it can 

 scarcely be doubted that relativity must figure in the general wave- 

 equation. 



The second limitation upon equation (20) is due to its origin in 



^ For the application of the relativistic equation to the hydrogen atom without 

 allowance for the spinning electron, see V. Fock, Zs.f. Phys., 38, pp. 262-269 (1926). 

 See also the first footnote on p. 688. 



