CONTEMPORARY ADVANCES IN PHYSICS 695 



article, we see that Schroedinger calculated the energy-values of the 

 Stationary States by conditions imposed upon the wave-lengths of the 

 waves, not upon their frequencies; the wave-patterns depend only 

 upon the wave-lengths, and the frequency of the light which an atom 

 emits in passing between two Stationary States depends only on the 

 difference between their energy-values. In relativistic mechanics, 

 energy is defined absolutely, and this difificulty never even threatens 

 to arise ; yet it is worth while to note that the ambiguity of the concept 

 "energy" in classical mechanics does not interfere with, nor is it re- 

 solved by, anything which has been observed in Nature and interpreted 

 by wave-mechanics. 



In relativistic mechanics, the wave-equation for the free-fiying 

 particle assumes the form: 



-r^ + TTT, (£ - ^?oV) = \ E = , . (197) 



dx'~ ^ ¥c' \ Vl - v'/c' 



The wave-length has the value /Wl — v'^c^/mov = h/mv; the frequency is 

 mocV/Wl — v'^/c'^; the speed of propagation of the waves is c-fv, 

 superior to the speed of light. 



I can no more than allude to the strangely suggestive fact, that in 

 general as well as in this special case the speed of the particle and the 

 speed of the associated waves are related to one another in the same 

 way as group-speed and wave-speed in ordinary optics. 



Attempt to Find a Meaning for the Symbol ^ 



Thirty-three years ago the Earl of Salisbury, invited by reason of his 

 eminence as a statesman to be the President of the British Association 

 for the Advancement of Science, observed the physicists of his day 

 involved in" their fervent speculations over the nature of the aether; 

 and of an address brilliant with felicitous phrases the best-remembered 

 words are those by which he described the outcome of their travail: 

 The main, if not the only, function of the word aether has been to furnish a 

 nominative case to the verb 'to undulate.' Quite the same thing could 

 now be said of the symbol ^, insofar as it serves to determine the 

 energy-values of the Stationary States of the systems devised to repre- 

 sent atoms. When it is used for this purpose, it vanishes just as the 

 final triumph is achieved. Like the variable under the sign of integra- 

 tion in a definite integral, it drops out of sight when the calculations 

 which it proposes are actually performed. Indeed it might be dis- 

 carded altogether during the process of calculating Eigenwerte and 

 energy-values; one might speak exclusively of the "differential opera- 

 tor" V2 — 8Tr~m{E — V)lh-; many mathematicians do so. 

 45 



