CONTEMPORARY ADVANCES IN PHYSICS 6cS3 



negative value to the parameter E, one would in general not be able 

 to find a solution which is finite both at the origin and at infinity. 

 This is the same situation as occurred in the theory of the linear 

 oscillator, where an arbitrary choice of a value for the parameter 

 there called C would in general have led to a solution implying infinite 

 amplitude at both ends of the "string." 



Schroedinger however discovered " that there is a series of Eigenwerte 

 for the parameter E, each of which (subject to a limitation to be 

 introduced below) entails a solution which is single-valued, continuous 

 and finite over the entire range of the variable r. 



These Eigenwerte are the following : 



En= - iTT-meyh'-n^; n = 1, 2, 3, 4, • • •. (177) 



llie consecutive permitted energy-values of the system of potential- 

 energy-function — e-Jr, the Stationary States of the model for the hydrogen 

 atom, are therefore specified by wave-mechanics as the quotients of the 

 fundamental factor — 2irme^!}i^ by the squares of the consecutive integers 

 from unity onward. 



These agree \\nt\\ experiment. The formula (177) is in fact the 

 renowned formula of Bohr, from w^hich the whole contemporary theory 

 of spectra sprang; a formula so successful that it is scarcely conceiv- 

 able that any alternative theory should ever win acceptance unless 

 by presenting the identical equation over again. 



Schroedinger's models for the hydrogen atom in its various Sta- 

 tionary States thus are imaginary fluids each pervading the whole of 

 space, and in each of which the wave-speed depends on the distance r 

 from a centre, according to a peculiar law — the law obtained by in- 

 serting into the formula (173) the appropriate value for E, chosen from 

 the sequence given in (177). If into (173) we were to put any value 

 chosen at random for the energy-constant E, we should be inventing 

 an imaginary fluid; but, in general, this fluid would not be capable of 

 sustaining a continuous stationary wave-pattern of finite amplitude. 

 Only when one of Bohr's sequence of energy-values is chosen do we get 

 a fluid able to resonate as a ball of actual physical substance can. 



The next task is to enquire into the wave-patterns in the imaginary 

 fluids corresponding to these various permitted energy -values. This is 

 much more difficult than the same problem for the imaginary strings 

 corresponding to the various permitted energy-values of the linear 

 oscillator, and the new complexities are not altogether due to the fact 

 that we now have three dimensions to deal with instead of one; they 



" Schroedinger, Ann. d. Phys., 79, pp. 361-376 (1926). For an alternative method 

 of proof see A. S. Eddington, Nature, 120, p. 117 (1927). 



