696 BELL SYSTEM TECHNICAL JOURNAL 



Schroedinger however conceived the daring, the admittedly tenta- 

 tive and still incomplete but very alluring, idea of seeking in ^ some 

 measure of the density of electric charge. Specifically, it occurred 

 to him to define the square of the amplitude of the oscillations of ^, 

 which the Eigenfunktionen prescribe as function of the coordinates— 

 to define this squared amplitude as the density of the electric charge, 

 spreading the electron as it were throughout space. 



Let us examine this idea, and see whither it leads. 



To avoid avoidable complexities as far as possible, I take the simplest 

 of all cases : the linear oscillator, represented by the imaginary "string" 

 stretched along the x-axis, possessed of a wave-speed varying as 

 Vl — x'^jU, real from the origin both ways as far as the points 

 X = ± L and imaginary thenceforward. I will also refer to the still 

 simpler "actual" case which served as an introduction to this one: 

 the problem of the stretched string, clamped at its extremities at 

 X = ± L, possessed of a uniform real wave-speed u at all points 

 between. 



In both these cases of the imaginary and the real string, the search 

 for the Eigenwerte and the Eigenfunktionen leads us to diverse natural 

 modes of vibration, executed with various frequencies I'o, vi, V2, n ' ■ - 

 and displaying stationary wave-patterns described by the Eigenfunk- 

 tionen: 



yi= fi{x)iAi cos iTTPit -\- Bi sin Iwi^it); i = 0,1,2,3 ••• (201) 



For the real string the functions fi(x) are sine-functions; for the 

 imaginary strings which are the model of the linear oscillator, they are 

 given by (155). I recall once more that in the latter case we have, not 

 distinct modes of oscillation of one string, but the fundamental modes of 

 as many strings as there are Stationary States. 



When the real string is vibrating in the ith mode, or when we are 

 dealing with the ith imaginary string, the function fi(x) is proportional 

 to its vibration-amplitude. The form of equation (201) shows that 

 this amplitude at any fixed point remains constant in time. 



If the square of the vibration-amplitude is to be regarded as the 

 density of electric charge along the string, it follows that when the 

 oscillator is in one of its stationary states, and the string therefore 

 vibrating in one of its modes, the density and the distribution of charge 

 remain everywhere constant. There would be no to-and-fro motion 

 of charges, and no tendency to radiation. 



Suppose now that the real string is vibrating simultaneously in two 

 modes, the ith and the jth; or that we have both the ith and the jth 

 imaginary string coexisting (this is where the model is clumsiest!). 



