680 BELL SYSTEM TECHNICAL JOURNAL 



According to Schroedinger the Stationary States of the linear oscillator 

 are distinguished by the energy-values which cause this equation to have 

 a solution finite at all values of the variable, infinity included. 



These are the values of the constant C which cause the parameter 

 C to take one of the Eigenwerte set down in (158). 



The energy-values of the Stationary States should therefore be 



£.= ^°(2«+l) 



1 3 5 ('*^^ 



= -xhvQ, -xhvo, -zhvo, • • •• 



The successive permitted energy-values of the linear simple-harmonic 

 oscillator of frequency vo, the energy-values of its consecutive Sta- 

 tionary States, are therefore specified by wave-mechanics as the 

 products of the fundamental factor hvo by the consecutive "half- 

 integers" 1/2, 3/2, 5/2, and so onward. 



The linear simple-harmonic oscillator thus furnishes an instance of 

 "half-quantum-numbers." In most of the earlier theories it was 

 either assumed or inferred that this "Planck" oscillator displayed 

 "whole quantum-numbers" — that its permitted energy-values were 

 the products of hvo by the successive integers 1, 2, 3, 4, • • • . However, 

 in the interpretation of certain features of band-spectra by the assump- 

 tion that the two atoms of a diatomic molecule vibrate as linear oscil- 

 lators along their line of centres, the half-quantum-numbers sometimes 

 led to better agreement with experience than did the whole-quantum- 

 numbers. 



The Eigenfunktionen corresponding to the consecutive Stationary 

 States are these: 



^„(.t) = const- e~^-''''"»'o^^"'Hn{27rx-\lmvo/h). (164) 



The first five of these Eigenfunktionen are exhibited in Fig. 2. These 

 curves may be regarded, if the reader so chooses, as the stationary- 

 wave patterns of "loops" and "nodes," exhibited by five resonating 

 strings along which the wave-speed varies according to the five laws 

 obtained by assigning the first five values given by (163) to the constant 

 E in the equation: 



u ^ . ^ (165) 



The various Stationary States of a linear oscillator are therefore imaged 

 not as the fundamental and the overtones of one and the same string, 

 but as the fundamental (and exclusive) nodes of vibration of distinct 



