506 Quantum Mechanical Basis of Molecular Spectra /27 : 2 



quantum numbers. This result is a direct consequence of the wave type 

 equations used to describe the transmission of particles. 



Quantum mechanics associates wave-like functions with all particles. 

 If one is sufficiently skilled in manipulating these mathematical wave 

 functions, one can deduce all the measurable characteristics of the 

 particle such as momentum and energy. These wave functions in 

 many ways resemble those describing the motion of vibrating strings. 

 Such a string has certain resonant modes, each of which can be repre- 

 sented by a suitable characteristic function. For each resonant mode, 



Characteristic 

 Shape 



Harmonic Number 



Characteristic -n r=r - 27r rt~ - i7T fft 



Frequency ^-j^T/p a> 2 - T jT/p a> 2 - T jT/p 



Ch F 7n C cTof C *^*"<T> •**' «i— ^C 3 ?) 1~* %=sin^) .*-* 



to = Ittv 

 Figure I. Eigenvalues and eigenfunctions for a string. 



there is a certain resonant or characteristic frequency. Any arbitrary 

 motion of the string can be described as sums of characteristic functions, 

 each multiplied by appropriate amplitudes. For example, a piano 

 string of length L, linear density p, and tension T, vibrates in resonant 

 modes which look something like those shown in Figure 1. Any 

 arbitrary motion of the string can be described by a wave function T 

 given by 



oo 



Y = 2 A n V n (5) 



B=l 



where the ^4 n 's are the appropriate amplitudes which are independent 

 of both x and t. 



This same mathematical reasoning is valid in quantum mechanics, 

 provided that minor changes are made in the terminology. Quantum 

 mechanicians usually use the words "quantum number" instead of 

 "harmonic number," and the German word "eigen" instead of the 

 English word "characteristic." Thus, they talk of eigenfrequencies, 

 eigenfunctions, and eigenvalues. 



In the case of the vibrating string, a knowledge of the eigenfrequencies 

 is insufficient to determine the energies associated with the different 



