CONTEMPORARY ADVANCES IN PHYSICS 691 



Interpretation of the Rotator by Wave- Mechanics 



The rotator or rotating body, the "spinning-top" as the Germans 

 often call it, is a very important object in the workshop of the builder 

 of atom-models. It is the accepted molecule-model used in theorizing 

 about the polarization of gases by electric and magnetic fields, and 

 the basis of the accepted molecule-model used in explaining the band- 

 spectra of diatomic and polyatomic gases. Most models devised for 

 the latter purpose combine the features of the rotator and the linear 

 oscillator; but for the present purpose it is sufficient to view these 

 separately, conceiving the rotator as a perfectly rigid whirling body. 



The treatment of the rotator by wave-mechanics is in one respect 

 admirably simple, but eventually we are led into the mazes of the 

 General Equation with its non-Euclidean geometry. One can how- 

 ever avoid the complexity long enough to benefit by the intelligible 

 feature, by considering first a rotator such as was invented more than 

 fifty years ago to account for the specific heats of diatomic gases such 

 as hydrogen — a dumbbell not permitted to spin around its own 

 axis-of-figure or line-of-centres, but revolving around some axis passing 

 through its center-of-mass perpendicular to its line-of-centres. The 

 orientation of such a dumbbell is specified by the angles 6 and which 

 define, in a polar coordinate-system, the direction in which its axis-of- 

 figure is pointing. The energy is exclusively kinetic, so that the term 

 containing V vanishes from the wave-equation, a circumstance which 

 is very helpful. Representing by A the moment of inertia of the 

 dumbbell about the axis of revolution, we find the wave-equation in 

 the form:-'^ 



V2^ + ^^^ lA = 0. (190) 



In this equation the Laplacian operator is to be expressed in the polar 

 coordinates 6 and 0, as it was expressed in equation (131), but without 

 the terms involving the third and missing coordinate r. We have 

 before us, therefore, the second of equations (133), with a specific 

 value for the constant there called X: 



— cosec 6 





^^. (i9i; 



Here, as there, the function \f/ must repeat itself whenever d is altered 

 by any multiple of w and 4> by any multiple of lir; for then we are 

 back at the same place, i.e. at the same orientation of the rotator. 



-•■• Schroedinger, Ann. d. Phys., 79, pp. 520-522 (1926). 



