The Symmetrical Top in the Undulatory Mechanics

Abstract

Schr\"odinger's method for determining the energy levels of an atomic system is applied to the case of the symmetrical top (moment of inertia about axis of symmetry $C$, the other one $A$). The energy values are found to be ${W}_{\mathrm{jn}}=\frac{{h}^{2}}{8{\ensuremath{\pi}}^{2}}\left[\frac{1}{A}j(j+1)+\left(\frac{1}{C}\ensuremath{-}\frac{1}{A}\right){n}^{2}\right]$ in agreement with the result obtained by Dennison from the matrix mechanics. The quantum numbers $j$ and $n$ must be integers restricted by $0\ensuremath{\leqq}j$, $|n|\ensuremath{\leqq}j$, while half-integral values are not permissible. The intensities of transitions are also calculated.