The Torsion Oscillator-Rotator in the Quantum Mechanics

Abstract

The theory of a symmetric rotator which in addition to the usual three degrees of rotational freedom has about the axis of symmetry also a degree of torsional freedom between two of its principal parts, is treated quantum mechanically. The potential energy is taken to be expressible in the form: $V=L(1\ensuremath{-}cosm{\ensuremath{\varphi}}^{\ensuremath{'}})$ where $L$ is proportional to the restoring torque, ${\ensuremath{\varphi}}^{\ensuremath{'}}$ is the angle of displacement, and $m$ the number of minima of the potential energy curve as ${\ensuremath{\varphi}}^{\ensuremath{'}}$ increases from zero to $2\ensuremath{\pi}$. The Schr\"odinger equation is found to be separable into two differential equations, one being the quantum mechanical equation for an ordinary symmetric rotator, and the other being of the form of Mathieu's equation: $\frac{{d}^{2}M}{d{x}^{2}}+(a+16qcos2x)$ $M=0$ in which $a$ is proportional to the energy parameter of the oscillator, $q$ proportional to the restoring torque, and $x$ to ${\ensuremath{\varphi}}^{\ensuremath{'}}$. It is found that the solutions to this equation must satisfy the condition: $M(x\ensuremath{-}m\ensuremath{\pi})=(\mathrm{exp}\ensuremath{-}\frac{2\ensuremath{\pi}\mathrm{iK}{{A}_{z}}^{\ensuremath{'}}}{{A}_{z}}) M(x)$ where $K$ is the quantum number of angular momentum about the axis of symmetry, ${{A}_{z}}^{\ensuremath{'}}$ and ${A}_{z}$ the moments of inertia of the lower part of the top and of the whole of the top respectively about the axis of symmetry. This ordinarily demands a general non-periodic solution to Mathieu's equation which, however, degenerates like the ordinary Mathieu functions to an exponential function as $q\ensuremath{\rightarrow}0$. A qualitative discussion is given about the manner in which the energy states in the limiting case where $q=0$ go over into the other limiting case where $q=\ensuremath{\infty}$, and the calculation of the intensities and the selection rules for the rotator are finally determined where $q=0$ and where $q=\ensuremath{\infty}$. These it is believed will also be valid at least in first approximation in the neighborhoods of these limiting cases where $q$ no longer is quite zero and not quite equal to infinity.