The Rotation of Molecules

Abstract

On the basis of the new quantum mechanics the rotational motion of the following molecules has been treated from the point of view of systems involving restraints; 1. the simple rotator in a plane, 2. the simple rotator in space and 3. molecules having an axis of symmetry. The quantum mechanical treatment of systems having restraints has been discussed and certain quantum conditions governing them have been obtained. The simple rotator in a plane is found to have an energy $W=({m}^{2}+m+\frac{\mathrm{I}}{2})\frac{{h}^{2}}{8{\ensuremath{\pi}}^{2}M{a}^{2}}$ and a total angular momentum $\ensuremath{\mu}=(m+\frac{\mathrm{I}}{2})\frac{h}{2\ensuremath{\pi}}.$ The quantum number $m$ may be whole or half numbered the corresponding normal states being $m=0$ and $m=\ensuremath{-}\frac{1}{2}$. The simple rotator in space has an energy $W=\frac{({m}^{2}+m+1){h}^{2}}{8{\ensuremath{\pi}}^{2}M{a}^{2}}$ and the square of its total angular momentum is ${\ensuremath{\mu}}^{2}=\frac{m(m+1)}{\frac{{h}^{2}}{4{\ensuremath{\pi}}^{2}}}$. The number $m$ must be integral and the normal state is $m=0$. Molecules having moments of inertia $A=B \mathrm{and} C$ are found to have an energy $W=\left\{\left(\frac{1}{A}\right)({m}^{2}+m+1)+\left(\frac{1}{C}\ensuremath{-}\frac{1}{A}\right){n}^{2}+\frac{1}{2C}\ensuremath{-}\frac{C}{4}{\left(\frac{1}{C}\ensuremath{-}\frac{1}{A}\right)}^{2}\right\}\frac{{h}^{2}}{8{\ensuremath{\pi}}^{2}}$ and the square of their total angular momentum is again ${\ensuremath{\mu}}^{2}=\frac{m(m+1){h}^{2}}{4{\ensuremath{\pi}}^{2}}$. The numbers $m$ and $n$ must be either both integral with normal state $m=n=0$, or both half integral with normal state $m=\frac{1}{2}$, $n=\ifmmode\pm\else\textpm\fi{}\frac{1}{2}$. The differences between these two solutions which may be expected to appear in the observed infra-red spectra are discussed. For all three examples the quantum theoretical amplitudes giving the transition probabilities have been obtained.