Quantum-Mechanically Correct Form of Hamiltonian Function for Conservative Systems

Abstract

Dirac showed that, if in the Hamiltonian $H$ momenta ${\ensuremath{\eta}}_{r}$ conjugate to the co-ordinates ${\ensuremath{\xi}}_{r}$ are replaced by $(\frac{h}{2\ensuremath{\pi}i})\frac{\ensuremath{\partial}}{\ensuremath{\partial}{\ensuremath{\xi}}_{r}}$, the Schr\"odinger equation appropriate to the coordinate system ${\ensuremath{\xi}}_{r}$ is $(H\ensuremath{-}E){\ensuremath{\psi}}_{\ensuremath{\xi}}=0$. Applied to coordinate systems other than cartesian this usually leads to incorrect results. The difficulty is here traced partially to the way in which ${\ensuremath{\psi}}_{\ensuremath{\xi}}$ is normalized and partly to the choice of $H$. In $H$ expressions such as $\mathrm{qp}{q}^{\ensuremath{-}1}p$ and ${p}^{2}$ are not equivalent, and the simplified form is generally incorrect. A formula satisfying all the requirements of quantum mechanics for a Hamiltonian of a conservative system, in an arbitrary coordinate system, is therefore developed $H=\frac{1}{2\ensuremath{\mu}}\ensuremath{\Sigma}\stackrel{r=n}{r=1}\ensuremath{\Sigma}\stackrel{s=n}{s=1}{g}^{\ensuremath{-}\frac{1}{4}}{p}_{r}{g}^{\frac{1}{2}}{g}^{\mathrm{rs}}{p}_{s}{g}^{\ensuremath{-}\frac{1}{4}}+U$ This formula is applied to a case of plane polar coordinates and leads to correct results.