Quantum-Mechanically Correct Form of Hamiltonian Function for Conservative Systems

Dirac showed that, if in the Hamiltonian $H$ momenta ${\eta }_r$ conjugate to the co-ordinates ${\xi }_r$ are replaced by $\left(\frac{h}{2\pi i}\right)\frac{\partial }{\partial {\xi }_r}$, the Schrödinger equation appropriate to the coordinate system ${\xi }_r$ is $\mathrm{}(H-E)\mathrm{}{\psi }_{\xi }=0\mathrm{}$. Applied to coordinate systems other than cartesian this usually leads to incorrect results. The difficulty is here traced partially to the way in which ${\psi }_{\xi }$ is normalized and partly to the choice of $H$. In $H$ expressions such as $\mathrm{qp}{q}^{-1\mathrm{}}p$ 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 $H=\frac{1}{2\mu }\Sigma \stackrel{r=n}{r=1\mathrm{}}\Sigma \stackrel{s=n}{s=1\mathrm{}}{g}^{-\frac{1}{4}}{p}_r{g}^{\frac{1}{2}}{g}^{\mathrm{rs}}{p}_s{g}^{-\frac{1}{4}}+U$ This formula is applied to a case of plane polar coordinates and leads to correct results.