Note on the Inequivalence of Classical and Quantum Hamiltonians

Abstract

By working directly from Newtonian equations of motion and representing acceleration $ẍ$ as $\mathrm{[(x,H),\hspace{0.17em}H]}$ , families of Hamiltonians may be constructed which are functions of the energy and which, while describing the same classical motions, are not connected by canonical transformations. One dimensional examples are given, and the formal rules of quantization executed when $H$ is $(\mathrm{energy}{)}^{\frac{1}{2}}$ to show that the quantum motion does not come out right. The reason is that the commutation rule $\mathrm{(x,p)\hspace{0.17em}=\hspace{0.17em}iħ}$ does not have universal standing; quantum theory in employing this rule is apparently requiring not merely that $H$ be a generator of motion, but also that it be a very particular one, viz., the energy when the system is conservative.