The Quantum-Mechanical Equations of Motion and Commutation Relations

It has been shown by Wigner that the equations of motion of a quantum-mechanical system with a hamiltonian H=1/2p²+V(q) need not imply the commutation relation $i[p,q]=1$. Particular examples are furnished by the free particle and the harmonic oscillator. On the other hand, Wigner points out that the above commutation rule does follow from the equations of motion if $V(q)$ is of the form $V=aq+b$ or V=aq³. In the present note, this latter result is generalized to the case of an arbitrary potential V=aqⁿ+b, where n is an odd integer; the last-named potentials are, in fact, particular cases of those satisfying a general criterion contained in the article.