Inequivalence of the classes of classical and quantum harmonic potentials: Proof by example

Abstract

There are an infinite number of potentials whose classical angular velocity ${\omega }_c$ is independent of energy. There are also an infinite number of potentials whose quantum eigenenergies are equally spaced, $\hslash {\omega }_q$. If ${\omega }_c={\omega }_q$, then in the WKB approximation these classes are the same. (Certain specific potentials belong to both of these classes exactly.) However, we show by a different specific example that in general these classes are not equivalent.