Quantum analog of the classical action integral: evaluation using approximate WKB and exact energy eigenvalues in familiar (and not so familiar) bound state problems

Abstract

Using both WKB approximation methods and complete eigenstate solutions, we examine four standard, textbook level, bound state problems in quantum mechanics. We show how the action integral from the Bohr–Sommerfeld quantization condition, extended to the WKB energy quantization condition, can be expressed in the language of semi-classical and exact quantum results. Relating classical time averaging and quantum expectation value methods, the action integral can be used to connect the average value of speed squared times period across such systems giving ⟨v2⟩nτn=(n+C~)(h/m), where n is the bound state quantum number ( n=0,1,2,...) supplemented by the Maslov correction ( C~). We then extend this analysis using exact results for an infinite hierarchy of potentials, one derived from the repeated supersymmetrization of the familiar infinite well, as a more mathematically sophisticated example. These discussions provide a novel pedagogical twist on h as the ‘quantum of action’, accessible at several levels of the undergraduate quantum mechanics curriculum.