On the semiclassical expansion in quantum mechanics for arbitrary Hamiltonians

Abstract

It is shown that if the quantum-mechanical propagator, satisfying the n-dimensional Schrödinger equation (H−ih/∂/∂tb) 〈qb,tb‖qa,ta〉 =0, with 〈qb,tb‖qa,tb〉 =δ (qb−qa) and arbitrary classical Hamiltonian Hc, admits a semiclassical (WKB) approximation, then the latter is of the form KWKB=const ×h/−n/2 (detM)1/2 exp(iSc/h/), where Sc is the classical action, Mij≡−∂2Sc/∂qai∂qbj, and K−1WKB(H−ih/∂/∂tb) KWKB =O (h/2). The restrictions on the correspondence rule chosen to pass from Hc to the operator H are spelled out in detail, and it is found that one has a considerable amount of leeway in choosing such a rule. Differential equations for higher-order corrections can be generated at will. This generalizes previously known partial results to arbitrary Hamiltonians. The arbitrariness of the Hamiltonian makes the method useful as a general tool in the theory of partial differential equations.