On the semiclassical expansion in quantum mechanics for arbitrary Hamiltonians

It is shown that if the quantum‐mechanical propagator, satisfying the n‐dimensional Schrödinger equation (H−ih/∂/∂t_b) 〈q_b,t_b‖q_a,t_a〉 =0, with 〈q_b,t_b‖q_a,t_b〉 =δ (q_b−q_a) and arbitrary classical Hamiltonian H_c, admits a semiclassical (WKB) approximation, then the latter is of the form K_WKB=const ×h/^−n/2 (detM)^1/2 exp(iS_c/h/), where S_c is the classical action, M_ij≡−∂²S_c/∂q_aⁱ∂q_b^j, and K⁻¹_WKB(H−ih/∂/∂t_b) K_WKB =O (h/²). The restrictions on the correspondence rule chosen to pass from H_c 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.