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.
Skip Nav Destination
Article navigation
April 1977
Research Article|
April 01 1977
On the semiclassical expansion in quantum mechanics for arbitrary Hamiltonians
Maurice M. Mizrahi
Maurice M. Mizrahi
Center for Naval Analyses of The University of Rochester, 1401 Wilson Boulevard, Arlington, Virginia 22209
Search for other works by this author on:
J. Math. Phys. 18, 786–790 (1977)
Citation
Maurice M. Mizrahi; On the semiclassical expansion in quantum mechanics for arbitrary Hamiltonians. J. Math. Phys. 1 April 1977; 18 (4): 786–790. https://doi.org/10.1063/1.523308
Download citation file:
Sign in
Don't already have an account? Register
Sign In
You could not be signed in. Please check your credentials and make sure you have an active account and try again.
Pay-Per-View Access
$40.00
Citing articles via
Almost synchronous quantum correlations
Thomas Vidick
Topological recursion of the Weil–Petersson volumes of hyperbolic surfaces with tight boundaries
Timothy Budd, Bart Zonneveld
Modified gravity: A unified approach to metric-affine models
Christian G. Böhmer, Erik Jensko
Related Content
On the WKB approximation to the propagator for arbitrary Hamiltonians
J. Math. Phys. (January 1981)
A differential Galois approach to path integrals
J. Math. Phys. (May 2020)
A rule based approach to the Design of Auditoria.
J Acoust Soc Am (April 2011)
Finite‐Larmor‐radius stabilization in a sharp‐boundary Vlasov‐fluid screw pinch
Phys. Fluids (April 1977)
Overview of drive by wire technologies in automobiles
AIP Conference Proceedings (November 2022)