On the WKB approximation to the propagator for arbitrary Hamiltonians

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Abstract
This paper presents a general expression for the WKB approximation to the propagator corresponding to an arbitrary Hamiltonian operator H. For example, if the correspondence rule used to pass from the classical Hamiltonian Hc to H is such that it associates aPiQ j +(1−a)Q jPi to piq j, then the formula gives KWKB=KVVexp {( 1/2 −a)ℱT(∂2Hc/∂qi∂pi )(qc(t),pc(t),t) dt}, where KVV≡(2πih/)−n/2 (det M)1/2exp(iSc/h/) is Van Vleck’s well-known formula, Sc being the action functional evaluated at the classical path (qc,pc) and Mij ≡−∂2Sc/∂qa i∂qbj. More generally, the formula presented here applies to any system with n degrees of freedom described by a function f(x,t) whose time evolution is given by (H(x,k∂/∂x,t)+k∂/∂t) f(x,t)=0, regardless of the form of H. The Schrödinger equation of quantum mechanics and the Fokker–Planck equation of diffusion are obvious examples. Many examples are discussed. This generalizes results obtained in a previous publication [J. Math. Phys. 18, 786–90 (1977)].

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