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American Journal of Physics

Path integrals and the WKB approximation

Barry R. Holstein, Arthur R. Swift

Published Sep 1, 1982• 35 citations
Publisher

Abstract

The WKB approximation is demonstrated to be formally identical to the semiclassical limit of the Feynman path integral propagator. In the limit that the path integral propagator is dominated by the classical particle trajectory, it is identical to the propagator calculated directly from WKB wave functions in the stationary phase approximation.

DOI
10.1119/1.12750

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The WKB approximation is demonstrated to be formally identical to the semiclassical limit of the Feynman path integral propagator. In the limit that the path integral propagator is dominated by the classical particle trajectory, it is identical to the propagator calculated directly from WKB wave functions in the stationary phase approximation.
We present a complete derivation of the semiclassical limit of the coherent-state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial-value representation for the semiclassical propagator, based on an initial Gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed Gaussian approximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent-state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.
The general method of path integrals in quantum mechanics introduced by R.P.Feynman is applied in this paper to discuss the amplitude of transition probability in quantum mechanics, that is, the subject of the propagator.Mauriel M.Mizrahi's semiclassical (WKB) approximation method has been used in studying the general expansion in detail for the propagator corresponding to an arbitrary Hamiltonian operator H. ?? where S_c is the action functional evaluated at the classical path (q_c, p_c) , ρ is a constant, ?? Maurice M.Mizrahi's solution of the K_(WKB) expansion has been generalized, and another linear independent particular solution is obtained.This is the solution when 0~2 = det M equals zero.The K_(WKB) can be applied to all systems corresponding to an arbitrary Hamiltonian operator.
The overcompleteness of the coherent states basis leads to a multiplicity of representations of Feynman's path integral. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. Two such semiclassical formulas were derived in \cite{Bar01} for the two corresponding path integral forms suggested by Klauder and Skagerstan in \cite{Klau85}. Each of these formulas involve trajectories governed by a different classical representation of the Hamiltonian operator: the P representation in one case and the Q representation in other. In this paper we construct a third representation of the path integral whose semiclassical limit involves directly the Weyl representation of the Hamiltonian operator, i.e., the classical Hamiltonian itself.
The overcompleteness of the coherent states basis leads to a multiplicity of representations of Feynman's path integral. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. Two such semiclassical formulas were derived in [1] for the two corresponding path integral forms suggested by Klauder and Skagerstan in [2]. Each of these formulas involve trajectories governed by a different classical representation of the Hamiltonian operator: the P representation in one case and the Q representation in other. In this paper we construct a third representation of the path integral whose semiclassical limit involves directly the Weyl representation of the Hamiltonian operator, i.e., the classical Hamiltonian itself.
It is widely believed that the WKB approximation to quantum barrier penetration is described in the path-integral formalism by a saddle-point approximation based on an expansion about an imaginary-time semiclassical path. This article examines the relationship between tunneling amplitudes and imaginary-time paths. It is shown, using a simple example, that for some tunneling amplitudes the dominant semiclassical path cannot be chosen to have purely imaginary time if all other variables are chosen to be real.