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Journal of Physics A Mathematical and General

Semiclassical approximation as a small-noise expansion

Antonio Defendi, M. Roncadelli

Published Nov 7, 1995• 6 citations
Publisher

Abstract

We discuss how the semiclassical approximation arises within the (classically improved) Langevin quantization. We derive a new representation of the semiclassical propagator (at imaginary time) in the form of a white noise average, which is valid over an arbitrary (imaginary) time interval. Although our result is ultimately equivalent to the standard representation of the semiclassical propagator, it nevertheless turns out to be more advantageous in dealing with certain problems that involve tunnelling and metastability.

DOI
10.1088/0305-4470/28/21/001

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We discuss how the semiclassical approximation arises within the (classically improved) Langevin quantization. We derive a new representation of the semiclassical propagator (at imaginary time) in the form of a white noise average, which is valid over an arbitrary (imaginary) time interval. Although our result is ultimately equivalent to the standard representation of the semiclassical propagator, it nevertheless turns out to be more advantageous in dealing with certain problems that involve tunnelling and metastability.
Recently, a novel quantization technique has been put forward, which yields the imaginary-time quantum-mechanical propagator as a noise average involving the solutions of a Langevin equation in configuration space with a Gaussian white noise. Contrary to previous attempts, this strategy does not require any knowledge about the ground-state quantum dynamics. Our aim is to show that this approach can be cast in a different form, which exhibits a manifest connection with classical mechanics (at imaginary time). The advantages of this strategy are pointed out and illustrated by taking the harmonic oscillator as an example.
We first point out that nonrelativistic quantum mechanics formulated at imaginary time s = it can formally be recognized as the Fokker-Planck description of a classical Brownian motion, which occurs (in general) in a medium where particles are absorbed and emitted. We next offer a new formulation of quantum dynamics, which is basically the Langevin treatment of that Brownian motion. More explicitly, we derive a noise average representation of the transition probability W ( X", s"|X', s' ) in terms of the solutions to a Langevin equation with a Gaussian white noise. Upon analytic continuation back to real time W ( X", s"|X', s' ) becomes the propagator of the original Schrödinger equation, namely we have ⟨ X", t"|X', t' ⟩ = W ( X", it"|X', it' ). The main advantage of this approach is to permit a systematic application to quantum-dynamical problems of the mathematical techniques developed in the theory of classical stochastic processes, even when vector potentials and/or time-dependent potentials are present.
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.
A semiclassical approximation to the time evolution of coherent states may be derived from a saddle point approximation to the exact quantum propagator, and in general it involves complex classical dynamics. We generalize previous one-dimensional results to ddd dimensions, and for the case d=2d=2d=2 we present several applications. We also consider other simple approximations that depend only on real classical trajectories, but are not initial value representations. These approximations are able to reproduce interference and tunneling effects and involve propagating a few classical initial conditions compatible with the quantum uncertainties.
Starting with path integrals in the SU(2) coherent state representation, the semiclassical approximation of the propagator for the spin system is investigated. By extending the idea of the semiclassical expansion method, which was developed in the usual phase-space path integrals, to the path integrals in the curved phase space, which is characteristic of the SU(2) coherent states, we obtain a closed form for the semiclassical propagator. As an application, we discuss the semiclassical quantization condition for the spin system.