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Journal of Mathematical Physics

New approach to semiclassical analysis in mechanics

Miguel Angel Alonso, G. W. Forbes

Published Apr 1, 1999• 13 citations
Publisher

Abstract

A new method is proposed for constructing approximate solutions to the Schrödinger equation. In place of the wave function, its Gaussian-windowed Fourier transform is used as the fundamental entity. This allows an intuitively attractive connection to be made with a family of classical trajectories and, at all times, the wave function is inferred from the present state of these trajectories. The fact that the connection between the wave function and the classical trajectories is consistently constructed in phase space allows this method to be free of the limitations of other methods.

DOI
10.1063/1.532829

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A new method is proposed for constructing approximate solutions to the Schrödinger equation. In place of the wave function, its Gaussian-windowed Fourier transform is used as the fundamental entity. This allows an intuitively attractive connection to be made with a family of classical trajectories and, at all times, the wave function is inferred from the present state of these trajectories. The fact that the connection between the wave function and the classical trajectories is consistently constructed in phase space allows this method to be free of the limitations of other methods.
We have recently published a new semiclassical method, generalized Gaussian wave packet dynamics, which extends Gaussian wave packet dynamics into complex phase space. Although we were able to give an accurate formulation of the method, we had at the time of writing that paper only an intuitive, heuristic understanding of the deeper causes which make the method work. A more mathematical understanding was needed. To close this gap we show in this paper the equivalence of the new method with a first order expansion of ℏ of the Schrödinger equation. We further prove that the new method is equivalent to the stationary phase approximation, using the usual WKB formula for the propagator. The latter equivalence enables us to show that all the symmetry properties of time-dependent quantum mechanics also hold in the new semiclassical theory. Finally, we provide some elaboration of the method, and clarify several issues that were not discussed before. With this new insight we are able to formulate a simple rule for the calculation of semiclassical wave functions that contain contributions from more than one branch. This corrects for the divergence of semiclassical wave functions near caustics, a problem that we encountered in the preceding paper.
A semiclassical approximation is derived by using a family of wavepackets to map arbitrary wavefunctions into phase space. If the Hamiltonian can be approximated as linear over each individual wavepacket, as often done when presenting Ehrenfest's theorem, the resulting approximation is a linear first-order partial differential equation on phase space, which will be referred to as the Schrödinger-Ehrenfest or SE equation. This advectively transports wavefunctions along classical trajectories, so that as a trajectory is followed in time the amplitude remains constant while the phase changes by the action divided by ℏ\hbarℏ. The wavefunction's squared-magnitude is a plausible phase space density and obeys Liouville's equation for the classical time evolution. This is a derivation of the Koopman-von~Neumann (KvN) formulation of classical mechanics, which previously was postulated but not derived. With the time-independent SE equation, continuity of the wavefunction requires the change of phase around any closed path in the torus covered by a classical trajectory to be an integer multiple of 2π2π2π, giving a standing wave picture of old quantum mechanics. While this applies to any system, for separable systems it gives Bohr-Sommerfeld quantization.
A new semiclassical approach that constructs the full semiclassical Green’s function propagation of any initial wave function directly from an ensemble of real trajectories, without root searching, is presented. Each trajectory controls a cell of initial conditions in phase space, but the cell area is not constrained by Planck’s constant. The method is shown to be accurate for rather long times in anharmonic oscillators, indicating the semiclassical time-dependent Green’s function is clearly worthy of more study. The evolution of wave functions in anharmonic potentials is examined and a spectrum from the semiclassical correlation function is calculated, comparing with exact fast Fourier transform results.
In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time−dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many−particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton’s equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State−specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H2 system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.
The quantum trajectories in the de Broglie-Bohm formulation of quantum mechanics depend on an additional quantum potential derived from the full wave solution of Schrödinger's equation. The task of supplying collectively all the correct quantum results strongly alters the characteristics of the corresponding classical trajectories, which underlie semiclassical approximations to the evolving wave function. Both classical and quantum trajectories are here considered to be conservative with no influence of an external environment, even though this is the source of eventual classicality in quantum systems, that is, decoherence. The concept of integrability, closely correspondent in classical and quantum mechanics, is not preserved by the quantum trajectories. General systems, in which classical chaotic motion participates, are much harder to treat semiclassically, but quantum trajectories can be chaotic even for integrable systems. This discrepancy between the character of classical and quantum trajectories in the de Broglie-Bohm interpretation does not clarify the singular classical-quantum transition.
A new technique is developed to generate semiclassical wave functions. The method uses only information already available from a standard semiclassical quantization of a system. Linear superpositions of Gaussian coherent states that lie along quantizing classical trajectories are used, with phases given by the action integrals plus a Maslov-type correction. Wave functions generated in this way suffer from none of the problems with caustics that primitive semiclassical wave functions encounter. The semiclassical wave functions are convenient for subsequent use in applications, e.g., molecular spectra. By generating wave functions for several simple systems, we show that under most circumstances these wave functions are very accurate approximations to the true quantum states.