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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Time-Dependent Formulation of the Semiclassical Approximation in Collision Theory. II

Sergej A. Lebedeff

Published Mar 15, 1970• 9 citations
Publisher

Abstract

In the present investigation the semiclassical approximation using the time-dependent formulation of the collision theory is extended. The previously used narrow-wave-packet approximation is improved by using, instead of expanding wave packets, wave packets that first contract and then expand. In addition, an integral equation is derived that allows one to obtain corrections to the narrow-wave-packet approximation. Several of these corrections are presented.

DOI
10.1103/physrevd.1.1583

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In the present investigation the semiclassical approximation using the time-dependent formulation of the collision theory is extended. The previously used narrow-wave-packet approximation is improved by using, instead of expanding wave packets, wave packets that first contract and then expand. In addition, an integral equation is derived that allows one to obtain corrections to the narrow-wave-packet approximation. Several of these corrections are presented.
A semiclassical approximation is presented which uses the wave-pocket formulation of collision theory. First an appropriate semiclassical approximation is derived by assuming that the wave packet is and remains narrow during the passage of the interaction region. After this, a process is devised to decompose a macroscopic wave packet into narrow wave packets. The wave function, after the collision, is obtained by summing up the contributions from all the narrow wave packets into which the macroscopic wave packet was decomposed. Finally, using this function, the SSS matrix and the scattering cross section are calculated.
This series of papers is concerned with the derivation of the equations of the classical picture of atomic collisions, $i\ensuremath{\hbar}\frac{d}{\mathrm{dt}}{d}_{i}(t)=\ensuremath{\Sigma}\stackrel{}{j}{h}_{\mathrm{ij}}(t){d}_{j}(t),$ which describe the "time" dependence of electronic-quantum-state amplitudes as the nuclei move along a classical trajectory. These equations are derived in two ways. In the first formulation, which coincides with the intuitive classical picture of the collision, the nuclear part of the wave function is treated as a superposition of narrow wave packets, each traveling along a classical trajectory. In the second formulation, a semiclassical approach is used. The validity and meaning of the two formulations are discussed and compared.
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.
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 theory of quantum amplitudes in the semiclassical, h/→0 limit is considerably extended by including overlaps between Gaussian coherent states (wave packets) in the primitive ’’semiclassical algebra.’’ Traditionally, semiclassical formulations have exclusively involved overlaps between eigenstates of Hermitian operators; these are delocalized in most representations. Paradoxically, the states which are most localized and particlelike in the classical limit (wave packets) are not eigenfunctions of Hermitian observables, and have been omitted from asymptotic, formal theories of the semiclassical limit. When incorporated into a generalized theory, the Gaussian wave packet states apppear to alleviate many of the practical difficulties encountered in implementation of the Hermitian, delocalized state formulation. We present the generalized theory here, and show how it contains and reduces to the Hermitian semiclassical theory in various limits.