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Physical Review A

Semiclassical Theory of Inelastic Collisions. I. Classical Picture and Semiclassical Formulation

J. B. Delos, Walter R. Thorson, Stephen K. Knudson

Published Aug 1, 1972• 246 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.6.709

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This series of papers is concerned with the derivation of the equations of the classical picture of atomic collisions, i\ensuremathℏddtdi(t)=\ensuremathΣjhij(t)dj(t),i\ensuremath{\hbar}\frac{d}{\mathrm{dt}}{d}_{i}(t)=\ensuremath{\Sigma}\stackrel{}{j}{h}_{\mathrm{ij}}(t){d}_{j}(t),i\ensuremathℏdtd​di​(t)=\ensuremathΣj​hij​(t)dj​(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.
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.
A useful and intuitive phase space picture of common semiclassical approximations and procedures is proposed and developed. Underlying the intuitive pictures is the semiclassical, C→0 limit of the Wigner distribution, which is discussed in an Appendix. Simple classical analog phase space diagrams are suggested which represent quantum amplitudes, completeness relations, stationary phase points and integrations, caustic singularities, uniformizations, and the interesting short and long time behavior of the semiclassical propagator. Semiclassical wave packet amplitudes, now known to be on equal footing with the more usual amplitudes between delocalized eigenstates of Hermitian operators, are included in the phase space pictures. It becomes apparent why the primitive semiclassical wave packet amplitudes are to some extent uniformizing, and numerical results are presented to support this conclusion. The power of the phase space picture as an aid in formulating new approximations is illustrated.
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.
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.
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.