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Physical Review A

Semiclassical theory for inelastic scattering

J. H. Jensen, L. D. Chang, Walter Kohn

Published Aug 1, 1989• 14 citations
Publisher

Abstract

We present a simple theory for obtaining the semiclassical approximation for inelastic scattering. The theory gives explicit expressions for the expectation values of a wide class of quantum operators, generally valid up to O(\ensuremath{\Elzxh}), which may be used to characterize the final distribution of a scattered particle. We illustrate the method with a one-dimensional model consisting of a particle scattering from a harmonic oscillator. In addition, we apply the theory to examine under what conditions the so-called trajectory approximation is and is not valid.

DOI
10.1103/physreva.40.1198

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We present a simple theory for obtaining the semiclassical approximation for inelastic scattering. The theory gives explicit expressions for the expectation values of a wide class of quantum operators, generally valid up to O(\ensuremath{\Elzxh}), which may be used to characterize the final distribution of a scattered particle. We illustrate the method with a one-dimensional model consisting of a particle scattering from a harmonic oscillator. In addition, we apply the theory to examine under what conditions the so-called trajectory approximation is and is not valid.
We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.
A simple model coupling a one-dimensional beam particle to a one-dimensional harmonic oscillator is used to explore complementarity and entanglement. This model, well-known in the inelastic scattering literature, is presented under three different conceptual approaches, with both analytical and numerical techniques discussed for each. In a purely classical approach, the final amplitude of the oscillator can be found directly from the initial conditions. In a partially quantum approach, with a classical beam and a quantum oscillator, the final magnitude of the quantum-mechanical amplitude for the oscillator's first excited state is directly proportional to the oscillator's classical amplitude of vibration. Nearly the same first-order transition probabilities emerge in the partially and fully quantum approaches, but conceptual differences emerge. The two-particle scattering wavefunction clarifies these differences and allows the consequences of quantum entanglement to be explored.
A semiclassical theory for inelastic scattering of a particle by a two-state system in the near-adiabatic limit is developed. The exponentially small transition amplitude is calculated. The theory is an extension of Pokrovskii and Khalatnikov's theory for above-barrier reflection (Zh. Eksp. Teor. Fiz. 40, 1713 (1961) [Sov. Phys.---JETP 13, 1207 (1961)]). The analysis involves studies of the WKB solutions in the complex coordinate plane along certain contours. The region of validity of the theory is established. Our result has the same form as the Landau-Zener-Stueckelberg formula; however, our theory is applicable to more general systems. Numerical comparisons with exact solutions are presented; the differences between our results and exact solutions become negligible as the adiabatic limit is approached.
Rather general expressions are derived which represent the semiclassical time-dependent propagator as an integral over initial conditions for classical trajectories. These allow one to propagate time-dependent wave functions without searching for special trajectories that satisfy two-time boundary conditions. In many circumstances, the integral expressions are free of singularities and provide globally valid uniform asymptotic approximations. In special cases, the expressions for the propagators are related to existing semiclassical wave function propagation techniques. More generally, the present expressions suggest a large class of other, potentially useful methods. The behavior of the integral expressions in certain limiting cases is analyzed to obtain simple formulas for the Maslov index that may be used to compute the Van Vleck propagator in a variety of representations.