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The Journal of Chemical Physics

On Semiclassical Scattering

John Adair Barker, C. H. J. Johnson

Published Apr 1, 1965• 6 citations
Publisher

Abstract

The behavior of the quantum-mechanical phase shifts for scattering by a potential well is examined in the classical limit. The phase shifts tend to zero in the classical limit for all cases in which the corresponding classical trajectory does not pass through the well. This disproves by counterexample the result of Curtiss and Powers which indicated that there should be a contribution to the phase shifts even in the classical limit from classically inaccessible but energetically possible regions between two inner classical turning points. The error in the latter result appears to arise from the neglect of bound states of positive energy which exist in the classical limit (and only in that limit).

DOI
10.1063/1.1696285

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The behavior of the quantum-mechanical phase shifts for scattering by a potential well is examined in the classical limit. The phase shifts tend to zero in the classical limit for all cases in which the corresponding classical trajectory does not pass through the well. This disproves by counterexample the result of Curtiss and Powers which indicated that there should be a contribution to the phase shifts even in the classical limit from classically inaccessible but energetically possible regions between two inner classical turning points. The error in the latter result appears to arise from the neglect of bound states of positive energy which exist in the classical limit (and only in that limit).
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.
Abstract An infinite potential well, truncated at finite height, provides a simple model for studying the effect of nonanalyticity on semiclassical approximations. An exact quantization condition for the bound states separates the effects associated with the untruncated well from those of the truncation. Because the truncation occurs beyond the classical turning points, it has no effect to any finite order in powers of Planck’s constant. The truncation contribution is exponentially small and depends on the potential in the classically forbidden region. The contribution associated with the well, when consistently approximated beyond all semiclassical orders, also leads to a small exponential, depending on the potential in the classically allowed region. Both exponentially small contributions can be extracted by asymptotic analysis, with explicit results in the simple case of a linear well. This combination of several different semiclassical techniques could be pedagogically useful as an exercise in teaching physical asymptotics at the postgraduate level.
This chapter discusses the connection between the classical and semiclassical domains of scattering. Scattering phenomena may be described via three regimes: the scattering of waves by objects with small, large, or comparable sizes with the wavelength of the incident (plane wave) radiation. All three regions can be related to three domains: the classical domain (geometrical optics, particle and particle/ray-like trajectories); the wave domain (physical optics, acoustic and electromagnetic waves, quantum mechanics); and the semiclassical domain (the vast intermediate region between the first and second domain). The chapter first provides an overview of classical and semiclassical scattering domains before beginning with an analysis of the semiclassical formulation. It also considers the radial equation, scattering by a one-dimensional potential barrier, and the radially symmetric problem. Solutions for phase shifts and the potential well are presented.
Semiclassical bound state theory is considered from the viewpoint of the phase space representations of quantum mechanics. In the small h/ limit, within a representation defined by the ordering chosen for the noncommuting position and momentum operators, we obtain: (1) equations of motion, which need not be classical and (2) correspondence principles in action-angle variables, regarded as those initial conditions on the equations of motion which dominate in the small h/ limit. The Weyl ordering is seen to generate the classical equations of motion and the normal correspondence result. Possible evaluation of semiclassical bound state energies is discussed by means of the eigenvalue approach to classical mechanics used by Prigogine and co-workers.
Quantum-mechanical calculations of the differential cross section for the small-angle elastic scattering of heavy particles are carried out to establish more definitely the region of validity of the classical approximation. Four results are discussed: (1) The Massey—Mohr phase-shift formula corresponds to the Kennard small-angle scattering formula in the semiclassical limit. (2) The Schiff approximation for the cross section is exactly the same as the semiclassical approximation at small angles, for any central potential. (3) At very small angles the semiclassical limit for the differential cross section varies as exp (—cθ2), where c is a function of velocity for which explicit expressions are given. (4) The first quantum deviation from the classical limit, which is proportional to ℏ2, can be combined with the preceding result to give a reasonable representation of the differential cross section over the entire range of small angles for which quantum deviations are appreciable. Detailed calculations for some specific systems are made, and it is shown that Wu's misgivings over the classical interpretation of the experimental results of Amdur and co-workers are unjustified.