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The Journal of Chemical Physics

Quantum Effects in Small-Angle Molecular-Beam Scattering

Edward A. Mason, Joseph T. Vanderslice, C. J. G. Raw

Published Apr 15, 1964• 75 citations
Publisher

Abstract

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.

DOI
10.1063/1.1725488

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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.
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 unified description of classical and quantal scattering is presented. This description is unified both in the mathematical language and in the conceptual pictures that are used. Statistical states and observables, represented in phase space, are emphasized while both the time dependent trajectory and stationary state beam pictures are used. Semiclassical-type approximation schemes to the generalized differential cross section are presented for the single and double potential cases. Connection is made with the Born and distorted wave Born approximations, for both quantum and classical mechanics. A similar semiclassical-type approximation scheme for the double potential internal state kinetic cross section is presented. Connection is also made with the 'constant acceleration approximation' of Oppenheim and Bloom. Zwanzig's projection operator method is used to express the kinetic cross sections in terms of memory effects for the interacting observables.
Chapter 18 is entirely devoted to the quantum theory of scattering, which is normally covered in more detail in a course on quantum field theory. We introduce the main concepts, including scattering amplitude, differential and total cross sections, and form factors, and derive the main formulas for elastic scattering, in a nonrelativistic framework; this allows us to appreciate how the information obtained from scattering experiments is used to explore the intricacies of quantum particles that are otherwise inaccessible. The Born approximation is studied and applied to scattering by a periodic potential, a weak potential, and Rutherford scattering. The partial wave expansion is derived and applied to obtain the optical theorem. The chapter concludes with a complementary section on resonant scattering.
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Quantum mechanical scattering theory has recently been formulated exclusively in terms of density operators. The present work interprets the resulting generalized collision cross sections as integrals overtime correlation functions of relative coordinate, two body operators. It is shown that a straight line trajectory approximation to these time correlation functions is identical to the Born approximation. A phase space representation of the quantum collision dynamics is formulated. In these terms, the quantum cross section can be reduced in a natural way to the corresponding classical cross section provided there is no 'singular' behaviour. It would appear that this formulation would be very appropriate for making semiclassical approximations for the differential cross sections.