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Modern Physics Letters A

UNIFORM SEMICLASSICAL APPROXIMATION FOR ABSORPTIVE SCATTERING SYSTEMS

Mahir Saleh Hussein, Mauricio Porto Pato

Published Apr 1, 1988• 1 citation
Publisher

Abstract

The uniform semiclassical approximation of the elastic scattering amplitude is generalized to absorptive systems. An integral equation is derived which connects the absorption-modified amplitude to the absorption-free one. Division of the amplitude into diffractive and refractive components is then made possible. Relevance of our formalism to the algebraic approach to scattering theory is pointed out.

DOI
10.1142/s0217732388000568

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The uniform semiclassical approximation of the elastic scattering amplitude is generalized to absorptive systems. An integral equation is derived which connects the absorption-modified amplitude to the absorption-free one. Division of the amplitude into diffractive and refractive components is then made possible. Relevance of our formalism to the algebraic approach to scattering theory is pointed out.
We show how to interpret the Ford-Wheeler semiclassical expression for the scattering amplitude as a integral, over both positive and negative impact parameters, of a quantity depending on the action. This formulation clarifies the relationship between the action and the phase of the semiclassical expression for the scattering amplitude. Using this formulation, we show that interference processes in both one- and two-channel elastic scattering are analogous to those occurring in the two-slit diffraction of light.
A procedure for the evaluation of one-dimensional uniform semiclassical wave functions is explored. This form of the approximate wave function is tested for the case of an exponential potential and is compared to the exact quantum mechanical and WKB wave functions. It is found that the uniform approximation provides a very accurate representation of the exact wave function, even at the turning points of the classical motion, where the WKB approximation diverges. It is demonstrated how correction terms can be constructed, leading to a formal expansion for the exact wave function. The individual terms of this expansion each involve a finite number of reflection points with uniform semiclassical propagation between these reflections. The analysis is extended to multisurface (nonadiabatic) problems, removing the turning point singularities which are generally present in semiclassical formulations of nonadiabatic problems.
The macroscopic linear light scattering theory is extended to the semiclassical level starting with a general discussion of spatially nonlocal scattering. An inte- gral relation for the gauge invariant transverse vector potential is given, stressing similarities and differences between the so-called photon-eye and electron-eye picture. The linear constitutive relations and the causal response tensor are key elements in the semiclassical light scattering theory. Elastic scattering processes are discussed from first principles by making use of the transverse Feynman propagator and the causal response tensor. In scattering from mesoscopic me- dia, e.g, the causal response tensor is a two-point function in direct space. In the spatial dispersion subfield this tensor becomes a one-point function, depending only on the vectorial distance between given source and observation points. In the spatial dispersion approximation a simple relation can be obtained between the selfconsistent and causal response tensors in the wave-vector-frequency do- main. Next, we turn to the scattering matrix concept in quantum physics. The treatment describes the transition from an initial to a final object-field tensor product. These states are connected by the S -matrix operator. A general ex- pression for this is obtained in the Interaction Picture. The reduced S -matrix which relates in a direct manner to the scattering between selected initial and final photon states is introduced. The chapter is finished by the establishment and a discussion of the weakly relativistic interaction Hamiltonian, a key ele- ment in the S -matrix operator.
Heretofore, the hypervirial theorems recently introduced by Hirschfelder have only been applied to bound-state systems. In this paper it is shown that, with certain modifications, these theorems can also be applied to free systems, and, in particular, to scattering problems. The new form of the general hypervirial theorem is derived, and the theory is illustrated with the problem of a particle scattered by a central field. The ordinary virial theorem is deduced, together with other results of physical interest. Both classical and quantum-mechanical formalisms are considered, and in some cases the semiclassical approximation links corresponding results.