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American Journal of Physics

Applications of the eikonal approximation in quantum mechanical scattering

Barry R. Holstein

Published May 1, 2023• 3 citations
Publisher

Abstract

Scattering is an important component of any quantum mechanics course. However, the scattering amplitude in the case of a general potential is often calculated using the simple Born approximation, which does not embed general properties such as unitarity or analyticity. We show that a relatively simple extension, the eikonal approximation, offers a significant improvement and demonstrate this in the case of the electromagnetic and gravitational interactions.

DOI
10.1119/5.0077649

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Scattering is an important component of any quantum mechanics course. However, the scattering amplitude in the case of a general potential is often calculated using the simple Born approximation, which does not embed general properties such as unitarity or analyticity. We show that a relatively simple extension, the eikonal approximation, offers a significant improvement and demonstrate this in the case of the electromagnetic and gravitational interactions.
The theory of scattering is a very important part of the quantum mechanics.The Born approximation and Eikonal approximation of the Scattering cross-section are calculated with the Gauss potential and Lenz-jensen potential under a particular incident energy.The results show that the bigger the Electronic energy is,the smaller the total scattering crossection is and the weaker the electron scattering of potential field is.There is little influence of potential to the energetic particle about the Lenz-jensen potential.And the majority of scattered particles are concentrated on a narrow range.
Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.
A development of the scattering amplitude in potential scattering which is based upon replacing a smooth potential by a sequence of potential steps is given. The Schrödinger equation is solved exactly in each interval over which the potential is constant and by matching boundary conditions an expression relating the coefficients of two linearly independent solutions at small radius to those at large radius is found. From this the scattering amplitude is calculated. The limits of validity are found to be dependent upon the value of the expression (r/2)ddrln(2mℏ2[E−V]). Limiting cases of the amplitude are shown to be a modified WKB approximation and the Born approximation. A particular application to the Yukawa potential is discussed. The discussion is restricted to attractive potentials.
A self-contained discussion of nonrelativistic quantum mechanical potential scattering in two dimensions is presented. The discussion includes, among other topics, partial wave decomposition in coordinate and momentum space, Lippmann–Schwinger integral equations of scattering for the scattering wavefunction and the transition operator, optical theorem, and the unitarity relation for the transition operator. The present definition of the scattering amplitude in terms of the asymptotic wavefunction differs from the usual definition. The present definition has certain advantages, for example, in writing the optical theorem and in studying the analytical properties of the scattering amplitude.
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.