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Progress of Theoretical Physics

Potential Scattering in One Dimension

T. Ishihara

Published Oct 1, 1975• 2 citations
Publisher

Abstract

General solution is derived in operator form on the one-dimensional potential scattering problem without any approximation. Backscattering is included explicitly in this formulation. Coordinate representation is presented and its relation to the eikonal approximation is discussed. Effects of backscattering are examined numerically in the case of a square-well potential.

DOI
10.1143/ptp.54.1106

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Chapter 9. Scattering in One Dimension

The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend

Sciendo eBooks•Sep 1, 2018•0 citations
General solution is derived in operator form on the one-dimensional potential scattering problem without any approximation. Backscattering is included explicitly in this formulation. Coordinate representation is presented and its relation to the eikonal approximation is discussed. Effects of backscattering are examined numerically in the case of a square-well potential.
Starting from the Lippmann-Schwinger equation, we give two kinds of new corrections in three-dimensional potential scattering. The one includes explicitly backscattering effects along the z-axis (to-and-fro motion). The other incorporates the Fresnel diffraction with the to-and-fro motion. This theory gives a clear physical image of the forward and backward propagation of a wave. Choice of the z-axis along ki + kf is used to facilitate calculation. Relation to the eikonal approximation is easily read from resultant expressions of scattering amplitude.
The problem of determining a scattering potential from scattered field data is addressed within the two integral equation formulation of scattering theory first proposed by Goldberger (1964). The lowest-order approximation within this formulation casts the inverse problem entirely in terms of the Heitler equation which relates the on-shell T matrix directly to the scattering potential. This lowest-order approximation is applied to the case of non-relativistic quantum mechanical inverse scattering from one-dimensional local potentials and is found to yield an exact solution for the case of a one-dimensional delta function potential.
Abstract We discuss a general description of the solutions to the 1D time-independent Schrödinger equation that does not a priori distinguish between scattering states and bound states and emphasizes and reinforces their relationship and connection to each other. This manuscript also introduces the concept of transfer matrices, which it presents as a logical extension of the traditional approach to evaluating 1D potentials. Using the transfer matrix method and a finite step approximation allows for a simple and straight-forward numerical solution of arbitrary 1D potentials. It also separates the process of solving the Schrödinger equation from selecting physically relevant solutions, which is a critical skill in quantum mechanics and is at the core of physics problems in general.
In this paper we extend our previous analysis of the scattering of wave packets in one dimension to the case of the square-well potential. The analytic properties of the general scattering solution are emphasized thereby making the analysis useful as introductory material for a more sophisticated S-matrix treatment. The square-well model is particularly interesting because of its application to the deuteron problem. Resonance scattering, barrier penetration, time delay, and line shape are discussed at the level of the first-year graduate student.
The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend