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American Journal of Physics

Quantum Theory of a Square Well Plus Delta Function Potential

W. Edward Gettys

Published May 1, 1973• 2 citations
Publisher

Abstract

The s-wave Schrödinger equation is solved for a square well potential with an attractive delta function at the well edge. This potential provides a number of soluble illustrations of the techniques of quantum mechanics. The bound state eigenvalue problem is solved as is the scattering problem for the cross section. The potential is capable of producing a sharp low energy resonance. The resonances are discussed using both R-matrix and S-matrix methods. Numerical examples are presented.

DOI
10.1119/1.1987329

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The s-wave Schrödinger equation is solved for a square well potential with an attractive delta function at the well edge. This potential provides a number of soluble illustrations of the techniques of quantum mechanics. The bound state eigenvalue problem is solved as is the scattering problem for the cross section. The potential is capable of producing a sharp low energy resonance. The resonances are discussed using both R-matrix and S-matrix methods. Numerical examples are presented.
The quantum-mechanical problem of the bound states corresponding to a sum of several delta-shell potentials is solved exactly as an illustration of the applications of the Lippmann-Schwinger formulation. We obtain the bound-state energies and illustrate their relation to the poles of the corresponding S matrix. In addition, the several-delta-shell-potentials problem also gives a good example for the study of localization of states in an elementary level, and we do this in some detail. It is found that for two well-separated delta-shell potentials, the ground state is localized in the stronger well whereas the excited state is localized in the weaker well. This result is generalized for n delta-shell potentials. A simple proof that n attractive delta-shell potentials cannot have more than n S-wave bound states is also given. Like other classical problems of quantum mechanics, namely the harmonic oscillator and the hydrogen atom, this problem of several delta-shell potentials serves as a good illustration of various physical and mathematical aspects of quantum mechanics.
The discretized Schrödinger equation is most often used to solve one-dimensional quantum mechanics problems numerically. While it has been recognized for some time that this equation is equivalent to a simple tight-binding model and that the discretization imposes an underlying bandstructure unlike free-space quantum mechanics on the problem, the physical implications of this equivalence largely have been unappreciated and the pedagogical advantages accruing from presenting the problem as one of solid-state physics (and not numerics) remain generally unexplored. This is especially true for the analytically solvable discretized finite square well presented here. There are profound differences in the physics of this model and its continuous-space counterpart which are direct consequences of the imposed bandstructure. For example, in the discrete model the number of bound states plus transmission resonances equals the number of atoms in the quantum well.
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.