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American Journal of Physics

The problem of several delta-shell potentials in the Lippmann–Schwinger formulation

A. K. Jain, S. K. Deb, C. S. Shastry

Published Feb 1, 1978• 11 citations

Abstract

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.

DOI: 10.1119/1.11375

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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.

Abstract In this paper, we propose a pedagogical presentation of the Lippmann–Schwinger equation as a powerful tool, so as to obtain important scattering information. In particular, we consider a one-dimensional system with a Schrödinger-type free Hamiltonian decorated with a sequence of N attractive Dirac delta interactions. We first write the Lippmann–Schwinger equation for the system and then solve it explicitly in terms of an N × N matrix. Then, we discuss the reflection and the transmission coefficients for an arbitrary number of centres and study the threshold anomaly for the N = 2 and N = 4 cases. We also study further features like the quantum metastable states and resonances, including their corresponding Gamow functions and virtual or antibound states. The use of the Lippmann–Schwinger equation simplifies our analysis enormously and gives exact results for an arbitrary number of Dirac delta potentials.

The quantum description of a free particle subject to a delta circular ring and a delta spherical shell is studied. We determine the bound eigenstates and energy spectra for different angular momentum states. The normalization constants of the bound states and the transmission and reflection coefficients of the scattering states are calculated in closed form. The ground state of the delta ring is always present regardless of the potential strength. In contrast, the delta shell requires a threshold of the delta strength for the first eigenstate to appear. For the ring and the shell, the wave functions tend to be degenerate at high energies, and the effect of angular momentum on the wave functions becomes negligible. This work generalizes the standard problem of the one-dimensional delta potential studied in textbooks.

Abstract We proceed in this chapter to discuss scattering theory and related aspects of the bound-state problem in quaternionic quantum mechanics. We begin with a detailed discussion of a simple one-dimensional example, the delta function potential model. From this model will emerge the result (Adler, 1988) that will be the principal theme of this and the succeeding two chapters: with the standard ray representative choice of Sec. 4.2, for short-range potentials the asymptotic scattering states and the S-matrix in quaternionic quantum mechanics are always complex ℂ(1, i).

The problem of N bosons interacting with each other via repulsive delta-function interactions and with an external, attractive, delta-function potential is solved, within the Hartree approximation, exactly. If the interparticle interactions are above a certain value, there is no bound state. Thus the bound state does not just expand to compensate for the increase in the repulsive Hartree potential. Moreover as the interaction strength is increased to that value, the ground-state wavefunction develops a pole at the position of the attractive potential.