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arXiv:quant-ph/9911059

Generalized One-Dimensional Point Interaction in Relativistic and Non-relativistic Quantum Mechanics

Takaomi Shigehara, Hiroshi Mizoguchi, Mishima, T., Taksu Cheon

Published Nov 13, 1999• 0 citations
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Abstract

We first give the solution for the local approximation of a four parameter family of generalized one-dimensional point interactions within the framework of non-relativistic model with three neighboring $δ$ functions. We also discuss the problem within relativistic (Dirac) framework and give the solution for a three parameter family. It gives a physical interpretation for so-called $ε$ potential. It will be also shown that the scattering properties at high energy substantially differ between non-relativistic and relativistic cases.

DOI
10.48550/arxiv.quant-ph/9911059

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We first give the solution for the local approximation of a four parameter family of generalized one-dimensional point interactions within the framework of non-relativistic model with three neighboring δδδ functions. We also discuss the problem within relativistic (Dirac) framework and give the solution for a three parameter family. It gives a physical interpretation for so-called εεε potential. It will be also shown that the scattering properties at high energy substantially differ between non-relativistic and relativistic cases.
We consider the one-dimensional quantum mechanical problem of defining interactions concentrated at a single point in the framework of the theory of distributions. The often ill-defined product which describes the interaction term in the Schr\"odinger and Dirac equations is replaced by a well-defined distribution satisfying some simple mathematical conditions and, in addition, the physical requirement of probability current conservation is imposed. A four-parameter family of interactions thus emerges as the most general point interaction both in the non-relativistic and in the relativistic theories (in agreement with results obtained by self-adjoint extensions). Since the interaction is given explicitly, the distributional method allows one to carry out symmetry investigations in a simple way, and it proves to be useful to clarify some ambiguities related to the so-called δ′\delta^\primeδ′ interaction.
There is a four-parameter family of point interactions in one-dimensional quantum mechanics. They represent all possible self-adjoint extensions of the kinetic energy operator. If time-reversal invariance is imposed, the number of parameters is reduced to three. One of these point interactions is the familiar function potential but the other generalized ones do not seem to be widely known. We present a pedestrian approach to this subject and comment on a recent controversy in the literature concerning the so-called interaction. We emphasize that there is little resemblance between the interaction and what its name suggests.
We consider a four-parameter family of point interactions in one dimension. This family is a generalization of the usual -function potential. We examine a system consisting of many particles of equal masses that are interacting pairwise through such a generalized point interaction. We follow McGuire who obtained exact solutions for the system when the interaction is the -function potential. We find exact bound states with the four-parameter family. For the scattering problem, however, we have not been so successful. This is because, as we point out, the condition of no diffraction that is crucial in McGuire's method is not satisfied except when the four-parameter family is essentially reduced to the -function potential.
A proposal is given, of how to implement point interactions and boundary conditions in the path integral. The starting point is a path-integral formulation of a Dirac particle in one dimension. The implementation of a point interaction yields, by means of a perturbation expansion, the corresponding Green function for a relativistic particle. In the non-relativistic limit several cases can be distinguished depending on which of the four possibilities of the point interaction has been chosen. By a proper combination of the various possibilities of implementing the point interaction, the whole range of the four-parameter family of boundary conditions for point interactions can be exploited, in the relativistic case as well as in the non-relativistic limit. In addition, making the strength of the point interaction infinitely repulsive yields boundary conditions at finite points on the real line. In particular, Dirichlet and Neumann boundary conditions emerge. The method is illustrated with some examples.
We propose a classification scheme for the complete family of 1D point interactions. To do so, we first review the solutions of the wavefunctions and Green functions of the problem. Second, we derive the exact time-dependent propagators in such a way that we can write the expressions for the K 's in a very compact form. As they should, such expressions do reduce to the known results in the literature according to the potential parameter values. Then, we analyse in general terms how the different point interactions scatter off arbitrary initially localized wave packets. Finally, we show that the physical features associated with the scattering process can be used to establish a classification procedure. Moreover, these physical characteristics are directly related to the potential parameters leading to the many formulae for the K 's. As an application, we present numerical calculations for Gaussian wave packets.