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American Journal of Physics

Regularization methods for delta-function potential in two-dimensional quantum mechanics

Su-Long Nyeo

Published Jun 1, 2000• 41 citations
Publisher

Abstract

The quantum mechanics of a bound particle in the delta-function potential in two dimensions is studied with a discussion of its regularization and renormalization. A simple regularization approach is considered with the introduction of several regularizing functions for defining the quantum system. More systematic regularization is introduced from the mathematical viewpoint of the theory of distributions. The renormalization scheme independence of the physical observable is demonstrated.

DOI
10.1119/1.19485

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The quantum mechanics of a bound particle in the delta-function potential in two dimensions is studied with a discussion of its regularization and renormalization. A simple regularization approach is considered with the introduction of several regularizing functions for defining the quantum system. More systematic regularization is introduced from the mathematical viewpoint of the theory of distributions. The renormalization scheme independence of the physical observable is demonstrated.
A path-integral approach for \ensuremath{\delta}-function potentials is presented. Particular attention is paid to the two-dimensional case, which illustrates the realization of a quantum anomaly for a scale-invariant problem in quantum mechanics. Our treatment is based on an infinite summation of perturbation theory that captures the nonperturbative nature of the \ensuremath{\delta}-function bound state. The well-known singular character of the two-dimensional \ensuremath{\delta}-function potential is dealt with by considering the renormalized path integral resulting from a variety of schemes: dimensional, momentum-cutoff, and real-space regularization. Moreover, compatibility of the bound-state and scattering sectors is shown.
When dealing with singular potentials, such as point interactions, the term V ( x )ψ( x ) in Schrödinger's equation is, in general, mathematically ill-defined. The traditional way of dealing with this difficulty, by employing regularization techniques, has led to some ambiguous results for the δ'( x ) potential in the literature. Here we propose a mathematically consistent approach to deal with the one-dimensional version of this problem by considering from the beginning the distributional nature of the whole interaction term. We show that the interaction is univocally determined from two simple mathematical requirements on the interaction distribution, together with the physical requirement of probability flux conservation and considerations of symmetry.
As in quantum field theory, for some singular two-body potentials, the momentum-space perturbative treatment of the three-dimensional quantum NNN-body problem is shown to exhibit ultraviolet divergences. Renormalization of these models leads to a scale and finite observables. For a Dirac delta potential we carry on this renormalization for two- and three-body systems. A new divergence, and hence a new scale, emerges for each added particle. For $N\ensuremath{\ge}3$ there are no divergences in one and two dimensions. The physical implications of these results are discussed.
It is well-known that a delta potential well in 1D has only one bound state but that in 3D it supports an {\it infinite} number of bound states with {\it infinite} binding energy for the lowest level. We show how this also holds for the less familiar 2D case, and then discuss why this makes 3D delta potential wells unphysical as models of interparticle interactions for condensed-matter many-body systems. However, both 2D and 3D delta wells can be ``regularized'' to support a single bound level which in turn renders them conveniently simple single-parameter interactions, e.g., for modeling the pair-forming dynamics of quasi-2D superconductors such as the cuprates, or in 3D of other superconductors and of neutral-fermion superfluids such as ultra-cold trapped Fermi gases.