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Journal of Nanhua University

An Application of Regularization Methods for Delta-Function Potential in Three-Dimensional Quantum Mechanics

HE Dong-hui

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

The quantum mechanics of a bound particle in the delta-function potential in three dimensions is studied with a discussion of its regularization and renormalization.A simple regularization approach is considered with introduction of several regularizing functions for defining the quantum system.Furthermore,more systematic regularization is introduced from the mathematical viewpoint of the theory of distributions.

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arXiv:2101.07876•Jan 18, 2021•0 citations
The quantum mechanics of a bound particle in the delta-function potential in three dimensions is studied with a discussion of its regularization and renormalization.A simple regularization approach is considered with introduction of several regularizing functions for defining the quantum system.Furthermore,more systematic regularization is introduced from the mathematical viewpoint of the theory of distributions.
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.
This paper presents several cases in which the effects of the addition of a delta function potential on bound states can be computed exactly. In the case of the one dimensional Schrödinger equation, a Green’s function technique is used to compute an exact implicit expression that gives the effect of the one−dimensional delta function potential on the bound states eigenvalues. It is also shown that the weak coupling limit of the implicit expression agrees with the perturbative treatment of the delta function. It is pointed out that this technique cannot be applied to obtain an exact treatment of the delta function potential δ3(r) in the three−dimensional Schrödinger equation. Instead, an alternate definition of the delta function in three dimensions as a limit of a sequence of square well potentials is used. Unlike the one dimensional case, the exact effect on the eigenvalue does not agree in the limit of weak coupling with the perturbative results. This indicates that the three−dimensional delta function potential should not be used to approximate strongly coupled short range potentials for which perturbation theory is inadequate.
Unlike the situation for the 1d Dirac delta derivative Schrodinger pseudo potential (SPP) and the 2d Dirac delta SPP, where the indeterminacy originates from a lack of scale in the first and both a lack of scale as well as the wave function not being well defined at the support of the generalized function SPP; the obstruction in 3d Euclidean space for the Schrodinger equation with the Dirac delta as a SPP only comes from the wave function (the L2L^2L2 bound sate solution) being singular at the compact point support of the Dirac delta function (measure). The problem is solved here in a completely mathematically rigorous manner with no recourse to renormalization nor regularization. The method involves a distributionally generalized version of the Schrodinger theory as developed by the author, which regards the formal symbol "HψHψHψ" as an element of the space of distributions, the topological dual vector space to the space of smooth functions with compact support. Two main facts come to light. The first is the bound state energy of such a system can be calculated in a well-posed context, the value of which agrees with both the mathematical and theoretical physics literature. The second is that there is then a rigorous distributional version of the Hellmann-Feynman theorem.