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American Journal of Physics

Interesting features of relativistic and nonrelativistic quantal bound states in one, two, and three dimensions

Basudeb Sahu, M. Z.Rahman Khan, C. S. Shastry, Bishwajyoti Dey, S. C. Phatak

Published Oct 1, 1989• 6 citations

Abstract

Various aspects of critically bound quantal ground states close to the threshold in one, two, and three dimensions are examined in nonrelativistic quantum mechanics using attractive square-well and delta-function potentials. The mathematical and physical reasons are presented for the nonoccurrence of the bound state in the case of three dimensions and its occurrence in one and two dimensions for infinitesimally small strengths and ranges of the potential. Also analyzed is the condition for critical binding in relativistic quantum mechanics using the Klein–Gordon equation in one, two, and three dimensions. Corresponding features for the case of the Dirac equation in one and three dimensions are briefly discussed.

DOI: 10.1119/1.15841

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Various aspects of critically bound quantal ground states close to the threshold in one, two, and three dimensions are examined in nonrelativistic quantum mechanics using attractive square-well and delta-function potentials. The mathematical and physical reasons are presented for the nonoccurrence of the bound state in the case of three dimensions and its occurrence in one and two dimensions for infinitesimally small strengths and ranges of the potential. Also analyzed is the condition for critical binding in relativistic quantum mechanics using the Klein–Gordon equation in one, two, and three dimensions. Corresponding features for the case of the Dirac equation in one and three dimensions are briefly discussed.

The Schrödinger, Klein–Gordon, and Dirac equations for the hydrogen atom are solved in D dimensions, where D may be noninteger. For D=1 the nondegenerate ground state with infinite binding energy is obtained as a limit as D tends to 1 for the Schrödinger equation. The corresponding solution of the Klein–Gordon equation is shown to be unacceptable, contradicting previous work. Unexpectedly, no bound state solutions of the Dirac equation with D=1 are found at all. A few comments are made on the D=3 and 2 cases.

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.

In a previous note, the nonrelativistic bound state wavefunction W(x) was written as Wo(x)W1(x) with Wo(x) being the ground state wavefunction. W1(x) satisfies an eigenvalue equation, similar to the Schrodinger equation, but with the potential term replaced with a coupling term: -1/2m d/dx d/dx W1(x) - 1/m d/dx W1(x) [ d/dx Wo(x)/Wo(x) ] = (E-Eo) W1 (x) ((1)) Mathematically, ((1)) may be compared to the hypergeometric differential equation and solutions found for the Schrodinger if coefficients meet certain criteria. A similar approach applies to the Klein Gordon and Dirac equations In this note, we wish to examine possible physical significance of Wo(x)W1(x). In addition, we wish to obtain an equation, analogous to ((1)) for both the Klein Gordon and Dirac equations to compare the coupling terms.