An exact treatment of the Dirac delta function potential in the Schrödinger equation

Abstract

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.