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Physical Review A

Boundary conditions across a δ-function potential in the one-dimensional Dirac equation

Chandana Roy

Published Apr 1, 1993• 26 citations
Publisher

Abstract

We have developed a unified perspective for boundary conditions concerned with (I) a direct relativistic treatment of a \ensuremath{\delta}-function potential and (II) the \ensuremath{\delta}-function limit of solutions for relativistic treatment of peaked potential. We have presented a critical discussion of our findings, especially in the context of (i) certain conditions related very importantly to this unified perspective and (ii) the difficulty in making a choice between the approaches (I) and (II) on the basis of mathematical analyses alone.

DOI
10.1103/physreva.47.3417

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We have developed a unified perspective for boundary conditions concerned with (I) a direct relativistic treatment of a \ensuremath{\delta}-function potential and (II) the \ensuremath{\delta}-function limit of solutions for relativistic treatment of peaked potential. We have presented a critical discussion of our findings, especially in the context of (i) certain conditions related very importantly to this unified perspective and (ii) the difficulty in making a choice between the approaches (I) and (II) on the basis of mathematical analyses alone.
#delta#'-function perturbations and Neumann boundary conditions are incorporated into the path integral formalism. The starting point is the consideration of the path integral representation for the one dimensional Dirac particle together with a relativistic point interaction. The non-relativistic limit yields either a usual #delta#-function or a #delta#'-function perturbation; making their strengths infinitely repulsive one obtains Dirichlet, respectively Neumann boundary conditions in the path integral. (orig.)
delta '-function perturbations and Neumann boundary conditions are incorporated into the path integral formalism. The starting point is consideration of the path integral representation for the one-dimensional Dirac particle together with a relativistic point interaction. The nonrelativistic limit yields either the usual delta -function or a delta '-function perturbation; making their strengths infinitely repulsive one respectively obtains either Dirichlet or Neumann boundary conditions in the path integral.
We revisit the discussion about the boundary condition at the origin in the Schrödinger radial equation for central potentials. We give a transparent and convincing reason for demanding the radial part R(r) of the wave function to be finite at r = 0, showing that if R(0) diverges the complete wave function ψ does not satisfy the full Schrödinger equation. If R(r) is singular, we show that the corresponding ψ follows an equation similar to Schrödinger's, but with an additional term involving the Dirac delta function or its derivatives at the origin. Although, in general, understanding some of our arguments requires certain knowledge of the theory of distributions, the important case of a behavior R ∝ 1/r near r = 0, which gives rise to a normalizable ψ, is especially simple: The origin of the Dirac delta term is clearly demonstrated by using a slight modification of the usual spherical coordinates. The argument can be easily followed by undergraduate physics students.
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.