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American Journal of Physics

Singular potentials in one dimension

M. Andrews

Published Nov 1, 1976• 101 citations
Publisher

Abstract

In quantum mechanics of one dimension it is shown for potentials which become infinite at a point but are continuous elsewhere, that the singularity acts as an impenetrable barrier if the potential is not integrable up to the singularity, but if the potential is integrable the behavior is not essentially different from that of a potential which does not become infinite.

DOI
10.1119/1.10585

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On the problem of quantum tunneling of a singular potential

G. A. Muradyan, H. R. Hakobyan, B. V. Balasanyan

The problem of quantum tunneling through the singular potential barrier $$V(x) = \left\{ {\begin{array}{*{20}c} {V_0 (b/x - x/a)^2 ,} & {0 < \left| x \right| \leqslant \sqrt {ab} } \\ {0,} & {\left| x \right| > \sqrt {ab} } \\ \end{array} } \right.$$ is discussed on the subject of the possibility to replace the singular behavior of the problem at the point x = 0 by a limiting process at the top of the truncated potential. The validity of such a replacement and, on this basis, the zero transparency of the quantum potential barrier are shown.

Journal of Contemporary Physics (Armenian Academy of Sciences)•Apr 1, 2016•1 citation
In quantum mechanics of one dimension it is shown for potentials which become infinite at a point but are continuous elsewhere, that the singularity acts as an impenetrable barrier if the potential is not integrable up to the singularity, but if the potential is integrable the behavior is not essentially different from that of a potential which does not become infinite.
The quantum mechanical problem of a particle moving in one dimension (1D) under the influence of a singular potential is reexamined. An attractive delta function potential of infinite strength and the 1D Coulomb potential is considered in particular. In both cases the singularity acts as an impenetrable barrier, thus dividing space into two independent regions. Analysis of the scattering states gives results in agreement with this conclusion. The implications that the bound states of the 1D Coulomb potential cannot be classified according to parity, even though the potential is symmetric, is also discussed herein.
Quantum tunneling of a nonrelativistic particle through a singular potential barrier V is studied on the line. The Hamiltonian is a self-adjoint extension of the operator H1=−d2/dx2+V(x). If H1 is essentially self-adjoint on its natural domain, the tunneling is forbidden for a class of potentials that includes all semiclassically impenetrable barriers. If H1 is not essentially self-adjoint, the Friedrichs extension of H1 yields no tunneling for another class of potentials which again includes the semiclassically impenetrable ones. In general, the occurrence of tunneling is not excluded and depends on the self-adjoint extension we choose as the Hamiltonian of our problem. As an example, we evaluate the transmission coefficient for all self-adjoint extensions of the operator H1 referring to V(x)=gx−2 with 0<g< 3/4 .
"We derive the expression V (x) u(x) = c δ(x − a) + v(x) u(x) (where V (x) is the potential, u(x) the wave function, c a constant and v(x) a finite potential function for x ≤ a), which is present in the one-dimensional Schr ̈ dinger equation on the whole real line when we have o an impenetrable barrier at x ≥ a, that is, an infinite step potential there. By studying the solution of this equation, we identify, connect and discuss three different Hamiltonian operators that describe the barrier. We extend these results by constructing an infinite square-well potential from two impenetrable barriers."
Abstract To get around the difficulty that the singular nature of the potential function V(x) = V0/|x|α introduces into physics, the regularization methods are used. However, they affect the singular nature of the problem, and so we discuss here how quantum tunneling behaves if the original singular nature of the Schrodinger equation remains unperturbed. For this purpose, I am starting from the precondition that the singular terms are mutually compensated in the current probability density and the current can be considered continuous. As a result, it is obtained that the mild-singular potential (with 0 < α < 1) has finite but unusual tunnel transparency, in particular, a non-zero value at zero energy of the incident particle. The transparency of a Coulomb potential well oscillates infinitely at zero energy, and only the strongly singular potential (with α > 1) repeats the property of the regularized prototypes: to be impenetrable.
The problem of quantum tunneling through the singular potential barrier $$V(x) = \left\{ {\begin{array}{*{20}c} {V_0 (b/x - x/a)^2 ,} & {0 < \left| x \right| \leqslant \sqrt {ab} } \\ {0,} & {\left| x \right| > \sqrt {ab} } \\ \end{array} } \right.$$ is discussed on the subject of the possibility to replace the singular behavior of the problem at the point x = 0 by a limiting process at the top of the truncated potential. The validity of such a replacement and, on this basis, the zero transparency of the quantum potential barrier are shown.