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Journal of Contemporary Physics (Armenian Academy of Sciences)

On the problem of quantum tunneling of a singular potential

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

Published Apr 1, 2016• 1 citation
Publisher

Abstract

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.

DOI
10.3103/s1068337216020055

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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.
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In quantum mechanics, the singularity of the potential is usually removed at first, the desired values are computed for the regularized potential, and then an inverse limit transition is made in the resulting expressions. In this paper, the regularization method is used in the quantum tunneling of a power-law potential of the form V(x)=V0/∣x∣αV(x) = {{V}_{0}}/{{\left| x \right|}^{\alpha }}V(x)=V0​/∣x∣α . Тhe reverse limit transition is possible for the range of power 0<α<10 < \alpha < 10<α<1 . The tunneling and reflection time of the wave packet is computed. It is shown that the McColl-Hartman paradox, well-known for ordinary potential barriers, is also valid for the moderately singular potential. The mathematical structure of the uncommon formation of paradox is presented.
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 .