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Journal of Contemporary Physics (Armenian Academy of Sciences)

Regularization of Quantum Tunneling of Singular Potential Barrier

G. A. Muradyan

Published Oct 1, 2019• 3 citations
Publisher

Abstract

The quantum tunneling of a singular potential is modeled, as a rule, by the method of regularization. It proceeds from the intermediate usage of a nonsingular-type preimage of the potential function. In this work, it has been ascertained that the preimages continuously differentiable at the point of singularity does not reproduce the singularity in the problem of quantum tunneling.

DOI
10.3103/s1068337219040017

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The quantum tunneling of a singular potential is modeled, as a rule, by the method of regularization. It proceeds from the intermediate usage of a nonsingular-type preimage of the potential function. In this work, it has been ascertained that the preimages continuously differentiable at the point of singularity does not reproduce the singularity in the problem of quantum tunneling.
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.
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.
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.
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 .
A Brief History of Quantum Tunneling Some Basic Questions Concerning Quantum Tunneling Simple Solvable Problems Time-Dependence of the Wave Function in One-Dimensional Tunneling Semiclassical Approximations Generalization of the Bohr - Sommerfeld Quantization Rule and its Application to Quantum Tunneling Gamow's Theory, Complex Eigenvalues, and the Wave Function of a Decaying State Tunneling in Symmetric and Asymmetric Local Potentials and Tunneling in Nonlocal and Quasi-Solvable Barriers Classical Descriptions of Tunneling Tunneling in Time-Dependent Barriers Decay Width and Scattering Theory The Method of Variable Reflection Amplitude Applied to Solve Multichannel Tunneling Problems Path Integral and Its Semi-Classical Approximation in Quantum Tunneling Heisenberg's Equations of Motion for Tunneling Wigner Distribution Function in Quantum Tunneling Decay Widths of Siegert States, Complex Scaling and Dilatation Transformation Multidimensional Quantum Tunneling Group and Signal Velocities Time-Delay, Reflection Time Operator and Minimum Tunneling Time More about Tunneling Time Tunneling of a System with Internal Degrees of Freedom Motion of a Particle in a Waveguide with Variable Cross Section and in a Space Bounded by a Dumbbell-Shaped Object Relativistic Formulation of Quantum Tunneling Inverse Problems of Quantum Tunneling Some Examples of Quantum Tunneling in Atomic and Molecular Physics Some Examples from Condensed Matter Physics Alpha Decay.