© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2102.12789 · Journal of Contemporary Physics (Armenian Academy of Sciences)

On Quantum Tunneling of a Singular Potential

A. Zh. Muradyan

Published Apr 1, 2021• 2 citations

Abstract

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.

DOI: 10.3103/s1068337221020122

Related papers

On the quantum tunneling of one dimensional coulomb singular potential barrier

Atom Zh. Muradyan, G. A. Muradyan

How quantum tunneling will behave when the singularity is preserved as much as possible is the main question of this paper. We get that the Coulomb sibgularity is reflected as infinitly accelerated oscillations in the transmission coefficient between zero and one when the incident particles energy approaches the zero boundary. At relatively high energies, the tunneling acqures the caracter inherent in regular potentials, and becomes completly transparent in the asymptote.

arXiv:2008.12957•Aug 29, 2020•1 citation

Quantum Tunneling Time of a Moderately Singular Potential: Regularization Method

A. Zh. Muradyan

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 $$V(x) = {{V}_{0}}/{{\left| x \right|}^{\alpha }}$$ . Тhe reverse limit transition is possible for the range of power $$0 < \alpha < 1$$ . 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.

Journal of Contemporary Physics (Armenian Academy of Sciences)•Sep 1, 2022•0 citations

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

Tunneling through a singular potential barrier

J. Dittrich, Pavel Exner

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 .

Journal of Mathematical Physics•Aug 1, 1985•56 citations

Exact solutions to fall of particle to singular potential: classical versus quantum cases

Michael I. Tribelsky

Exact solutions describing a fall of a particle to the centre of a non-regularized singular potential in classical and quantum cases are obtained and compared. We inspect the quantum problem with the help of the conventional Schrödinger’s equation. During the fall, the wave function spatial localization area contracts into a single zero-dimensional point. For the fall-admitting potentials, the Hamiltonian is non-Hermitian. Because of that, the wave function norm occurs time-dependent. It demands an extension to this case of the continuity equation and rules for mean value calculations. Surprisingly, the quantum and classical solutions exhibit striking similarities. In particular, both are self-similar as the particle energy equals zero. The characteristic spatial scales of the quantum and classical self-similar solutions obey the same temporal dependence. We present arguments indicating that these self-similar solutions are attractors to a broader class of solutions, describing the fall at finite energy of the particle.

arXiv:2202.12615•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Sep 1, 2023•4 citations

Regularization of Quantum Tunneling of Singular Potential Barrier

G. A. Muradyan

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.

Journal of Contemporary Physics (Armenian Academy of Sciences)•Oct 1, 2019•3 citations

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) = {{V}_{0}}/{{\left| x \right|}^{\alpha }}

. Тhe reverse limit transition is possible for the range of power

0 < \alpha < 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 .

Exact solutions describing a fall of a particle to the centre of a non-regularized singular potential in classical and quantum cases are obtained and compared. We inspect the quantum problem with the help of the conventional Schrödinger’s equation. During the fall, the wave function spatial localization area contracts into a single zero-dimensional point. For the fall-admitting potentials, the Hamiltonian is non-Hermitian. Because of that, the wave function norm occurs time-dependent. It demands an extension to this case of the continuity equation and rules for mean value calculations. Surprisingly, the quantum and classical solutions exhibit striking similarities. In particular, both are self-similar as the particle energy equals zero. The characteristic spatial scales of the quantum and classical self-similar solutions obey the same temporal dependence. We present arguments indicating that these self-similar solutions are attractors to a broader class of solutions, describing the fall at finite energy of the particle.