© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Journal of Physics A Mathematical and General

An asymptotic formula for energy eigenvalues in the bound-state Aharonov-Bohm effect

Ranjith Ananda Samandra, W P Healy

Published Nov 27, 1998• 2 citations

Abstract

An asymptotic formula for the energy eigenvalues of a charged particle contained in a cylindrical shell is derived. The formula is used to estimate the eigenvalues when there is a cylindrically symmetric static magnetic-induction field present and the system exhibits the bound-state Aharonov-Bohm effect. It is shown that the errors involved form an sequence with terms labelled by the radial quantum number. Numerical approximations obtained from the asymptotic formula are compared to those obtained from analytical solutions of Schrödinger's equation in the following two cases. (i) There is no magnetic-induction field in the shell itself but there is a field inside the inner cylinder. (ii) There is a constant magnetic-induction field in the shell as well as a (possibly different) field inside the inner cylinder. In these cases, the radial eigenfunctions are expressible in terms of Bessel functions and confluent hypergeometric functions, respectively. The numerical results are consistent with the asymptotic character of the formula for the energy eigenvalues.

DOI: 10.1088/0305-4470/31/47/014

Related papers

Aharonov-Bohm Effect on Landau States in Annular Cylindrical Boxes

E. Ley‐Koo, G. Villa‐Torres, Dmitrii Kouznetsov

The Schrodinger equation for an electron inside an annular cylindrical box and in the presence of an axial uniform magnetic field is solved in two comparative situations: i) when the magnetic induction ~ B0 = kB0 is the same in the central perforation and in the box, and ii) when its values in the perforation ~ Bi = kBi and in the box ~ B0 = kB0 are different. The Aharonov-Bohm effect on the Landau states of the confined electron is exhibited through the analysis of the dependence of the energy eigenvalues and eigenfunctions on the difference of the magnetic flux in the perforation as B i i B0 changes.

Chinese Journal of Physics•Apr 1, 2002•3 citations

Revisiting a Coulomb-type potential inside a nonconducting cylinder in a rotating reference frame

K. Bakke

We explore a mesoscopic system made of a long nonconducting cylinder of inner radius [Formula: see text] where a spinless electron interacts with an attractive [Formula: see text] potential. We analyze the asymptotic behavior of the radial wave function by assuming that there is an impenetrable potential wall located at [Formula: see text], and thus, discuss the effects of rotation on the energy levels, as well as the Aharonov–Bohm effect for bound states. Further, we include a non-Hermitian operator and show that the non-Hermitian Hamiltonian has real energy eigenvalues. We also discuss the effects of rotation and the Aharonov–Bohm effect from the real energy eigenvalues. We extend our discussion to the revival time.

International Journal of Modern Physics A•Feb 7, 2025•2 citations

Aharonov–Bohm effect from an attractive r−2 scalar potential together with a r−1 scalar potential

J. Carvalho Veloso, K. Bakke

By confining a point charge to an attractive [Formula: see text] potential together with a [Formula: see text] scalar potential, we study the Aharonov–Bohm effect for bound states. In view of the presence of the centrifugal term and the attractive [Formula: see text] potential, we analyze the case where the attractive [Formula: see text] potential is the dominant term, and that one where the centrifugal term is the dominant term. In both cases, we show that the bound states arise around the inner radius of the long nonconducting cylinder.

International Journal of Modern Physics A•Oct 23, 2025•0 citations

Missing He–McKellar–Wilkens geometric quantum phase

K. Bakke

We discuss the appearance of a missing geometric quantum phase from the interaction of the permanent electric dipole moment of a neutral particle with the magnetic field produced by a uniform distribution of the magnetic charges inside an infinitely long non-conductor cylinder with an inner radius r a . This geometric quantum phase is associated with the missing magnetic charge per unit length in a region with a cylindrical shape of radius r a and gives rise to an analogue of the He–McKellar–Wilkens geometric quantum phase. By searching for bound state solutions, we show that an Aharonov–Bohm-type effect arises from the influence of the missing geometric quantum phase on the eigenvalues of energy.

Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Mar 1, 2022•4 citations

Scattering of a charged particle from a hard cylindrical solenoid: Aharonov-Bohm Effect

Oktay Yilmaz

The scattering amplitude of a charged particle from a long hard cylinderical solenoid is derived by solving the time independent Schrödinger equation on a double connected plane. It is a summation over the angular momentum quantum number (partial wave summation). It is shown that only negative mechanical angular momenta contribute to the amplitude when the radius of the solenoid goes to zero limit without varying the magnetic induction flux (Flux line). Original Aharonov-Bohm result is obtained with this limit.

arXiv:1311.6924•Chinese Journal of Physics•Nov 27, 2013•2 citations

Aharonov-Bohm effect in quantum-to-classical correspondence of the Heisenberg principle

De‐Hone Lin, Jee‐Gong Chang, Chi‐Chuan Hwang

The exact energy spectrum and wave function of a charged particle moving in the Coulomb field and Aharonov-Bohm's magnetic flux are solved by the nonintegrable phase factor. The universal formula for the matrix elements of the radial operator ${r}^{\ensuremath{\alpha}}$ of arbitrary power $\ensuremath{\alpha}$ is given by an analytical solution. The difference between the classical limit of matrix elements of inverse radius in quantum mechanics and the Fourier components of the corresponding quantity for the pure Coulomb system in classical mechanics is examined in reference to the correspondence principle of Heisenberg. Explicit calculation shows that the influence of nonlocal Aharonov-Bohm effect exists even in the classical limit. The semiclassical quantization rule for systems containing the topological effect is presented in the light of Heisenberg's corresponding principle.

Physical Review A•Apr 22, 2003•4 citations

An asymptotic formula for the energy eigenvalues of a charged particle contained in a cylindrical shell is derived. The formula is used to estimate the eigenvalues when there is a cylindrically symmetric static magnetic-induction field present and the system exhibits the bound-state Aharonov-Bohm effect. It is shown that the errors involved form an sequence with terms labelled by the radial quantum number. Numerical approximations obtained from the asymptotic formula are compared to those obtained from analytical solutions of Schrödinger's equation in the following two cases. (i) There is no magnetic-induction field in the shell itself but there is a field inside the inner cylinder. (ii) There is a constant magnetic-induction field in the shell as well as a (possibly different) field inside the inner cylinder. In these cases, the radial eigenfunctions are expressible in terms of Bessel functions and confluent hypergeometric functions, respectively. The numerical results are consistent with the asymptotic character of the formula for the energy eigenvalues.

The Schrodinger equation for an electron inside an annular cylindrical box and in the presence of an axial uniform magnetic field is solved in two comparative situations: i) when the magnetic induction ~ B0 = kB0 is the same in the central perforation and in the box, and ii) when its values in the perforation ~ Bi = kBi and in the box ~ B0 = kB0 are different. The Aharonov-Bohm effect on the Landau states of the confined electron is exhibited through the analysis of the dependence of the energy eigenvalues and eigenfunctions on the difference of the magnetic flux in the perforation as B i i B0 changes.

We explore a mesoscopic system made of a long nonconducting cylinder of inner radius [Formula: see text] where a spinless electron interacts with an attractive [Formula: see text] potential. We analyze the asymptotic behavior of the radial wave function by assuming that there is an impenetrable potential wall located at [Formula: see text], and thus, discuss the effects of rotation on the energy levels, as well as the Aharonov–Bohm effect for bound states. Further, we include a non-Hermitian operator and show that the non-Hermitian Hamiltonian has real energy eigenvalues. We also discuss the effects of rotation and the Aharonov–Bohm effect from the real energy eigenvalues. We extend our discussion to the revival time.

By confining a point charge to an attractive [Formula: see text] potential together with a [Formula: see text] scalar potential, we study the Aharonov–Bohm effect for bound states. In view of the presence of the centrifugal term and the attractive [Formula: see text] potential, we analyze the case where the attractive [Formula: see text] potential is the dominant term, and that one where the centrifugal term is the dominant term. In both cases, we show that the bound states arise around the inner radius of the long nonconducting cylinder.

We discuss the appearance of a missing geometric quantum phase from the interaction of the permanent electric dipole moment of a neutral particle with the magnetic field produced by a uniform distribution of the magnetic charges inside an infinitely long non-conductor cylinder with an inner radius r a . This geometric quantum phase is associated with the missing magnetic charge per unit length in a region with a cylindrical shape of radius r a and gives rise to an analogue of the He–McKellar–Wilkens geometric quantum phase. By searching for bound state solutions, we show that an Aharonov–Bohm-type effect arises from the influence of the missing geometric quantum phase on the eigenvalues of energy.

The scattering amplitude of a charged particle from a long hard cylinderical solenoid is derived by solving the time independent Schrödinger equation on a double connected plane. It is a summation over the angular momentum quantum number (partial wave summation). It is shown that only negative mechanical angular momenta contribute to the amplitude when the radius of the solenoid goes to zero limit without varying the magnetic induction flux (Flux line). Original Aharonov-Bohm result is obtained with this limit.

The exact energy spectrum and wave function of a charged particle moving in the Coulomb field and Aharonov-Bohm's magnetic flux are solved by the nonintegrable phase factor. The universal formula for the matrix elements of the radial operator ${r}^{\ensuremath{\alpha}}$ of arbitrary power $\ensuremath{\alpha}$ is given by an analytical solution. The difference between the classical limit of matrix elements of inverse radius in quantum mechanics and the Fourier components of the corresponding quantity for the pure Coulomb system in classical mechanics is examined in reference to the correspondence principle of Heisenberg. Explicit calculation shows that the influence of nonlocal Aharonov-Bohm effect exists even in the classical limit. The semiclassical quantization rule for systems containing the topological effect is presented in the light of Heisenberg's corresponding principle.