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Physical Review A

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

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

Published Apr 22, 2003• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.67.042109

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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\ensuremathα{r}^{\ensuremath{\alpha}}r\ensuremathα of arbitrary power \ensuremathα\ensuremath{\alpha}\ensuremathα 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.
In the quantum field,due to Bohr correspondence principle,quantum mechanics go to classical mechanics in large quantum number.Based upon Heisenberg correspondence principle,quantum matrix element of a Hermitian operator reduces to the coefficient of Fourier expansion of the corresponding classical quantity in the classical limit.Using Heisenberg correspondence principle,quantum-classical correspondence of particles in magnetic is studied.Applying Heisenberg correspondence principle in relativistic realm,the exact wave functions of Klein-Gordon equation are obtained for particles in magnetic with quantum matrix element in a rectangular coordinate system of new representation.It turns out that in the classical limit matrix elements of quantum operators reduces to the classical solutions and with classical approach,the coordinate is cyclical variation with time,the orbit of particles is a circle,and the movement is uniform circular motion in uniform magnetic field.
A useful and universal formula for the expectation value of the radial operator in the presence of the Aharonov-Bohm flux and the Coulomb Field is established. We find that the expectation value <rλ>< r^λ><rλ> (−∞≤λ≤∞)(-\infty \leq λ\leq \infty)(−∞≤λ≤∞) is greatly affected due to the non-local effect of the magnetic flux although the Aharonov-Bohm flux does not have any dynamical significance in classical mechanics. In particular, the quantum fluctuation increases in the presence of the magnetic flux due to the Aharonov-Bohm effect. In addition, the Virial theory in quantum mechanics is also constructed for the spherically symmetric system under the Aharonov-Bohm effect.
The Aharonov-Bohm (AB) effect is a pure quantum effect that implies a measurable phase shift in the wave function for a charged particle that encircles a magnetic flux located in a region inaccessible to the particle. Classically, such a nonlocal effect appears to be impossible since the Lorentz force depends on only the magnetic field at the location of the particle. In quantum mechanics, the Hamiltonian, and thus the Schr\"odinger equation, has a local coupling between the current due to the particle and the electromagnetic vector potential A\mathbf{A}A, which extends to the entire space beyond the region with finite magnetic field. This has sometimes been interpreted as meaning that in quantum mechanics A\mathbf{A}A is in some sense more ``fundamental'' than B\mathbf{B}B in spite of the former being gauge dependent and thus unobservable. Here we shall, with a general proof followed by a few examples, demonstrate that the AB effect can be fully accounted for by considering only the gauge-invariant B\mathbf{B}B field, as long as it is included as part of the quantum action of the entire isolated system. The price for the gauge-invariant formulation is that we must give up locality---the AB phase for the particle will arise from the change in the action for the B\mathbf{B}B field in the region inaccessible to the particle.
We investigate the scattering of a charged quantum particle in a helically twisted background that induces an effective Coulomb-like interaction, in the presence of an Aharonov-Bohm (AB) flux. Starting from the nonrelativistic Schrödinger equation in the twisted metric, we derive the radial equation and show that, after including the AB potential, it can be mapped onto the same Kummer-type differential equation that governs the planar 2D2D2D Coulomb +++ AB problem, with a geometry-induced Coulomb strength and the azimuthal quantum number shifted as m→m−λm\to m-λm→m−λ. We construct the exact scattering solutions, obtain closed expressions for the partial-wave SSS matrix and phase shifts, and derive the corresponding scattering amplitude, differential cross section, and total cross section. We also show that the pole structure of the SSS matrix is consistent with the bound-state quantization previously obtained for the helically twisted Coulomb-like problem.
We establish systematic consolidation of the Aharonov–Bohm (AB) and Aharonov–Casher effects including their scalar counterparts. Their formal correspondences in acquiring topological phases are revealed on the basis of the gauge symmetry in non-simply-connected spaces and the adiabatic condition for the state of magnetic dipoles. In addition, investigation of basic two-body interactions between an electric charge and a magnetic dipole clarifies their appropriate relative motions and discloses physical interrelations between the effects. Based on the two-body interaction, we also construct an exact microscopic description of the AB effect, where all the elements are treated on equal footing, i.e., magnetic dipoles are described quantum-mechanically and electromagnetic fields are quantized. This microscopic analysis not only confirms the conventional (semiclassical) results and the topological nature but also allows one to explore the fluctuation effects due to the precession of the magnetic dipoles with the adiabatic condition relaxed.
The Aharonov-Bohm effect is a physical phenomenon in which the quantum state of a charged particle acquires a phase shift that is directly proportional to the magnetic flux, ΦΦΦ, due to a (classical) magnetic field, B{\mathbf B}B, which is confined in a spatial region from which the magnetic field cannot escape. Even though the charged particle is not allowed to interact with the magnetic field, it accumulates a phase shift that affects the interference pattern produced. Not surprisingly, this apparent nonlocality is puzzling and counter intuitive. In this work, we provide an explanation that explains the physics underlying this apparent nonlocality. We find that the role of the confined magnetic field is to impart a puncture in the configuration space, R2\mathbb{R}^2R2, of the charge. Therefore, the quantum state corresponding to the charged quantum particle acquires the phase shift due to its response to the modified topology of the configuration space, R2−{0}\mathbb{R}^2-\{0\}R2−{0}, corresponding to the charge.