Simplified Path Integral Approach to the Aharonov-Bohm Effect

William O. Straub

Published Mar 1, 2014• 0 citations

Abstract

In classical electrodynamics the vacuum is defined as a region where there are no electric or magnetic fields. In such a region, a charged particle (such as an electron) will feel no effect — the Lorentz force is zero. The space ex-ternal to a perfect (i.e., infinite) solenoid can be considered an electromagnetic vacuum, since ~E and ~B vanish there. While a non-zero vector potential ~A does exist outside the solenoid, it can exert no influence on the particle, and as such cannot be directly detected or quantified classically. However, in 1959 Aharonov and Bohm predicted that a vector field would exert a purely quantum-mechanical effect on the phase of the particle’s wave function, which in principle should be detectable. The predicted phase shift was not observed experimentally until 1986, when Tonomura brilliantly verified the effect using a microscopic solenoid. This paper provides a simplified explanation of the Aharonov-Bohm prediction using a path-integral approach that is suitable for the advanced undergraduate. 1.

Related papers

The Aharonov-Bohm Effect: An Exploration of Quantum Interference and Electromagnetic Potentials

Bahman Zohuri

The AB (Aharonov-Bohm) effect is a pivotal quantum mechanical phenomenon that illustrates the fundamental role of the electromagnetic vector potential A in determining the phase of a charged particle's wave function, even in regions where the magnetic field B is zero.This effect demonstrates that quantum particles are influenced not only by the fields directly present but also by the potentials associated with those fields.In the AB effect, an electron beam is split into two paths, with one path encircling a solenoid and the other bypassing it.Despite the absence of a magnetic field in the regions traversed by the beams, the vector potential A associated with the magnetic flux  through the solenoid induces a phase shift in the electron's wave function.This phase shift, quantified by qc     , manifests as a change in the interference pattern observed in the detection screen.The phenomenon underscores the principle of gauge invariance in QED (quantum electrodynamics), where physical observables remain invariant under local gauge transformations of the vector and scalar potentials.This reinforces the notion that the vector potential A has a profound impact on quantum systems, beyond its classical role.This article outlines the AB effect, including its theoretical framework, experimental observations, and implications.The focus on the role of the vector potential in quantum mechanics provides a comprehensive understanding of this important phenomenon.

Journal of Energy and Power EngineeringJun 28, 20241 citation

An interpretation for Aharonov-Bohm effect with classical electromagnetic theory

Gaobiao Xiao

The magnetic Aharonov-Bohm effect shows that charged particles may be affected by the vector potential in regions without any electric or magnetic fields [1]. The Aharonov-Bohm effect was experimentally confirmed [2-3] and has been found in many situations [4-6]. A common explanation is based on quantum mechanics, which states that the wavefunctions associated with the charges will accumulate a phase shift due to the vector potential. However, consensus about its nature and interpretation has not been achieved [7-14]. We here propose a simple but reasonable interpretation based on the theory for electromagnetic radiation and couplings [15]. The energy associated with a pulse radiator is divided into a Coulomb-velocity energy and a radiative energy, together with a macroscopic Schott energy accounting for the energy exchange between them. All these energies are expressed with terms including the potentials, so are the mutual coupling energies. There exists a force acting on the moving charges even though the fields completely vanish. This force makes the charges pass through the magnetic solenoid in different velocity with different path length, causing a phase shift the same as that obtained with quantum mechanics. The theory is originally aimed for providing an interpretation for electromagnetic radiation and mutual coupling. It is derived directly from the Maxwell theory with no modification but only substitution and reorganization.

arXiv:2201.12292Jan 26, 20223 citations

The electric Aharonov-Bohm effect

Ricardo Weder

The seminal paper of Aharonov and Bohm [Phys. Rev. 115, 485 (1959)]10.1103/PhysRev.115.485 is at the origin of a very extensive literature in some of the more fundamental issues in physics. They claimed that electromagnetic fields can act at a distance on charged particles even if they are identically zero in the region of space where the particles propagate, that the fundamental electromagnetic quantities in quantum physics are not only the electromagnetic fields but also the circulations of the electromagnetic potentials; what gives them a real physical significance. They proposed two experiments to verify their theoretical conclusions. The magnetic Aharonov-Bohm effect, where an electron is influenced by a magnetic field that is zero in the region of space accessible to the electron, and the electric Aharonov-Bohm effect where an electron is affected by a time-dependent electric potential that is constant in the region where the electron is propagating, i.e., such that the electric field vanishes along its trajectory. The Aharonov-Bohm effects imply such a strong departure from the physical intuition coming from classical physics that it is no wonder that they remain a highly controversial issue after more than fifty years, in spite of the fact that they are discussed in most of the text books in quantum mechanics. The magnetic case has been studied extensively. The experimental issues were settled by the remarkable experiments of Tonomura et al. [Phys. Rev. Lett. 48, 1443 (1982); Phys. Rev. Lett. 56, 792 (1986)] with toroidal magnets, that gave a strong evidence of the existence of the effect, and by the recent experiment of Caprez et al. [Phys. Rev. Lett. 99, 210401 (2007)]10.1103/PhysRevLett.99.210401 that shows that the results of the Tonomura et al. experiments cannot be explained by the action of a force. The theoretical issues were settled by Ballesteros and Weder [Commun. Math. Phys. 285, 345 (2009)10.1007/s00220-008-0579-1; J. Math. Phys. 50, 122108 (2009)10.1063/1.3266176; Commun. Math. Phys. 303, 175 (2011)]10.1007/s00220-010-1166-9 who rigorously proved that quantum mechanics predicts the experimental results of Tonomura et al. and of Caprez et al. The electric Aharonov-Bohm effect has been much less studied. Actually, its existence, that has not been confirmed experimentally, is a very controversial issue. In their 1959 paper Aharonov and Bohm proposed an ansatz for the solution to the Schrödinger equation in regions where there is a time-dependent electric potential that is constant in space. It consists in multiplying the free evolution by a phase given by the integral in time of the potential. The validity of this ansatz predicts interference fringes between parts of a coherent electron beam that are subjected to different potentials. In this paper we prove that the exact solution to the Schrödinger equation is given by the Aharonov-Bohm ansatz up to an error bound in norm that is uniform in time and that decays as a constant divided by the velocity. Our results give, for the first time, a rigorous proof that quantum mechanics predicts the existence of the electric Aharonov-Bohm effect, under conditions that we provide. We hope that our results will stimulate the experimental research on the electric Aharonov-Bohm effect.

arXiv:1006.1385Journal of Mathematical PhysicsMay 1, 201116 citations

Classical Electrodynamics without Fields and the Aharonov-Bohm effect

Eugene V. Stefanovich

The Darwin-Breit Hamiltonian is applied to the Aharonov-Bohm experiment. In agreement with the standard Maxwell-Lorentz theory, the force acting on electrons from infinite solenoids or ferromagnetic rods vanishes. However, the interaction energies and phase factors of the electron wave packets are non-zero. This allows us to explain the Aharonov-Bohm effect without involvement of electromagnetic potentials, fields, and topological properties of space.

arXiv:0803.1326Mar 9, 20081 citation

From Primitives to the Aharonov–Bohm Phase

Allen Proxmire

In 1959, Yakir Aharonov and David Bohm pointed out an experiment that should be impossible according to the standard intuitions of classical electromagnetism. A charged particle is sent through a double‑slit apparatus with a long thin solenoid placed between the two paths. The solenoid contains a magnetic field; the regions traversed by the particle do not. In classical mechanics, the particle should be entirely unaffected — no force, no torque, no Lorentz coupling. The interference pattern should be unchanged whether the solenoid is on or off. It isn't. When the solenoid is energized, the interference pattern shifts by Δφ = (q/ℏ) Φ, where Φ = ∫B·dA is the magnetic flux threading the solenoid. The particle’s wavefunction registers a region of magnetic field through which the particle never travels. Chambers performed the experiment in 1960; Tonomura confirmed it definitively with superconducting measurements in 1986. The effect is now standard textbook material and one of the deepest empirical phenomena in quantum mechanics. The puzzle is structural. In classical electromagnetism, the magnetic vector potential A was treated as a mathematical convenience — a quantity from which B = ∇×A could be computed but with no independent physical reality. The AB effect breaks this picture: the interference shift depends on ∮A·dx around the closed path, a gauge‑invariant integral that is non‑zero in regions where B = 0. The vector potential carries physical information that the magnetic field alone does not. Standard QM accommodates this by accepting that A is the physically meaningful quantity in the quantum context. This works mathematically but leaves a deeper question open: why should A have physical reality at all? This walkthrough shows how the Event Density framework provides an answer through Theorem 17. T17 establishes that gauge fields are the continuum‑level appearance of the substrate’s rule‑type connection — the substrate object that carries rule‑type label information from one substrate region to another. The gauge potential is not a mathematical convenience; it is the substrate’s parallel‑transport infrastructure for the rule‑type relabeling group. The AB phase is the integral of this substrate connection around a non‑contractible loop in a multiply‑connected substrate channel — a substrate‑level monodromy with observable consequences whenever the substrate’s topology supports closed loops the wavefunction can wind around. Stokes’ theorem connects the monodromy to the enclosed flux: Δφ = qΦ/ℏ. The structural payoff: the AB phase is not a mysterious nonlocal effect of the gauge potential. It is the substrate‑level monodromy of the rule‑type connection around a closed loop in a multiply‑connected substrate region. The gauge potential’s physical reality is not a postulate; it is what T17 establishes the substrate to provide.

May 12, 20260 citations

The covariant, time-dependent Aharonov–Bohm effect

Douglas Singleton, Elias C. Vagenas

We discuss two possible covariant generalizations of the Aharonov–Bohm effect – one expression in terms of the space–time line integral of the four-vector potential and the other expression in terms of the space–time “area” integral of the electric and magnetic fields written in terms of the Faraday 2-form. These expressions allow one to calculate the Aharonov–Bohm effect for time-dependent situations. In particular, we use these expressions to study the case of an infinite solenoid with a time varying flux and find that the phase shift is zero due to a cancellation of the Aharonov–Bohm phase shift with a phase shift coming from the Lorentz force associated with the electric field, E=−∂tA, outside the solenoid. This result may already have been confirmed experimentally.

arXiv:1305.1498Physics Letters BMay 9, 201345 citations