In the so-called electric Aharonov-Bohm effect, a quantum interference pattern shift is produced when electrons move in an electric field-free region but, at the same time, under the presence of a time-dependent electric potential. Analogous fringe shifts are observed in interference experiments where electrons, travelling through an electrostatic field, suffer a classical lag effect produced by electric forces. Since these experiments are reported mainly in specialized journals, it could be rather difficult and time consuming for a student to attain a comprehensive overview of the subject. Therefore, particular attention has been addressed to reviewing the theory, to describing a couple of experiments and to the interpretation of the results. We believe that students can be suitably introduced to exploring the “paradoxical” aspects of the interaction of electrons with electric potentials and fields.
In the so-called electric Aharonov-Bohm effect, a quantum interference pattern shift is produced when electrons move in an electric field-free region but, at the same time, under the presence of a time-dependent electric potential. Analogous fringe shifts are observed in interference experiments where electrons, travelling through an electrostatic field, suffer a classical lag effect produced by electric forces. Since these experiments are reported mainly in specialized journals, it could be rather difficult and time consuming for a student to attain a comprehensive overview of the subject. Therefore, particular attention has been addressed to reviewing the theory, to describing a couple of experiments and to the interpretation of the results. We believe that students can be suitably introduced to exploring the “paradoxical” aspects of the interaction of electrons with electric potentials and fields.
Classical electromagnetic lag effects can give rise to quantum interference pattern shifts such as that observed experimentally in the Aharonov-Bohm effect involving electrons passing a solenoid. This paper presents an extensive comparison between interference pattern shifts based upon classical electromagnetic fields, and based upon classical electromagnetic potentials as suggested by Aharonov and Bohm. Stress is placed upon the difference between two types of interference pattern shifts: those involving deflection of the entire interference pattern and those involving a deflection of only the double-slit pattern while leaving the single-slit envelope undisplaced. The first type of shift is produced by a classical deflecting force. The second type of shift can be produced by classical electromagnetic lag effects, and is also the type of shift associated with the Aharonov-Bohm effect. The two types are confused in the literature. A new experiment is proposed which shows the relationship between a classical lag effect, due to electrostatic fields on electrons passing along different paths, and the associated quantum interference pattern shifts. The experiment is analyzed in detail using the WKB approximation in the Schr\"odinger equation and also semiclassical ideas. The classical limit for the situation illustrates the Bohr correspondence principle, showing the relative lag between the electron wave packets becoming a measurable classical lag with a disappearance of the interference pattern as the lag becomes large compared to the wave-packet dimensions. For small shifts, the phase change predicted for the new experiment is identical with the scalar potential effect proposed by Aharonov and Bohm for a slightly different, time-varying experimental arrangement. The theoretical and experimental differences for large phase shifts are noted. The possibility is raised that a new classical electromagnetic lag effect may occur for electrons passing a small solenoid. Using a particular model for energy conservation, the predicted lag effect can be calculated and is associated with a quantum interference pattern shift of the same magnitude as predicted by Aharonov and Bohm based upon the electromagnetic vector potential. Thus the possibility exists that the experiments of Chambers and of M\"ollenstedt and Bayh may not confirm the ideas of Aharonov and Bohm on the vector potential in quantum theory. Several experiments are suggested which allow confirmation that the Aharonov-Bohm effect indeed involves local effects of the classical electromagnetic potential, rather than local electromagnetic fields leading to a new classical lag effect and hence to the observed quantum interference pattern shift.
This technical note gives a self-contained exposition of the Aharonov–Bohm effect, aimed at advanced undergraduate students. Starting from the role of electromagnetic potentials in classical electrodynamics, it introduces gauge freedom, the quantum Hamiltonian with minimal coupling, and the gauge covariance of the Schrödinger equation. It then derives the Aharonov–Bohm phase for both the time-dependent scalar configuration and the magnetic configuration, showing how a charged particle acquires an observable phase shift while travelling only through regions where the electromagnetic field vanishes. The final part reviews the experimental evidence, from the early electron-interference experiments of Chambers and of Möllenstedt and Bayh to the 1986 toroidal-magnet experiment of Tonomura and collaborators. In that experiment, superconducting shielding confined the magnetic flux completely and closed the leakage-field loophole, while flux quantization produced the characteristic half-fringe shift. The note is expository and pedagogical: it does not present original research results and has not been peer reviewed. Originally developed from study notes prepared for the PHYS02 scientific talk at Lady Margaret Hall, Oxford, in July 2026; an earlier version of the notes has been publicly available on GitHub since August 2026. Corrections and comments are more than welcome at giancarminesparso@gmail.com.
This article describes the Aharonov-Bohm (AB) effect of electron waves travelling in free space and its application to the observation of gauge fields (vector potentials). The AB effect is inconceivable in classical physics since an observable effect is produced on electrons passing through field-free spaces. Electrons can be affected only by Lorentz forces due to electromagnetic fields. The situation is different in quantum mechanics, where electrons show wave properties: the concept of force is no longer relevant, so electric field E and magnetic field B, defined as forces acting on a unit charge, take on secondary meanings. “Phase shifts” come into play, and the primary physical entities become neither E nor B but electrostatic potential V and vector potential A. These potentials interact with electron waves and shift their phases.
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.
In the Aharonov–Bohm (AB) effect, electrons form an interference pattern whose bright and dark fringes shift position when the magnetic flux confined within a nearby solenoid is changed, even though the electrons travel only through regions where the magnetic field is zero. Operationally, the experiment compares electron interference patterns obtained under different solenoid current (or enclosed flux) conditions, revealing a measurable phase difference—observed as fringe displacement—between the two electron paths despite the absence of any force acting on the electrons along those paths. Each electron arrives as a dot, but many dots build up the interference pattern. The effect tracks the enclosed flux (current level), with on/off as an illustrative extreme. What is measured is phase difference / fringe displacement, i.e., a relative effect between the two paths. This Aharonov-Bohm effect is famous in modern physics and widely regarded as proof that electromagnetic potentials possess physical reality even in regions where fields vanish. This paper reinterprets the AB effect by reframing it using the SP3 (space-phase) theory.
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.