The effects of the scalar and vector potentials in quantum mechanics, which were pointed out recently by Aharonov and Bohm, are discussed from the point of view of the consistency of the quantum-mechanical description of interference experiments. A well-known requirement for this consistency is that if any measuring device is introduced that can be used to determine which path the particle has taken, it must have the effect of eliminating the interference phenomenon. Two conceptual experiments are discussed, corresponding to the two phase effects noted by Aharonov and Bohm. In each case it is found that the phase effect is of just the magnitude required to destroy the interference pattern when the circumstances are such that no pattern should be observed.
The effects of the scalar and vector potentials in quantum mechanics, which were pointed out recently by Aharonov and Bohm, are discussed from the point of view of the consistency of the quantum-mechanical description of interference experiments. A well-known requirement for this consistency is that if any measuring device is introduced that can be used to determine which path the particle has taken, it must have the effect of eliminating the interference phenomenon. Two conceptual experiments are discussed, corresponding to the two phase effects noted by Aharonov and Bohm. In each case it is found that the phase effect is of just the magnitude required to destroy the interference pattern when the circumstances are such that no pattern should be observed.
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.
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.
The Aharonov-Bohm (AB) effect highlights the fundamental role of electromagnetic potentials in quantum mechanics. While extensively studied in the static case, the impact of a time-varying magnetic flux on the electron's phase shift remains an open and debated question. In this paper, we derive the AB phase shift for a time-dependent magnetic vector potential and show that it is proportional to the time average of enclosed magnetic flux. Our analysis reveals that the AB phase is continuously accumulated as the electron traverses its path, challenging the conventional view that it emerges instantaneously at the point of interference. This generalized AB effect may provide deeper insight into the role of gauge-dependent potentials in quantum mechanics and also suggest novel experimental tests using alternating or pulsed magnetic flux.
Abstract The Aharonov-Bohm (AB) effect is an important discovery in quantum theory and electrodynamics. It serves as a surprising quantum phenomenon in which an electrically charged particle can be affected by an electromagnetic potential, despite being confined to a region in which both the magnetic field and electric field are zero. This fact gives the electromagnetic potentials greater significance in quantum physics than in classical physics. Actually the original AB effect belongs to an ``electromagnetism-type". A certain type of vector potential is crucial for building the corresponding type of AB effect. In this work, we focus on the ``spin-type" AB effect. First, we derive the spin vector potential by considering a particle with a spin operator. Second, based on the spin vector potential we propose the spin AB effect, which is a gedanken double-slit interference experiment and is possible to observed in the lab. Third, we discuss the connections between the spin vector potential and some important spin interactions and predict a new type of spin-orbital interaction. Our result advances the study of the potential theory and opens a new avenue of the spin AB effect.
Beginning with the basic notions of quantum theory, impossibility of `trajectory' description for particles that ensues from uncertainty principle is discussed. Why the observed tracks in bubble/cloud chambers are not really the `trajectories' of high energy particles, rather they are simply the trails of the atoms/molecules excited or ionized in the direction of the high momenta of the incoming particles, are highlighted. Thereafter, the notion of symmetry and its application to the non-relativistic Schrodinger equation have been delineated. The demand for U(1) gauge invariance and the resulting `gauge miracle', that automatically leads to the correct interaction terms describing the electromagnetic force between the charge particles and the field, have been elaborated upon. A simple treatment, but with explicit derivation, of the Aharonov-Bohm effect, is presented, that underlines a strange phenomena - sites, impenetrable by charge particles but are threaded with magnetic field lines, conspire with a `Trojan Horse', namely the magnetic vector potential outside, to cause measurable shift in the double-slit interference fringes. This shift in the interference pattern is non-classical since the charge particles move in field-free regions. The crucial Aharonov-Bohm (AB) phase that makes its entry in the above bizarre effect is also deployed to derive the observed magnetic flux quantisation in superconductors as well as the Dirac result which implies that the existence of a single magnetic monopole anywhere in the universe would entail quantisation of the product of a particle's electric charge and the monopole's magnetic charge. Nontrivial consequences of AB phase follow whenever the physical region accessible, from the point of view of the wavefunction of a charge particle, is multiply-connected.
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.