The Aharonov-Bohm effect has recently been questioned on theoretical and experimental grounds. Such discussions have suffered from ambiguities which resulted from their focusing on scattering states. It is noted here that the bound-state problem is much simpler. To avoid the Aharonov-Bohm effect in theory requires us to abandon the most fundamental ideas of quantum mechanics. The quantization of flux in superconducting rings and Josephson junctions is a powerful experimental confirmation.
The Aharonov-Bohm effect has recently been questioned on theoretical and experimental grounds. Such discussions have suffered from ambiguities which resulted from their focusing on scattering states. It is noted here that the bound-state problem is much simpler. To avoid the Aharonov-Bohm effect in theory requires us to abandon the most fundamental ideas of quantum mechanics. The quantization of flux in superconducting rings and Josephson junctions is a powerful experimental confirmation.
Ever since Yakir Aharonov and David Bohm pointed out, in 1959, that electrons propagating around a magnetic field through a field-free vacuum should exhibit a quite surprising interference effect, its experimental and theoretical investigation has been contributing to our understanding of the fundamental character of the quantum theory. The recent appearance of this Aharonov–Bohm effect in a rather different context may eventually have a profound effect on our view of the solid state on a “mesoscopic” scale—between the microscopic and the truly macroscopic.
The Aharonov–Bohm effect is a fundamental issue in physics. It describes the physically important electromagnetic quantities in quantum mechanics. Its experimental verification constitutes a test of the theory of quantum mechanics itself. The remarkable experiments of Tonomura et al. [“Observation of Aharonov-Bohm effect by electron holography,” Phys. Rev. Lett 48, 1443 (1982) and “Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave,” Phys. Rev. Lett 56, 792 (1986)] are widely considered as the only experimental evidence of the physical existence of the Aharonov–Bohm effect. Here we give the first rigorous proof that the classical ansatz of Aharonov and Bohm of 1959 [“Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485 (1959)], that was tested by Tonomura et al., is a good approximation to the exact solution to the Schrödinger equation. This also proves that the electron, that is, represented by the exact solution, is not accelerated, in agreement with the recent experiment of Caprez et al. in 2007 [“Macroscopic test of the Aharonov–Bohm effect,” Phys. Rev. Lett. 99, 210401 (2007)], that shows that the results of the Tonomura et al. experiments can not be explained by the action of a force. Under the assumption that the incoming free electron is a Gaussian wave packet, we estimate the exact solution to the Schrödinger equation for all times. We provide a rigorous, quantitative error bound for the difference in norm between the exact solution and the Aharonov–Bohm Ansatz. Our bound is uniform in time. We also prove that on the Gaussian asymptotic state the scattering operator is given by a constant phase shift, up to a quantitative error bound that we provide. Our results show that for intermediate size electron wave packets, smaller than the ones used in the Tonomura et al. experiments, quantum mechanics predicts the results observed by Tonomura et al. with an error bound smaller than 10−99. It would be quite interesting to perform experiments with electron wave packets of intermediate size. Furthermore, we provide a physical interpretation of our error bound.
The Aharonov-Bohm effect is a quantum mechanical effect, that is, has no classical counterpart. The effect was predicted in 1959 in a seminal paper of Y. Aharonov and D. Bohm [AB59] in which they demonstrated that a beam of electrons is affected by the existence of the electric/magnetic field even though electrons travel through field-free regions. Aharonov and Bohm carried out two hypothetical experiments to support their claim that potentials are more fundamental than fields and they are respon-sible of the effect. Since then, the debate has arisen around whether potentials are mathematical tools or fundamental entities in physics. Different arguments have been set to explain the results predicted by Aharonov and Bohm and experimentally confirmed. Amongst these arguments, the first argument adopted by Aharonov and Bohm was that potentials are physically significant. Many have claimed that fields do have non-local features, i.e. action at a distance. Others have claimed that topological effects may interpret the effect in which potentials are modeled as connections in higher-dimensional fiber bun-dle geometries. The most recent argument has been proposed by Vaidman [Vai12] who claimed that the the composite system is represented by one state, an entangled state, and due to the electromagnetic interactions part of this state is changed, hence, the total state. In the present essay, I will discuss the latter argument as well as reviewing some other arguments. i
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.
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.