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Tests of the Aharonov-Bohm effect

Adam Caprez

Published Jan 1, 2009• 1 citation
OpenAlex

Abstract

The Aharonov-Bohm effect was first proposed in 1959, and has stimulated discussion and controversy since the start. Seen by many as a purely quantum mechanical effect, it nevertheless involves such issues as gauge invariance and relativistic dynamics. This multi-faceted nature has led it to be considered a keystone of modern quantum theory. Over the past half-century, interest has remained strong in what many believe is still an open debate as to the purely quantum nature. Additionally, discovery of similar effects for particles other than electrons has further driven curiosity. To better understand the issues such as electromagnetic mass and relativistic effects involved in the Aharonov-Bohm effect, a theoretical study of a simpler two particle system was done. The goal was to understand the manner in which mass of the system behaved strictly classically as compared to a relativistically. As a result of this, a Gedanken experiment is presented which serves as a test for covariance. The two particle system was extended to a spherical shell interacting with a single particle, and a second thought experiment put forth to explore a coupling between electromagnetism and gravitation. In the course of searching for simpler systems which involved the same issues as the Aharonov-Bohm effect, a system presented in the Feynman Lectures of Physics was found to be appropriate. We conduct a complete relativistic analysis of this system as a step towards a full relativistic analysis of the Aharonov-Bohm effect. Given the history and significance of the Aharonov-Bohm effect, it is perhaps surprising that relevant experimental tests had not been completed. For the first time, we searched for time delays as an electron passes a macroscopic solenoid. Such time delays are characteristic of classical forces acting on the electron. No such delays were found, seemingly confirming the standard viewpoint. It is still possible a classical explanation may exist for microscopic solenoids, though our experiment has served to place an upper limit for any such explanation. The definitive experiment concerning the Aharonov-Bohm Effect has yet to be completed. Proposed by Zeilinger, it consists of showing the dispersionless nature of the effect. To accomplish this requires an electron interferometer capable of enclosing a larger area and operating at lower energies than any which currently exist. We are attempting to construct such an interferometer using a hybrid approach with a nanofabricated grating and electron bi-prism. We have shown that experimentally that this combination can produce a large beam separation and still retain sufficient coherence to function as an interferometer.

Related papers

More variations on Aharonov-Bohm

Herman Batelaan, Akira Tonomura

Batelaan and Tonomura reply: Werner Ehrenberg and Raymond Siday did propose the magnetic version of what is now called the Aharonov-Bohm (AB) effect, 1 1. A. Tonomura, Electron Holography, 2nd ed., Springer, New York (1999), p. 63. as Peter Sturrock and Timothy Groves point out. We had included this reference in the early versions of our manuscript. However, limited space directed the focus of the paper to the “effect without a force” discussion, rather than a historic perspective.Alexander Ershkovich notes that in the Hamilton-Jacobi formulation of classical mechanics, both the action and the Hamiltonian depend on the vector potential; he ponders whether the AB effect might have a classical manifestation. Newton’s formulation of classical mechanics is equivalent to the Hamilton-Jacobi formulation. Because the absence of a field means the absence of a force in Newton’s formulation, classical trajectories are unaffected. That result is not expected to change in the equivalent Hamilton-Jacobi formulation. Thus the AB effect is usually considered to be a pure quantum effect. On the other hand, we may interpret “classical mechanics” in a broader sense, such as in general relativistic classical mechanics. In electrodynamics classical trajectories are not affected by a localized magnetic field through which they do not pass. However, considering that the energy content of a current-carrying solenoid is larger than that of one without current, the trajectory is clearly affected gravitationally, at least in principle. Though not due to the AB effect, that result elucidates that a generalized description may lead to other insights. The Hamilton-Jacobi equation may be an example of a theoretical vehicle by which to explore generalizations such as relativistic effects, separation of variables, multiple particle effects, or the classical limit of the de Broglie-Bohm theory. C. Alden Mead recollects interesting statements made by Bohm. We agree fully with Bohm’s statement that “it would be much more revolutionary for this effect to be wrong than for it to be right.” Attempts to disprove the AB effect should be seen for what they are, outright attempts at finding limits to the validity of quantum mechanics itself. And although quantum mechanics is unfinished with respect to, say, de-coherence theory and quantum gravity, the AB effect appears to be well within its validity range.We do not share Bohm’s astonishment that, as Mead relates, “certain physicists refused to accept the AB effect and even went to great lengths to try to disprove it.” Rather, to risk overusing a platitude, extraordinary phenomena should be exposed to extraordinary scrutiny. Failed attempts to disprove an idea often provide insight into its fundamental character. In that context, we reiterate the main message of our article. Many facets of the AB effect—for example, the electric version, the dispersionless nature, relativistic momentum conservation, the relation to the Mott-Schwinger effect, and the AB effects for other than electromagnetic gauge-invariant theories—need exploration. We predict a bright future for the AB effect, with many surprises to come.REFERENCESection:ChooseTop of pageREFERENCE <<1. A. Tonomura, Electron Holography, 2nd ed., Springer, New York (1999), p. 63. Google ScholarCrossref© 2010 American Institute of Physics.

Physics Today•Apr 1, 2010•1 citation

The Aharonov-Bohm effect: A quantum or a relativistic phenomenon?

K. Wilhelm, B. N. Dwivedi

The Aharonov-Bohm effect is considered by most authors as a quantum effect, but a generally accepted explanation does not seem to be available. The phenomenon is studied here under the assumption that hypothetical electric dipole distributions configured by moving charges in the solenoid act on the electrons as test particles. The relative motions of the interacting charged particles introduce relativistic time dilations. The massless dipoles are postulated as part of an impact model that has recently been proposed to account for the far-reaching electrostatic forces between charged particles described by Coulomb's law. The model provides a quantitative explanation of the Aharonov-Bohm effect.

arXiv:1501.05261•Aug 23, 2014•1 citation

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:1006.1385•Journal of Mathematical Physics•May 1, 2011•16 citations

Wherein Lies the Momentum in Aharonov-Bohm Quantum Interference Experiment - A Classical Physics Perspective

Ashok K. Singal

In the Aharonov-Bohm setup, a double-slit experiment, coherent beams of electrons passing through two slits form an interference pattern on the observing screen. However, when a long but thin solenoid of current is introduced behind the slits between the two electron beams, an extra phase difference between them appears, as shown by a shift in the interference pattern. This happens even though there is no magnetic field outside the solenoid, at the location of the beams. This is known as Aharonov-Bohm effect, though the idea was mooted apparently a decade earlier, and the effect sometimes is called Ehrenberg-Siday-Aharonov-Bohm effect. This mysterious effect, purportedly arises owing to an electromagnetic momentum, attributed to the presence at the location of either beam, a vector potential due to the solenoid of current even when there exists no magnetic field outside the solenoid. The first experimental confirmation came soon and It has since been amply verified using clever experimental setups, leaving hardly any doubts that the observed effect is real. However, on the theoretical side the picture is not so clear and a satisfactory physical explanation of the existence of momentum, at least under the aegis of classical electromagnetism, is still missing since inception of the idea more than half a century back. It has remained a puzzle, how just potential, thought to be a mere mathematical tools for calculating electromagnetic field, can give rise to an electromagnetic momentum in a system, in lieu of field itself. We here show that a subtle momentum can be seen to lie in the product of the drift velocities of the current carrying charges and the mass equivalent of their non-localized potential energies in the electric field of the interfering electrons, which manifests, from a classical point of view, a linear momentum in the system. It is this hard-to-pinpoint, additional momentum, reflected through an extra phase difference between the interfering beams of electrons, which exhibits from a classical physics perspective, the presence of an elusive, long sought-after electromagnetic momentum in the system.

•Jan 18, 2023•1 citation

The Aharonov–Bohm effect and Tonomura et al. experiments: Rigorous results

Miguel Ballesteros, Ricardo Weder

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.

arXiv:0903.2609•Journal of Mathematical Physics•Dec 1, 2009•25 citations

Misinterpretation of the Aharonov-Bohm Effect

Timothy H. Boyer

It is pointed out that accounts in the literature sometimes misinterpret the Aharonov-Bohm effect, involving the shift in interference pattern for electrons passing a long solenoid. The quantum description and experimental verification of the effect are reviewed, and it is emphasized that both theory and experiment indicate that there is no deflection of the average momentum of the electrons passing the solenoid. There is no average force on the electrons due to the solenoid. It is remarked that the Aharonov-Bohm effect involves the shift of double-slit interference fringes which is independent of the quantum unit of action ħ and yet allows a natural classical limit.

American Journal of Physics•Jan 1, 1972•25 citations
The Aharonov-Bohm effect was first proposed in 1959, and has stimulated discussion and controversy since the start. Seen by many as a purely quantum mechanical effect, it nevertheless involves such issues as gauge invariance and relativistic dynamics. This multi-faceted nature has led it to be considered a keystone of modern quantum theory. Over the past half-century, interest has remained strong in what many believe is still an open debate as to the purely quantum nature. Additionally, discovery of similar effects for particles other than electrons has further driven curiosity. To better understand the issues such as electromagnetic mass and relativistic effects involved in the Aharonov-Bohm effect, a theoretical study of a simpler two particle system was done. The goal was to understand the manner in which mass of the system behaved strictly classically as compared to a relativistically. As a result of this, a Gedanken experiment is presented which serves as a test for covariance. The two particle system was extended to a spherical shell interacting with a single particle, and a second thought experiment put forth to explore a coupling between electromagnetism and gravitation. In the course of searching for simpler systems which involved the same issues as the Aharonov-Bohm effect, a system presented in the Feynman Lectures of Physics was found to be appropriate. We conduct a complete relativistic analysis of this system as a step towards a full relativistic analysis of the Aharonov-Bohm effect. Given the history and significance of the Aharonov-Bohm effect, it is perhaps surprising that relevant experimental tests had not been completed. For the first time, we searched for time delays as an electron passes a macroscopic solenoid. Such time delays are characteristic of classical forces acting on the electron. No such delays were found, seemingly confirming the standard viewpoint. It is still possible a classical explanation may exist for microscopic solenoids, though our experiment has served to place an upper limit for any such explanation. The definitive experiment concerning the Aharonov-Bohm Effect has yet to be completed. Proposed by Zeilinger, it consists of showing the dispersionless nature of the effect. To accomplish this requires an electron interferometer capable of enclosing a larger area and operating at lower energies than any which currently exist. We are attempting to construct such an interferometer using a hybrid approach with a nanofabricated grating and electron bi-prism. We have shown that experimentally that this combination can produce a large beam separation and still retain sufficient coherence to function as an interferometer.
Batelaan and Tonomura reply: Werner Ehrenberg and Raymond Siday did propose the magnetic version of what is now called the Aharonov-Bohm (AB) effect, 1 1. A. Tonomura, Electron Holography, 2nd ed., Springer, New York (1999), p. 63. as Peter Sturrock and Timothy Groves point out. We had included this reference in the early versions of our manuscript. However, limited space directed the focus of the paper to the “effect without a force” discussion, rather than a historic perspective.Alexander Ershkovich notes that in the Hamilton-Jacobi formulation of classical mechanics, both the action and the Hamiltonian depend on the vector potential; he ponders whether the AB effect might have a classical manifestation. Newton’s formulation of classical mechanics is equivalent to the Hamilton-Jacobi formulation. Because the absence of a field means the absence of a force in Newton’s formulation, classical trajectories are unaffected. That result is not expected to change in the equivalent Hamilton-Jacobi formulation. Thus the AB effect is usually considered to be a pure quantum effect. On the other hand, we may interpret “classical mechanics” in a broader sense, such as in general relativistic classical mechanics. In electrodynamics classical trajectories are not affected by a localized magnetic field through which they do not pass. However, considering that the energy content of a current-carrying solenoid is larger than that of one without current, the trajectory is clearly affected gravitationally, at least in principle. Though not due to the AB effect, that result elucidates that a generalized description may lead to other insights. The Hamilton-Jacobi equation may be an example of a theoretical vehicle by which to explore generalizations such as relativistic effects, separation of variables, multiple particle effects, or the classical limit of the de Broglie-Bohm theory. C. Alden Mead recollects interesting statements made by Bohm. We agree fully with Bohm’s statement that “it would be much more revolutionary for this effect to be wrong than for it to be right.” Attempts to disprove the AB effect should be seen for what they are, outright attempts at finding limits to the validity of quantum mechanics itself. And although quantum mechanics is unfinished with respect to, say, de-coherence theory and quantum gravity, the AB effect appears to be well within its validity range.We do not share Bohm’s astonishment that, as Mead relates, “certain physicists refused to accept the AB effect and even went to great lengths to try to disprove it.” Rather, to risk overusing a platitude, extraordinary phenomena should be exposed to extraordinary scrutiny. Failed attempts to disprove an idea often provide insight into its fundamental character. In that context, we reiterate the main message of our article. Many facets of the AB effect—for example, the electric version, the dispersionless nature, relativistic momentum conservation, the relation to the Mott-Schwinger effect, and the AB effects for other than electromagnetic gauge-invariant theories—need exploration. We predict a bright future for the AB effect, with many surprises to come.REFERENCESection:ChooseTop of pageREFERENCE <<1. A. Tonomura, Electron Holography, 2nd ed., Springer, New York (1999), p. 63. Google ScholarCrossref© 2010 American Institute of Physics.
The Aharonov-Bohm effect is considered by most authors as a quantum effect, but a generally accepted explanation does not seem to be available. The phenomenon is studied here under the assumption that hypothetical electric dipole distributions configured by moving charges in the solenoid act on the electrons as test particles. The relative motions of the interacting charged particles introduce relativistic time dilations. The massless dipoles are postulated as part of an impact model that has recently been proposed to account for the far-reaching electrostatic forces between charged particles described by Coulomb's law. The model provides a quantitative explanation of the Aharonov-Bohm effect.
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 setup, a double-slit experiment, coherent beams of electrons passing through two slits form an interference pattern on the observing screen. However, when a long but thin solenoid of current is introduced behind the slits between the two electron beams, an extra phase difference between them appears, as shown by a shift in the interference pattern. This happens even though there is no magnetic field outside the solenoid, at the location of the beams. This is known as Aharonov-Bohm effect, though the idea was mooted apparently a decade earlier, and the effect sometimes is called Ehrenberg-Siday-Aharonov-Bohm effect. This mysterious effect, purportedly arises owing to an electromagnetic momentum, attributed to the presence at the location of either beam, a vector potential due to the solenoid of current even when there exists no magnetic field outside the solenoid. The first experimental confirmation came soon and It has since been amply verified using clever experimental setups, leaving hardly any doubts that the observed effect is real. However, on the theoretical side the picture is not so clear and a satisfactory physical explanation of the existence of momentum, at least under the aegis of classical electromagnetism, is still missing since inception of the idea more than half a century back. It has remained a puzzle, how just potential, thought to be a mere mathematical tools for calculating electromagnetic field, can give rise to an electromagnetic momentum in a system, in lieu of field itself. We here show that a subtle momentum can be seen to lie in the product of the drift velocities of the current carrying charges and the mass equivalent of their non-localized potential energies in the electric field of the interfering electrons, which manifests, from a classical point of view, a linear momentum in the system. It is this hard-to-pinpoint, additional momentum, reflected through an extra phase difference between the interfering beams of electrons, which exhibits from a classical physics perspective, the presence of an elusive, long sought-after electromagnetic momentum in the system.
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.
It is pointed out that accounts in the literature sometimes misinterpret the Aharonov-Bohm effect, involving the shift in interference pattern for electrons passing a long solenoid. The quantum description and experimental verification of the effect are reviewed, and it is emphasized that both theory and experiment indicate that there is no deflection of the average momentum of the electrons passing the solenoid. There is no average force on the electrons due to the solenoid. It is remarked that the Aharonov-Bohm effect involves the shift of double-slit interference fringes which is independent of the quantum unit of action ħ and yet allows a natural classical limit.