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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

The Aharonov-Bohm effect revisited

H. Fearn

Published Jun 1, 1995• 4 citations
Publisher

Abstract

Schrodinger's equation for an electron in a vector potential A has a one-to-one correspondence with the wave equation for an electric field in a moving medium. First the author points out a few errors in the literature concerning the Aharonov-Bohm (AB) effect then she explores the optical analogy further. The author finds that curl A plays a similar role to curl v where v is the velocity of the moving medium.

DOI
10.1088/1355-5111/7/3/002

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The article about Aharonov-Bohm effects is interesting and comprehensive. The primary and best-known effect shows that the vector potential A of the electromagnetic field is a physical reality rather than a mathematical artifice. That reality was implicit in the Schrödinger equation, as the Hamiltonian H depends on A instead of the electric field E and magnetic field B = ∇ × A. But the same statement also refers to the Hamilton-Jacobi equation in classical mechanics, so one may expect that a similar effect exists in classical physics as well.Indeed, classical mechanics is governed by the fundamental Hamilton-Jacobi equation for the action S: δS/δt + H = 0, which naturally follows from William Hamilton’s principle of least action. Both H and S depend on A (even with E = B = 0). Thus it is quite possible that the vector potential is also a physical reality in classical physics. Erwin Schrödinger arrived at his famous equation in 1926 by using a mechanics-optics analogy, the so-called eikonal equation. Hamilton in 1834 proved that eikonal and Hamilton-Jacobi equations are equivalent, so that the Schrödinger equation actually follows from Hamilton-Jacobi.Had Hamilton known that classical mechanics does not always hold, quantum mechanics might have appeared a century earlier. 1 1. H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 2nd ed., Addison-Wesley, Reading, MA (1980), p.484 . The analogy between the equations of classical and quantum mechanics provides a reason to search for experiments that might prove whether a classical analogue of the Aharonov-Bohm effect exists.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 2nd ed., Addison-Wesley, Reading, MA (1980), p.484 . Google Scholar© 2010 American Institute of Physics.

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Writing about what is usually termed the Aharonov-Bohm (AB) effect, Peter Sturrock and Timothy Groves argue (Physics Today, April 2010, page 8) that the same physics was discovered a decade earlier 1 1. W. Ehrenberg, R. E. Siday, Proc. Phys. Soc. B 62, 8 (1949). and should rightly be called the Ehrenberg-Siday effect. I agree that Werner Ehrenberg and Raymond Siday deserve recognition for their anticipation of AB. Indeed, in recent talks celebrating the 50th anniversary of AB, I began by describing the unfairly neglected paper by Ehrenberg and Siday. Nevertheless, I have come to a different conclusion from Sturrock and Groves: The expression “Aharonov-Bohm effect” is justified, for two reasons.First, although there is no doubt that the work by Ehrenberg and Siday anticipated how inaccessible magnetic flux can influence electron interference, that was as a curiosity, at the end of a paper whose main emphasis was the Hamiltonian analysis of electron optics. By contrast, Yakir Aharonov and David Bohm emphasized from the start, as an essential and general aspect of quantum mechanics, the physical influence of inaccessible fields that act nonlocally through the vector potential.Second, Ehrenberg and Siday’s semi-classical approximation—essentially applying the Dirac magnetic phase factor to electrons traveling on either side of the flux—implies a wavefunction that is multivalued and therefore not the correct solution of Schrödinger’s equation. The lack of a single-valued wavefunction leaves their prediction open to doubt. By contrast, Aharonov and Bohm derived the exact single-valued solution for waves scattered by a flux line. Recently, it was shown that the Ehrenberg-Siday approximation corresponds to the first terms in a “many-whirls” representation that treats the exact AB wavefunction as a topological sum over paths circling the flux. 2 2. M. V. Berry, J. Phys. A (in press). Attribution of credit is a delicate matter. It tends to excite strong feelings, and I write about it reluctantly. But the Ehrenberg-Siday paper does seem to exemplify the unfortunate phenomenon identified by Alfred North White-head in a 1916 address to the British Association for the Advancement of Science: “Everything of importance has been said before, by someone who did not discover it.”In a companion letter in the April 2010 issue, Alexander Ershkovich correctly points out that the AB effect is present in classical Hamiltonian mechanics, even though remote magnetic fields cannot influence Newtonian trajectories. He advocates a “search for experiments that might prove … a classical analogue of the Aharonov-Bohm effect.” Such an experiment exists already. In the classical physics of waves on a moving medium, the flow velocity acts like the vector potential in quantum mechanics, so the flow vorticity acts like the magnetic field; the analogy is precise. Fine details of the AB wavefunction were observed in ripples on the surface of water swirling irrotationally into a bathtub vortex, 3 3. M. V. Berry, R. G. Chambers, M. D. Large, C. Upstill, J. C. Walmsley, Eur. J. Phys. 1, 154 (1980). whose core is the analogue of inaccessible magnetic flux. REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. W. Ehrenberg, R. E. Siday, Proc. Phys. Soc. B 62, 8 (1949). Google ScholarCrossref2. M. V. Berry, J. Phys. A (in press). Google Scholar3. M. V. Berry, R. G. Chambers, M. D. Large, C. Upstill, J. C. Walmsley, Eur. J. Phys. 1, 154 (1980). Google ScholarCrossref© 2010 American Institute of Physics.

Physics Today•Jul 30, 2010•12 citations
Schrodinger's equation for an electron in a vector potential A has a one-to-one correspondence with the wave equation for an electric field in a moving medium. First the author points out a few errors in the literature concerning the Aharonov-Bohm (AB) effect then she explores the optical analogy further. The author finds that curl A plays a similar role to curl v where v is the velocity of the moving medium.
The electromagnetic wave equations in a moving medium may be approximated by a form similar to that of the Schrödinger equation for a particle in an electromagnetic field, with the velocity v of the medium and the vorticity ∇×v playing the roles of the vector potential and magnetic field, re- spectively. A purely classical optical analogue of the Aharonov–Bohm effect follows by consider- ation of the interference pattern produced by two beams, each of which propagates in a region with zero vorticity, but such that the flux of the vorticity through the closed loop defined by the optical paths does not vanish. Fizeau’s experiment (1851) on the velocity of light in moving media may be regarded as an example of such a situation.
Aharonov–Bohm effect is a quantum mechanical phenomenon that attracted the attention of many physicists and mathematicians since the publication of the seminal paper of Aharonov and Bohm [1] in 1959. We consider different types of Aharonov–Bohm effects such as the magnetic AB effect, electric AB effect, combined electromagnetic AB effect, AB effect for the Schrödinger equations with Yang–Mills potentials, and the gravitational analog of AB effect. We shall describe different approaches to prove the AB effect based on the inverse scattering problems, the inverse boundary value problems in the presence of obstacles, spectral asymptotics, and the direct proofs of the AB effect.
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 article about Aharonov-Bohm effects is interesting and comprehensive. The primary and best-known effect shows that the vector potential A of the electromagnetic field is a physical reality rather than a mathematical artifice. That reality was implicit in the Schrödinger equation, as the Hamiltonian H depends on A instead of the electric field E and magnetic field B = ∇ × A. But the same statement also refers to the Hamilton-Jacobi equation in classical mechanics, so one may expect that a similar effect exists in classical physics as well.Indeed, classical mechanics is governed by the fundamental Hamilton-Jacobi equation for the action S: δS/δt + H = 0, which naturally follows from William Hamilton’s principle of least action. Both H and S depend on A (even with E = B = 0). Thus it is quite possible that the vector potential is also a physical reality in classical physics. Erwin Schrödinger arrived at his famous equation in 1926 by using a mechanics-optics analogy, the so-called eikonal equation. Hamilton in 1834 proved that eikonal and Hamilton-Jacobi equations are equivalent, so that the Schrödinger equation actually follows from Hamilton-Jacobi.Had Hamilton known that classical mechanics does not always hold, quantum mechanics might have appeared a century earlier. 1 1. H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 2nd ed., Addison-Wesley, Reading, MA (1980), p.484 . The analogy between the equations of classical and quantum mechanics provides a reason to search for experiments that might prove whether a classical analogue of the Aharonov-Bohm effect exists.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 2nd ed., Addison-Wesley, Reading, MA (1980), p.484 . Google Scholar© 2010 American Institute of Physics.
It is frequently stated that the electromagnetic vector potential acquires a fundamental role in quantum physics, whereas classically it only represents a convenient, but by no means necessary, way of representing the electromagnetic field. Here we argue that this is a historical accident due to the fact that the electromagnetic field was discovered before photons, while the electron itself was discovered first as a particle, before it became clear that it must also be treated as a wave and therefore as an excitation of the underlying electron field. We illustrate the fact that the vector potential ought to play a fundamental role classically using the Aharonov-Bohm effect. This effect is considered as the strongest argument for the role the vector potential plays in quantum physics, however, here we offer a fully classical account of it. This is a consequence of the fact that any account, be it classical or quantum, must involve the vector potential in order to preserve the local nature of the Aharonov-Bohm (as well as all the other) phases.
Writing about what is usually termed the Aharonov-Bohm (AB) effect, Peter Sturrock and Timothy Groves argue (Physics Today, April 2010, page 8) that the same physics was discovered a decade earlier 1 1. W. Ehrenberg, R. E. Siday, Proc. Phys. Soc. B 62, 8 (1949). and should rightly be called the Ehrenberg-Siday effect. I agree that Werner Ehrenberg and Raymond Siday deserve recognition for their anticipation of AB. Indeed, in recent talks celebrating the 50th anniversary of AB, I began by describing the unfairly neglected paper by Ehrenberg and Siday. Nevertheless, I have come to a different conclusion from Sturrock and Groves: The expression “Aharonov-Bohm effect” is justified, for two reasons.First, although there is no doubt that the work by Ehrenberg and Siday anticipated how inaccessible magnetic flux can influence electron interference, that was as a curiosity, at the end of a paper whose main emphasis was the Hamiltonian analysis of electron optics. By contrast, Yakir Aharonov and David Bohm emphasized from the start, as an essential and general aspect of quantum mechanics, the physical influence of inaccessible fields that act nonlocally through the vector potential.Second, Ehrenberg and Siday’s semi-classical approximation—essentially applying the Dirac magnetic phase factor to electrons traveling on either side of the flux—implies a wavefunction that is multivalued and therefore not the correct solution of Schrödinger’s equation. The lack of a single-valued wavefunction leaves their prediction open to doubt. By contrast, Aharonov and Bohm derived the exact single-valued solution for waves scattered by a flux line. Recently, it was shown that the Ehrenberg-Siday approximation corresponds to the first terms in a “many-whirls” representation that treats the exact AB wavefunction as a topological sum over paths circling the flux. 2 2. M. V. Berry, J. Phys. A (in press). Attribution of credit is a delicate matter. It tends to excite strong feelings, and I write about it reluctantly. But the Ehrenberg-Siday paper does seem to exemplify the unfortunate phenomenon identified by Alfred North White-head in a 1916 address to the British Association for the Advancement of Science: “Everything of importance has been said before, by someone who did not discover it.”In a companion letter in the April 2010 issue, Alexander Ershkovich correctly points out that the AB effect is present in classical Hamiltonian mechanics, even though remote magnetic fields cannot influence Newtonian trajectories. He advocates a “search for experiments that might prove … a classical analogue of the Aharonov-Bohm effect.” Such an experiment exists already. In the classical physics of waves on a moving medium, the flow velocity acts like the vector potential in quantum mechanics, so the flow vorticity acts like the magnetic field; the analogy is precise. Fine details of the AB wavefunction were observed in ripples on the surface of water swirling irrotationally into a bathtub vortex, 3 3. M. V. Berry, R. G. Chambers, M. D. Large, C. Upstill, J. C. Walmsley, Eur. J. Phys. 1, 154 (1980). whose core is the analogue of inaccessible magnetic flux. REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. W. Ehrenberg, R. E. Siday, Proc. Phys. Soc. B 62, 8 (1949). Google ScholarCrossref2. M. V. Berry, J. Phys. A (in press). Google Scholar3. M. V. Berry, R. G. Chambers, M. D. Large, C. Upstill, J. C. Walmsley, Eur. J. Phys. 1, 154 (1980). Google ScholarCrossref© 2010 American Institute of Physics.