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arXiv:0709.0466 · Physical Review A

Comment on “Relativistic Aharonov-Bohm effect in the presence of planar Coulomb potentials”

C. R. Hägen

Published Mar 27, 2008• 14 citations
arXivHTMLPDF

Abstract

It is shown that the principal results of a recent work by Khalilov [Phys. Rev. A 71, 012105 (2005)] are in conflict with earlier work of the present author [Phys. Rev. Lett. 64, 503 (1990)] and also with a well known result concerning helicity conservation in a magnetic field. This can be attributed to the fact that Khalilov requires wave function regularity at the origin even when the relevant potential is singular at that point.

DOI
10.1103/physreva.77.036101

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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.

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Comment on “Aharonov-Casher and Scalar Aharonov-Bohm Topological Effects”

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In this Comment we point out (i) that the Hamiltonian, Eq. (17) in the Letter(Phys. Rev. Lett. 108, 070405 (2012)), is not a relativistic Hamiltonian, (ii) then that the conditions in the Letter are irrelevant for a topological AC and SAB effects, and (iii) conclusively that the non-relativistic Hamiltonian employed by Peshkin and Lipkin (Phys. Rev. Lett. 74, 2847 (1995)) has the same $U(1)_{mm}$ gauge structure for a fixed spin and then is not wrong, but their incorrect interpretation of the spin autocorrelations led to the incorrect conclusion.

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Abstract We focus on new expressions for the Aharonov–Bohm (AB) phase proposed recently by C. Marletto and V. Vedral (Phys. Rev. Lett. 125, 040401 (2020)) and by our team (A. L. Kholmetskii et al., Ann. Phys., 434, 168627 (2021)). We emphasize the mismatch between the standard expression for the electric component of the AB phase and the expression by Marletto and Vedral for this component, which, as we show, has fundamental roots and indicates the need to use different expressions for the Hamiltonian of the radiative component and bound component of the electromagnetic (EM) field. We further show that the corrected expression for the AB phase of Marletto and Vedral and our expression for the AB phase do not coincide with its standard representation through the potentials of EM field (Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485) and contain additional terms, which straightforwardly allow to re-express the AB phase via electric and magnetic fields instead of scalar and vector potentials. The recognition of the field-dependent origin of the AB effect also requires to re-express the Hamiltonian of a charged particle through the EM field, which we show can be done in the only possible way. We discuss the physical meaning of the AB effect with the primary role of the EM field and also discuss some general quantum mechanical implications.

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Relativistic Aharonov-Bohm effect in the presence of planar Coulomb potentials

V. R. Khalilov

Exact analytic solutions are found to the Dirac equation in $2+1$ dimensions for a combination of an Aharonov-Bohm potential and the Lorentz three-vector and scalar Coulomb potentials. By means of the solutions obtained the relativistic quantum Aharonov-Bohm effect is studied for the free (in the presence of a Lorentz three-vector Coulomb potential) and bound fermion states. We obtain the total scattering amplitude in a combination of the Aharonov-Bohm and Lorentz three-vector Coulomb potentials as a sum of two scattering amplitudes. This modifies the expression for the standard Aharonov-Bohm cross section due to the interference of these two amplitudes with each other. We discuss that the observable quantities can be the phases of electron wave functions or the energies of bound states.

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Aptly named Aharonov-Bohm effect has classical analogue, long history

Murray Peshkin

I am an author of the 1961 paper that first called the Aharonov-Bohm effect by that name. 1 1. M. Peshkin, I. Talmi, L. J. Tassie, Ann. Phys. 12, 426 (1961). The term has since appeared in more than 1000 published papers. However, it has been suggested a few times, most recently by Peter Sturrock and Timothy Groves, that the name does not do justice to Werner Ehrenberg and Raymond Siday, who found in 1949 that the motion of electrons can be influenced by magnetic fields confined to regions that the electrons do not enter. 2 2. W. Ehrenberg, R. E. Siday, Proc. Phys. Soc. B 62, 8 (1949). I argue that the name Aharonov-Bohm effect is appropriate, although for a reason that I did not fully appreciate in 1961.The 1959 Aharonov-Bohm paper profoundly changed the way we think about electromagnetic fields in quantum mechanics. 3 3. Y. Aharonov, D. Bohm, Phys. Rev. 115, 485 (1959). The gauge-invariant part of the vector potential was promoted to a real physical field, not just a convenient device for summarizing certain information about the electric and magnetic fields. In the words of C. N. Yang, “The electromagnetic field strength fµv in quantum mechanics underdescribes electromagnetism, as the Bohm-Aharonov effect demonstrates … information about the phase factor exp[ie∮(A⋅dx)/ℏc] for all closed loops correctly describes electromagnetism.” 4 4. C. N. Yang, in Proceedings of the International Symposium on Foundations of Quantum Mechanics in the Light of New Technology, Tokyo, 1983, S. Kamefuchi et al. , eds., Physical Society of Japan, Tokyo (1984), p. 5.See also T. T. Wu, C. N. Yang, Phys. Rev. D 12, 3845 (1975). A century earlier, James Clerk Maxwell changed the way we think about action at a distance by identifying what we now call the Maxwell fields as real physical things that contain energy and momentum and that enable microscopic conservation laws; they are not just mathematical functions that summarize the necessary information about the past motions of charges. Later those physical fields had to be quantized. Yakir Aharonov and David Bohm found that in quantum mechanics the Maxwell fields in a multiply connected region do not contain all the physics; the vector potential must also be endowed with reality to make sense of the subtler interactions in quantum mechanics. Neither Aharonov and Bohm nor Ehrenberg and Siday were the first to observe that magnetic fields in places where a charged particle’s wave function vanishes may influence the motion of that particle. 5 5. P. A. M. Dirac, Proc. R. Soc. London A 133, 60 (1931) F. London, Nature 140, 793 (1937). Paul Dirac, in his 1931 paper on magnetic monopoles, noted that the electron’s wavefunction must vanish on singular flux lines but said nothing about that raising any question of nonlocality. Fritz London pointed out in the 1930s that the motion of electrons in a superconducting ring depends on an external magnetic flux through the hole in the ring, where the electrons cannot go. He wrote in his 1937 paper, “The most stable state of a ring has no current, unless an external magnetic field is applied.” Like Dirac, he said nothing about a nonlocal action of the magnetic field being surprising or unusual. Of course, London, like Dirac, was focusing on something else.Ehrenberg and Siday were also focusing on something else—electron optics—when they found in 1949 that the motion of an electron can depend on the magnetic field in a region from which the electron is excluded. They chose not to mention that curious phenomenon in the abstract of their paper, although they did explicitly say elsewhere that it was “curious.”None of those earlier authors went on to conclude that such a phenomenon implies that the vector potential has to be seen as a real physical field in quantum mechanics. Only Aharonov and Bohm did that.REFERENCESSection:ChooseTop of pageREFERENCES <<1. M. Peshkin, I. Talmi, L. J. Tassie, Ann. Phys. 12, 426 (1961). Google ScholarCrossref, ISI2. W. Ehrenberg, R. E. Siday, Proc. Phys. Soc. B 62, 8 (1949). Google ScholarCrossref3. Y. Aharonov, D. Bohm, Phys. Rev. 115, 485 (1959). Google ScholarCrossref, ISI4. C. N. Yang, in Proceedings of the International Symposium on Foundations of Quantum Mechanics in the Light of New Technology, Tokyo, 1983, S. Kamefuchi et al. , eds., Physical Society of Japan, Tokyo (1984), p. 5. Google ScholarSee also T. T. Wu, C. N. Yang, Phys. Rev. D 12, 3845 (1975). Google ScholarCrossref, ISI5. P. A. M. Dirac, Proc. R. Soc. London A 133, 60 (1931) Google ScholarCrossref F. London, Nature 140, 793 (1937). , Google ScholarCrossref© 2010 American Institute of Physics.

Physics Today•Aug 1, 2010•1 citation
It is shown that the principal results of a recent work by Khalilov [Phys. Rev. A 71, 012105 (2005)] are in conflict with earlier work of the present author [Phys. Rev. Lett. 64, 503 (1990)] and also with a well known result concerning helicity conservation in a magnetic field. This can be attributed to the fact that Khalilov requires wave function regularity at the origin even when the relevant potential is singular at that point.
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 this Comment we point out (i) that the Hamiltonian, Eq. (17) in the Letter(Phys. Rev. Lett. 108, 070405 (2012)), is not a relativistic Hamiltonian, (ii) then that the conditions in the Letter are irrelevant for a topological AC and SAB effects, and (iii) conclusively that the non-relativistic Hamiltonian employed by Peshkin and Lipkin (Phys. Rev. Lett. 74, 2847 (1995)) has the same U(1)mmU(1)_{mm}U(1)mm​ gauge structure for a fixed spin and then is not wrong, but their incorrect interpretation of the spin autocorrelations led to the incorrect conclusion.
Abstract We focus on new expressions for the Aharonov–Bohm (AB) phase proposed recently by C. Marletto and V. Vedral (Phys. Rev. Lett. 125, 040401 (2020)) and by our team (A. L. Kholmetskii et al., Ann. Phys., 434, 168627 (2021)). We emphasize the mismatch between the standard expression for the electric component of the AB phase and the expression by Marletto and Vedral for this component, which, as we show, has fundamental roots and indicates the need to use different expressions for the Hamiltonian of the radiative component and bound component of the electromagnetic (EM) field. We further show that the corrected expression for the AB phase of Marletto and Vedral and our expression for the AB phase do not coincide with its standard representation through the potentials of EM field (Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485) and contain additional terms, which straightforwardly allow to re-express the AB phase via electric and magnetic fields instead of scalar and vector potentials. The recognition of the field-dependent origin of the AB effect also requires to re-express the Hamiltonian of a charged particle through the EM field, which we show can be done in the only possible way. We discuss the physical meaning of the AB effect with the primary role of the EM field and also discuss some general quantum mechanical implications.
Exact analytic solutions are found to the Dirac equation in 2+12+12+1 dimensions for a combination of an Aharonov-Bohm potential and the Lorentz three-vector and scalar Coulomb potentials. By means of the solutions obtained the relativistic quantum Aharonov-Bohm effect is studied for the free (in the presence of a Lorentz three-vector Coulomb potential) and bound fermion states. We obtain the total scattering amplitude in a combination of the Aharonov-Bohm and Lorentz three-vector Coulomb potentials as a sum of two scattering amplitudes. This modifies the expression for the standard Aharonov-Bohm cross section due to the interference of these two amplitudes with each other. We discuss that the observable quantities can be the phases of electron wave functions or the energies of bound states.
I am an author of the 1961 paper that first called the Aharonov-Bohm effect by that name. 1 1. M. Peshkin, I. Talmi, L. J. Tassie, Ann. Phys. 12, 426 (1961). The term has since appeared in more than 1000 published papers. However, it has been suggested a few times, most recently by Peter Sturrock and Timothy Groves, that the name does not do justice to Werner Ehrenberg and Raymond Siday, who found in 1949 that the motion of electrons can be influenced by magnetic fields confined to regions that the electrons do not enter. 2 2. W. Ehrenberg, R. E. Siday, Proc. Phys. Soc. B 62, 8 (1949). I argue that the name Aharonov-Bohm effect is appropriate, although for a reason that I did not fully appreciate in 1961.The 1959 Aharonov-Bohm paper profoundly changed the way we think about electromagnetic fields in quantum mechanics. 3 3. Y. Aharonov, D. Bohm, Phys. Rev. 115, 485 (1959). The gauge-invariant part of the vector potential was promoted to a real physical field, not just a convenient device for summarizing certain information about the electric and magnetic fields. In the words of C. N. Yang, “The electromagnetic field strength fµv in quantum mechanics underdescribes electromagnetism, as the Bohm-Aharonov effect demonstrates … information about the phase factor exp[ie∮(A⋅dx)/ℏc] for all closed loops correctly describes electromagnetism.” 4 4. C. N. Yang, in Proceedings of the International Symposium on Foundations of Quantum Mechanics in the Light of New Technology, Tokyo, 1983, S. Kamefuchi et al. , eds., Physical Society of Japan, Tokyo (1984), p. 5.See also T. T. Wu, C. N. Yang, Phys. Rev. D 12, 3845 (1975). A century earlier, James Clerk Maxwell changed the way we think about action at a distance by identifying what we now call the Maxwell fields as real physical things that contain energy and momentum and that enable microscopic conservation laws; they are not just mathematical functions that summarize the necessary information about the past motions of charges. Later those physical fields had to be quantized. Yakir Aharonov and David Bohm found that in quantum mechanics the Maxwell fields in a multiply connected region do not contain all the physics; the vector potential must also be endowed with reality to make sense of the subtler interactions in quantum mechanics. Neither Aharonov and Bohm nor Ehrenberg and Siday were the first to observe that magnetic fields in places where a charged particle’s wave function vanishes may influence the motion of that particle. 5 5. P. A. M. Dirac, Proc. R. Soc. London A 133, 60 (1931) F. London, Nature 140, 793 (1937). Paul Dirac, in his 1931 paper on magnetic monopoles, noted that the electron’s wavefunction must vanish on singular flux lines but said nothing about that raising any question of nonlocality. Fritz London pointed out in the 1930s that the motion of electrons in a superconducting ring depends on an external magnetic flux through the hole in the ring, where the electrons cannot go. He wrote in his 1937 paper, “The most stable state of a ring has no current, unless an external magnetic field is applied.” Like Dirac, he said nothing about a nonlocal action of the magnetic field being surprising or unusual. Of course, London, like Dirac, was focusing on something else.Ehrenberg and Siday were also focusing on something else—electron optics—when they found in 1949 that the motion of an electron can depend on the magnetic field in a region from which the electron is excluded. They chose not to mention that curious phenomenon in the abstract of their paper, although they did explicitly say elsewhere that it was “curious.”None of those earlier authors went on to conclude that such a phenomenon implies that the vector potential has to be seen as a real physical field in quantum mechanics. Only Aharonov and Bohm did that.REFERENCESSection:ChooseTop of pageREFERENCES <<1. M. Peshkin, I. Talmi, L. J. Tassie, Ann. Phys. 12, 426 (1961). Google ScholarCrossref, ISI2. W. Ehrenberg, R. E. Siday, Proc. Phys. Soc. B 62, 8 (1949). Google ScholarCrossref3. Y. Aharonov, D. Bohm, Phys. Rev. 115, 485 (1959). Google ScholarCrossref, ISI4. C. N. Yang, in Proceedings of the International Symposium on Foundations of Quantum Mechanics in the Light of New Technology, Tokyo, 1983, S. Kamefuchi et al. , eds., Physical Society of Japan, Tokyo (1984), p. 5. Google ScholarSee also T. T. Wu, C. N. Yang, Phys. Rev. D 12, 3845 (1975). Google ScholarCrossref, ISI5. P. A. M. Dirac, Proc. R. Soc. London A 133, 60 (1931) Google ScholarCrossref F. London, Nature 140, 793 (1937). , Google ScholarCrossref© 2010 American Institute of Physics.