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arXiv:2306.02112 · Journal of Physics A Mathematical and Theoretical

Concerning the direction of the Aharonov–Bohm deflection

Timothy H. Boyer

Published Oct 10, 2023• 3 citations
arXivHTMLPDF

Abstract

Abstract The interaction of a solenoid with a passing charged particle can be treated within classical or quantum physics. If charged particles pass around both sides of a solenoid, there is an experimentally-observed Aharonov–Bohm deflection of the double-slit particle interference pattern between charges passing on opposite sides. Such a deflection can be obtained by a classical force calculation. Although the magnitude of the angular deflection agrees between the classical force calculation and the quantum topological theory, the direction of the predicted deflection is opposite. Here we point out the simple basis for the direction of the deflection based upon classical electrodynamics and based upon quantum theory. Also, we mention some deflection analogs, both the electrostatic deflection of the particle interference pattern and the optical analog of the classical calculation. The deflection direction involves an experimental question which is addressed rarely if ever. In the deflection direction, there is a direct experimental confrontation connected with the long-standing controversy involving the interpretation of the Aharonov–Bohm phase shift.

DOI
10.1088/1751-8121/ad0201

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Abstract The interaction of a solenoid with a passing charged particle can be treated within classical or quantum physics. If charged particles pass around both sides of a solenoid, there is an experimentally-observed Aharonov–Bohm deflection of the double-slit particle interference pattern between charges passing on opposite sides. Such a deflection can be obtained by a classical force calculation. Although the magnitude of the angular deflection agrees between the classical force calculation and the quantum topological theory, the direction of the predicted deflection is opposite. Here we point out the simple basis for the direction of the deflection based upon classical electrodynamics and based upon quantum theory. Also, we mention some deflection analogs, both the electrostatic deflection of the particle interference pattern and the optical analog of the classical calculation. The deflection direction involves an experimental question which is addressed rarely if ever. In the deflection direction, there is a direct experimental confrontation connected with the long-standing controversy involving the interpretation of the Aharonov–Bohm phase shift.
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.
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 long‐standing controversy regarding the Aharonov‐Bohm phase shift is reviewed. The shifts of both optical and particle interference patterns are summarized. It is pointed out that a line of electric dipoles and a line of magnetic dipoles (a long solenoid) both produce experimentally observed phase shifts similar to that produced by introducing a rectangular block of glass behind one slit of a double‐slit interference pattern; the double‐slit pattern is shifted while the single‐slit envelope remains undisplaced. The quantum explanation for the magnetic interference pattern shift introduced by Aharonov and Bohm in 1959 involves completely different ideas from those suggested by a semiclassical analysis. Experiments planned by Caprez, Barwick, and Batelaan should clarify the connections between classical and quantum theories in connection with the Aharonov‐Bohm phase shift.
In 1984 Aharonov and Casher proposed that a beam of neutral particles with magnetic dipole moments passing around opposite sides of a line charge will undergo a relative quantum phase shift. This is currently under experimental investigation. Aharonov and Casher claimed that the magnetic dipole particles should undergo the phase shift despite experiencing no classical force. They suggested that this new effect has a ``duality'' with the solenoid Aharonov-Bohm effect where charged particles passing around a magnetic solenoid experience a phase shift despite, it is claimed, experiencing no classical force. Here it is pointed out that a magnetic dipole particle passing a line charge does indeed experience a classical electromagnetic force in the usual electric-current model for a magnetic dipole. This force will produce a relative lag between dipoles passing on opposite sides of the line charge, and the classical lag then leads to a quantum phase shift in exact agreement with that calculated by Aharonov and Casher. Thus actually the proposed Aharonov-Casher effect has a transparent explanation as a classical lag effect. It is emphasized that the solenoid Aharonov-Bohm effect is currently the only phase-shift effect which cannot be explained in an obvious fashion as arising from a classical lag and it is again proposed that also this shift may actually involve a classical electromagnetic lag effect. A natural experimental consequence suggested by the lag point of view is the breakdown of the interference pattern when the lag becomes comparable to the wave-packet coherence length.
The traditional Aharonov Bohm effect (AB effect) attributes the light and dark fringes on screen to phase interference of electron matter wave. It presumes that the magnetic field outside the solenoid equals zero and particles receive no force. This system has long served as key support for wave particle duality. This paper completely abandons the hypothesis of wave interference and quantum phase. It establishes the model of electron vibrating point particle to reveal the real physical mechanism of AB effect. Fringes are not superposition product of waves. They are formed by differential deflection of charged electrons under magnetic force and Coulomb repulsion, followed by regular superposition of two upper and lower particle beams. Both magnetic force and Coulomb repulsion are long range forces. They only differ in effective action range and variation rule of force with distance. Comparative experiments using direct current solenoid and direct current long straight wire can falsify the unrealistic assumption that magnetic field outside coil cancels out to zero. Supporting single side shielding verification experiment can directly test the correctness of orthodox quantum interpretation. Electrons have no wave property. As long as point particles with inherent vibration are modulated by field, they can form fringe similar to interference fringe. Photons are also point particles with inherent vibration. When photons pass through slits, they interact with slit wall and produce deflection, so as to form photon diffraction and interference fringes. The hypothesis of wave particle duality is invalid.
In 1959 Aharonov and Bohm suggested that an electron passing around a long solenoid would pick up a phase shift dependent on the magnetic field of the solenoid, even though the electrons themselves pass through a region of space which has a zero magnetic field. It has long been held that this result is purely quantum and is derived in many well known quantum mechanics text books using the Schrodinger equation and vector potential. Here the same phase shift is derived from a purely classical force, but relativistic transformations are taken into account. The force is in the direction of motion of the electron (or opposite) leading to a phase advance (or lag) and we obtain precisely the phase shift thought previously to be purely quantum. The only quantum result used here is the de Broglie wavelength of the particle, in order to get two slit like interference and the phase shift. We employ a stack of dipoles as the solenoid and note the same force on the electron in two different frames of reference. We shall consider the solenoid stationary and the electron moving, and then consider the electron rest frame and consider the solenoid moving in the opposite direction.