In the well-known Aharonov-Bohm and Ehrenberg-Siday effects, the diffraction envelope of an electron is not affected by Lorentz forces in a region of zero magnetic field; instead, a phase difference—created by the magnetic vector potential—shifts the constructive- and destructive-interference fringe positions when electron wavefunctions are overlapped. This shift in fringe positions has been previously derived as resulting from potential-induced phase differences, without being clear on the physical mechanism behind it. In this paper, we show that the de Broglie wavelength of the electron is changed locally by its interaction with the vector potential. The vector potential thus acts as a quantum “phase plate,” changing the phase difference between interfering electron wavefunctions in a non-dispersive, gauge-invariant manner.
The Aharonov-Bohm effect – where electron-interference fringe positions are shifted based on the presence of a non-local magnetic field – has not yet been utilised to its full potential in quantum device design. In this paper, we show that the de Broglie wavelength of the electron is changed locally by its interaction with the magnetic vector potential. The vector potential thus acts as a quantum “phase plate”, changing the phase difference between interfering electron wavefunctions in a non-dispersive, gauge-invariant manner. This model is then shown to be applicable to the design and optimisation of phase-dependent quantum devices such as superconducting quantum interference devices (SQUIDs), quantum sensors, and so on.
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 AB (Aharonov-Bohm) effect is a pivotal quantum mechanical phenomenon that illustrates the fundamental role of the electromagnetic vector potential A in determining the phase of a charged particle's wave function, even in regions where the magnetic field B is zero.This effect demonstrates that quantum particles are influenced not only by the fields directly present but also by the potentials associated with those fields.In the AB effect, an electron beam is split into two paths, with one path encircling a solenoid and the other bypassing it.Despite the absence of a magnetic field in the regions traversed by the beams, the vector potential A associated with the magnetic flux through the solenoid induces a phase shift in the electron's wave function.This phase shift, quantified by qc , manifests as a change in the interference pattern observed in the detection screen.The phenomenon underscores the principle of gauge invariance in QED (quantum electrodynamics), where physical observables remain invariant under local gauge transformations of the vector and scalar potentials.This reinforces the notion that the vector potential A has a profound impact on quantum systems, beyond its classical role.This article outlines the AB effect, including its theoretical framework, experimental observations, and implications.The focus on the role of the vector potential in quantum mechanics provides a comprehensive understanding of this important phenomenon.
Recent experiments with the Aharonov-Bohm geometry have shown that in addition to an electron-interference fringe shift, there is also a lateral displacement of the electron diffraction envelope. In this paper, we derive an electron displacement force based on a second-order expansion of the magnetic vector potential. The analysis illustrates the conservation of canonical angular momentum, where the mechanical angular momentum and field angular momentum sum to a constant of the motion; the azimuthal force required to change the mechanical momentum is thus supplied by changes in field momentum associated with the second-order vector potential term. Our results are consistent with all known Aharonov-Bohm experiments, including interference fringe shifts, lateral displacements, and the absence of longitudinal time delays.
When an electron beam passes outside a confined magnetic flux, the motional magnetic field of the electrons overlaps the confined flux. The potential energy of the magnetic dipolar interaction of the motional magnetic field with the confined magnetic flux causes the velocity of the electrons to change slightly, which shifts the phase of the electron de Broglie wave. When portions of an electron beam pass on different sides of a confined flux and are then mixed, they produce an interference pattern. There is a similar magnetic dipolar interaction, with resulting velocity and phase changes, when a beam of magnetic particles interacts with the motional magnetic flux produced by their crossing a strong electric field. These are classical electromagnetic dipolar (not Lorentz) interactions that causes changes in Berry's geometrical phase, but are not ESAB or AC effects. Data that were suggested to demonstrate the ESAB and AC effects have been quantitatively interpreted in terms of motional magnetic fields and velocity changes.
The Aharonov-Bohm (AB) effect highlights the fundamental role of electromagnetic potentials in quantum mechanics, manifesting as a phase shift for a charged particle in field-free regions. While well-established for static magnetic fluxes, the effect's behavior under time-varying fluxes remains an open and debated question. Employing the WKB method, we derive the AB phase shift for a time-dependent magnetic vector potential, demonstrating that for circular paths in the quasistatic regime, it is proportional to the time-averaged enclosed magnetic flux,
ΔφAB=T1∫0TeΦ(t)dt, with the total phase shift, including kinetic contributions, equaling
eΦ(0). For non-circular paths, the phase shift depends on both the flux history and path geometry, revealing the effect's hybrid nature involving gauge potentials and induced electric fields. We verify the consistency of our gauge choice with Maxwell's equations and discuss the implications for local versus nonlocal interpretations of the AB effect. We also generalize the results to scenarios with nonzero external magnetic fields, where the enclosed flux is through the actual electron paths, and for circular paths of radius
R, the AB phase shift is also proportional to the time average of the enclosed flux
Φenc(R,t), with the total phase shift depending only on the initial enclosed flux
eΦenc(R,0); for general non-circular paths, the external magnetic field affects trajectories and phase accumulation through the Lorentz force, leading to additional path dependence. These findings clarify the role of gauge-dependent potentials and induced fields in the generalized AB effect, offering new theoretical insights and potential applications in quantum technologies.