We simulated time-varying Aharanov-Bohm effect (ABE) under time-independent magnetic field using Matlab, taking into account the influence of time-varying magnetic field on charged particles moving in a two-dimensional plane. According to our models, we discovered the time evolution of wave function and interference patterns. We also developed two mathematical models to analyze this phenomenon, of which the first mode focuses on the time evolution of the wave function, governed by the time-dependent Schrödinger equation incorporating electromagnetic potentials, while the second one is responsible for simulating the resulting interference patterns by modifying the intensity of wave packet based on the frequency-dependent phase shift caused by the sinusoidal magnetic field in the solenoid. Our results align with theoretical expectations and provide new insights into the frequency-dependent nature of the time-varying ABE.
The canonical Aharonov–Bohm effect is usually studied with time-independent potentials. In this work, we investigate the Aharonov–Bohm phase acquired by a charged particle moving in time-dependent potentials. In particular, we focus on the case of a charged particle moving in the time-varying field of a plane electromagnetic wave. We work out the Aharonov–Bohm phase using both the potential (i.e.
∮Aμdxμ ) and the field (i.e.
21∫Fμνdσμν ) forms of the Aharonov–Bohm phase. We give conditions in terms of the parameters of the system (frequency of the electromagnetic wave, the size of the space–time loop, amplitude of the electromagnetic wave) under which the time-varying Aharonov–Bohm effect could be observed.
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.
The Aharonov-Bohm (AB) effect demonstrates that charged particles can acquire a phase shift from a magnetic vector potential, even when propagating through regions devoid of magnetic fields. This dissertation develops time-domain numerical simulators to systematically investigate this effect across 2D and 3D coordinate systems. In 2D models, the Schrodinger equation is integrated using the Crank-Nicolson scheme. The Cartesian simulation reproduces flux-dependent fringe shifts, establishes grid resolution criteria, and demonstrates the necessity of the Coulomb gauge to avoid singular gauge artifacts. A 2D cylindrical model then verifies the coordinate-independence of the simulated AB phase shift. For 3D systems, a Strang-splitting scheme is introduced. Applied to a finite-length solenoid with elliptic-integral vector potentials, it utilizes FFT-based azimuthal diagonalization to isolate the pure AB phase inversion by initializing the wavepacket at the equatorial plane under small magnetic flux. Under fringing fields, the scheme captures the transition to classical Lorentz deflection, illustrating the quantum-classical correspondence. Finally, a spherical-coordinate model featuring a spherical-shell barrier employs a regularized wavefunction 𝑢 = 𝑟𝜓 to efficiently capture pre-slit phase inversions. To summarize, these platforms demonstrate the AB effect and provide a versatile framework for investigating quantum interference in realistic 3D geometries.
This paper introduces an unconditionally stable, accurate, and efficient algorithm to solve the time-dependent Schrödinger equation for a particle moving in an inhomogeneous magnetic field. The algorithm is used to simulate a fundamental thought experiment of quantum mechanics: the magnetic Aharonov–Bohm effect. The calculations demonstrate that the Aharonov–Bohm prediction of the phase shift holds with great precision, even when the dimensions of the interferometer are comparable to the de Broglie wavelength of the particle.
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.
Stokes' theorem is investigated in the context of the time-dependent Aharonov-Bohm effect—the two-slit quantum interference experiment with a time varying solenoid between the slits. The time varying solenoid produces an electric field which leads to an additional phase shift which is found to exactly cancel the time-dependent part of the usual magnetic Aharonov-Bohm phase shift. This electric field arises from a combination of a non-single valued scalar potential and/or a 3-vector potential. The gauge transformation which leads to the scalar and 3-vector potentials for the electric field is non-single valued. This feature is connected with the non-simply connected topology of the Aharonov-Bohm set-up. The non-single valued nature of the gauge transformation function has interesting consequences for the 4-dimensional Stokes' theorem for the time-dependent Aharonov-Bohm effect. An experimental test of these conclusions is proposed.