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American Journal of Physics

Finite and half-infinite solenoids and the Aharonov-Bohm effect

Joel Franklin, Gopal Goel

Published Oct 18, 2019• 2 citations
Publisher

Abstract

In quantum mechanics, the magnetic field has a physical effect beyond its local presence. For axially symmetric vector potentials, A=A(s,z)ϕ̂, it is the magnetic flux that plays a central role in both interference experiments (as in the Aharonov-Bohm effect) and in bound state spectra for a particle constrained to move in a circle. The flux dependence is evident in various configurations, and we have chosen two applications to highlight the non-local nature of the quantum mechanical magnetic field dependence: A finite solenoid and a Dirac string with a non-zero radius. The finite solenoid is interesting because it represents the actual experimental apparatus for measuring the Aharonov-Bohm effect. The Dirac string provides an electric charge quantization mechanism that depends on where the quantization argument is applied, yielding different fundamental charge units for different locations.

DOI
10.1119/1.5124814

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In quantum mechanics, the magnetic field has a physical effect beyond its local presence. For axially symmetric vector potentials, A=A(s,z)ϕ̂, it is the magnetic flux that plays a central role in both interference experiments (as in the Aharonov-Bohm effect) and in bound state spectra for a particle constrained to move in a circle. The flux dependence is evident in various configurations, and we have chosen two applications to highlight the non-local nature of the quantum mechanical magnetic field dependence: A finite solenoid and a Dirac string with a non-zero radius. The finite solenoid is interesting because it represents the actual experimental apparatus for measuring the Aharonov-Bohm effect. The Dirac string provides an electric charge quantization mechanism that depends on where the quantization argument is applied, yielding different fundamental charge units for different locations.
In 1959, Yakir Aharonov and David Bohm pointed out an experiment that should be impossible according to the standard intuitions of classical electromagnetism. A charged particle is sent through a double‑slit apparatus with a long thin solenoid placed between the two paths. The solenoid contains a magnetic field; the regions traversed by the particle do not. In classical mechanics, the particle should be entirely unaffected — no force, no torque, no Lorentz coupling. The interference pattern should be unchanged whether the solenoid is on or off. It isn't. When the solenoid is energized, the interference pattern shifts by Δφ = (q/ℏ) Φ, where Φ = ∫B·dA is the magnetic flux threading the solenoid. The particle’s wavefunction registers a region of magnetic field through which the particle never travels. Chambers performed the experiment in 1960; Tonomura confirmed it definitively with superconducting measurements in 1986. The effect is now standard textbook material and one of the deepest empirical phenomena in quantum mechanics. The puzzle is structural. In classical electromagnetism, the magnetic vector potential A was treated as a mathematical convenience — a quantity from which B = ∇×A could be computed but with no independent physical reality. The AB effect breaks this picture: the interference shift depends on ∮A·dx around the closed path, a gauge‑invariant integral that is non‑zero in regions where B = 0. The vector potential carries physical information that the magnetic field alone does not. Standard QM accommodates this by accepting that A is the physically meaningful quantity in the quantum context. This works mathematically but leaves a deeper question open: why should A have physical reality at all? This walkthrough shows how the Event Density framework provides an answer through Theorem 17. T17 establishes that gauge fields are the continuum‑level appearance of the substrate’s rule‑type connection — the substrate object that carries rule‑type label information from one substrate region to another. The gauge potential is not a mathematical convenience; it is the substrate’s parallel‑transport infrastructure for the rule‑type relabeling group. The AB phase is the integral of this substrate connection around a non‑contractible loop in a multiply‑connected substrate channel — a substrate‑level monodromy with observable consequences whenever the substrate’s topology supports closed loops the wavefunction can wind around. Stokes’ theorem connects the monodromy to the enclosed flux: Δφ = qΦ/ℏ. The structural payoff: the AB phase is not a mysterious nonlocal effect of the gauge potential. It is the substrate‑level monodromy of the rule‑type connection around a closed loop in a multiply‑connected substrate region. The gauge potential’s physical reality is not a postulate; it is what T17 establishes the substrate to provide.
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 role of the Aharonov-Bohm effect is analyzed for the bound states of a charged particle in the superposition of two fields: A uniform magnetic field plus the vector potential of a solenoid. Exact solutions are given for a solenoid of zero size and the corrections due to finite size are discussed. It is shown that a series of new "betatron resonances" appear in the spectrum, dependent on the flux quantum number $f=\frac{e\ensuremath{\Phi}}{\mathrm{hc}}$. The fundamental betatron resonance remains narrow and unshifted for solenoid radii large enough to broaden and shift the cyclotron resonance.
"It is a known fact that there is no magnetic field outside an infinite solenoid carrying a constant current, althoughthere is magnetic vector potential non-null outside it. The existence of the Aharonov-Bohm effect (AB) is usuallyconsidered as proof of the relevance of the vector potential for quantum mechanics. In this paper we will see that thereis a non null electric field outside an infinite resistive solenoid carrying a constant current and its possible relevanceto the analysis of the Aharonov-Bohm effect. When calculating, we introduce the Galilean invariance of Maxwell´sequations and then we obtain the contribution to the AB effect due to electric potential."
In the context of the Aharonov-Bohm effect, it is well-known that the local action of the vector potential or the non-local action of the magnetic field are properties attributable to quantum physics, which can explain the observed Aharonov-Bohm phase. Here we point out that, surprisingly, these properties are also attributable to classical physics and that this result follows from examining the electromagnetic angular momentum of a static electric charge outside an infinitely-long solenoid of finite radius carrying a uniform magnetic flux. This angular momentum may be expressed in three equivalent forms. The first form shows that there is an electromagnetic angular momentum even in regions for which the magnetic field vanishes but the potential does not, which stresses the physical nature of the vector potential in classical physics. The second form shows that there is an electromagnetic angular momentum produced by the non-local action of the magnetic field. The third form shows that the Aharonov-Bohm phase is proportional to the magnitude of the electromagnetic angular momentum, which sheds light on the physical meaning of this phase.
The Aharonov-Bohm (AB) effect is a pure quantum effect that implies a measurable phase shift in the wave function for a charged particle that encircles a magnetic flux located in a region inaccessible to the particle. Classically, such a nonlocal effect appears to be impossible since the Lorentz force depends on only the magnetic field at the location of the particle. In quantum mechanics, the Hamiltonian, and thus the Schr\"odinger equation, has a local coupling between the current due to the particle and the electromagnetic vector potential A\mathbf{A}A, which extends to the entire space beyond the region with finite magnetic field. This has sometimes been interpreted as meaning that in quantum mechanics A\mathbf{A}A is in some sense more ``fundamental'' than B\mathbf{B}B in spite of the former being gauge dependent and thus unobservable. Here we shall, with a general proof followed by a few examples, demonstrate that the AB effect can be fully accounted for by considering only the gauge-invariant B\mathbf{B}B field, as long as it is included as part of the quantum action of the entire isolated system. The price for the gauge-invariant formulation is that we must give up locality---the AB phase for the particle will arise from the change in the action for the B\mathbf{B}B field in the region inaccessible to the particle.