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Journal of Physics A Mathematical and Theoretical

Kirchhoff diffraction optics and the nascent Aharonov–Bohm effect: a theorem

J H Hannay

Published Aug 12, 2010• 4 citations
Publisher

Abstract

Textbook diffraction optics, Kirchhoff diffraction, is connected to the Aharonov–Bohm effect of quantum mechanics by an easy theorem proved here. The connection is between the Kirchhoff wave field and the Aharonov–Bohm quantum wave field in the limit of zero flux: the 'nascent' Aharonov–Bohm effect. The diffracting opaque screen of Kirchhoff optics is replaced in the quantum mechanics by a magnetic flux line, or loop, in the shape of the boundary edge of the screen. The gauge must be chosen appropriately: a delta function on that surface, spanning the boundary edge, which matches the screen.

DOI
10.1088/1751-8113/43/35/354001

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Textbook diffraction optics, Kirchhoff diffraction, is connected to the Aharonov–Bohm effect of quantum mechanics by an easy theorem proved here. The connection is between the Kirchhoff wave field and the Aharonov–Bohm quantum wave field in the limit of zero flux: the 'nascent' Aharonov–Bohm effect. The diffracting opaque screen of Kirchhoff optics is replaced in the quantum mechanics by a magnetic flux line, or loop, in the shape of the boundary edge of the screen. The gauge must be chosen appropriately: a delta function on that surface, spanning the boundary edge, which matches the screen.
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.
Quantum-mechanical scattering off a magnetic vortex is considered, and the optical theorem is derived. The vortex core is assumed to be impermeable to scattered particles, and its transverse size is taken into account. We show that the scattering Aharonov–Bohm effect is independent of the choice of boundary conditions from the variety of the Robin ones. The behaviour of the scattering amplitude in the forward direction is analysed, and the persistence of the Fraunhofer diffraction in the short-wavelength limit is shown to be crucial for maintaining the optical theorem in the quasiclassical limit.
Since the Aharonov-Bohm effect is the purely quantum effect that has no analogues in classical physics, its persistence in the quasiclassical limit seems to be hardly possible. Nevertheless, we show that the scattering Aharonov-Bohm effect does persist in the quasiclassical limit owing to the diffraction, i.e. the Fraunhofer diffraction in the case when space outside the enclosed magnetic flux is Euclidean, and the Fresnel diffraction in the case when the outer space is conical. Hence, the enclosed magnetic flux can serve as a gate for the propagation of short-wavelength, almost classical, particles. In the case of conical space, this quasiclassical effect which is in principle detectable depends on the particle spin.
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.
The Aharonov-Bohm (AB) equation is reformulated by means of the gauge transformation. A scattering theoretical approach yields an integral equation of the form of Kirchhoff-Huygens. The method to solve this equation is analogous to the one previously used to study the dislocation mediated AB effect. It is reconfirmed that the shadow of the magnetic-flux filament appears in the down stream of the electronic flux. Expression for the large angle scattering obtained by AB is improved and the present expression is shown to agree with numerical solution.