Aharonov-Bohm
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bulletIntroduction
bulletComputer experiment
bulletNecessary conditions

Introduction

In classical mechanics the motion of a charged particle is not affected by the presence of magnetic fields in regions from which the particle is excluded. The motion of classical particles emitted by the source S is not affected by the magnetic field B because the particles can not enter the region of space where the magnetic field is present. For a quantum charged particle there can be an observable phase shift in the interference pattern recorded at the detector D. This phase shift results from the fact that although the magnetic field is zero in the space accessible to the particle, the associated vector potential is not. The phase shift depends on the flux enclosed by the two alternative sets of paths a and b. But the overall envelope of the diffraction pattern is not displaced indicating that no classical magnetic force acts on the particles. The Aharonov-Bohm effect demonstrates that the electromagnetic potentials, rather than the electric and magnetic fields, are the fundamental quantities in quantum mechanics.

Computer experiment

The following animation shows the diffraction of a Gaussian wave packet by two slits. In this case there is no magnetic field. 

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The angular distribution I (q ) recorded by a detector placed far away from a source characterizes the interference pattern. For zero magnetic field the simulation yields an angular distribution I (q ) that is in excellent agreement with the Fraunhofer diffraction pattern of a double slit. 

The next animation illustrates the Aharonov-Bohm effect. The magnetic field is confined to the red area and is chosen such that the shift in the interference pattern is as large as possible.

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Inspection of the angular distribution shows that compared to the zero field case the maxima and the minima are interchanged:

The diffraction pattern is shifted by an angle f. In general f ( B ) = f ( B + 2B' ), where B' is the magic field for which f ( B ) = 180. The interchange of maxima and minima can also be demonstrated by superimposing the animations for zero and magic magnetic field. In the animation that follows only the blue colored wave packet is affected by the vector potential.

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Necessary conditions

The necessary conditions to observe the Aharonov-Bohm effect, i.e. a shift of the diffraction pattern that varies periodically with B, are: 

  1. There must be at least two interfering alternatives for the particle to arrive at the detector, and 
  2. At least two of these interfering alternatives must enclose a shielded magnetic field and must be topologically distinct. 

The next two animations illustrate the importance of the first condition. Each animation shows the superposition of the waves for the case with (in blue) and without (in red) a magnetic field. The field B = B'. 

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The Aharonov-Bohm effect vanishes as the number of interfering alternatives decreases. As illustrated in the following animation, increasing the width of and the distance between the two slits reduces the amount of scattering and therefore also the Aharonov-Bohm effect.

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Turning the double slit into a single slit changes the topology of the space accessible to the particle. In the absence of a magnetic field the interference pattern is that of a single slit. 

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When the magnetic field inside the slit is turned on the particle experiences the Lorentz force, resulting in a displacement of the diffraction pattern as a whole. However, this displacement is not a periodic function of the applied field. 

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Inspection of the angular distribution shows that compared to the zero field case the maximum is shifted:

In this simply-connected geometry there is no systematic interchange of the maxima and minima of the diffraction pattern. The following animation shows the superposition of the B = 0 (in red) and B = B' (in blue) simulations. 

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For a single-slit geometry the Aharonov-Bohm effect is suppressed because there are no topologically distinct paths for the particle to pass through the slit.


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