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experiments:aharonov-bohm

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# The Aharonov-Bohm Experiment

## Layman

Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.

## Student

For quite some time it was felt that such phase changes in the wavefunction were of no physical significance since all of the physically measurable quantities associated with the charge q depend only on the squared modulus $|ψ|^2$ and this is the same for $ψ$ and $e^{iqΩ} ψ$. However, in 1959, Aharonov and Bohm [AB] suggested that, while the phase of a single charge may well be unmeasurable, the relative phase of two charged particles that interact should have observable consequences. They proposed an experiment that went roughly as follows: A beam of electrons is split into two partial beams that pass around opposite sides of a solenoid (this is a very tightly wound coil of wire through which a current passes, creating a magnetic field that is confined inside the coil ). Beyond the solenoid the beams are recombined and detected at a screen. The result is a typical interference pattern that manifests itself experimentally as a variation from point to point on the screen of the prob- ability of detecting a particle there. One observes this interference pattern when there is no current flowing through the coil, so that the magnetic field in the solenoid is zero, and then again when there is a current and hence a nonzero magnetic field inside the coil. Since the electrons pass around the coil and the magnetic field is confined inside the coil, any shift in the interference pattern in these two cases cannot be attributed to the magnetic field (which the electrons do not encounter). The vector potential, on the other hand, is generally nonzero outside the solenoid even though the magnetic field in this region is always zero. One could then only conclude that this vector potential induces different phase shifts on the two partial beams before they are recombined and that these relative phase changes account for the altered interference pattern. This experiment has, in fact, been performed (first by R. G. Chambers in 1960) with results that confirmed the expectations of Aharonov and Bohm.page 6 in Topology, Geometry and Gauge Fields: Foundations by Naber

## Researcher

Topologically, the solenoid is a defect. There are no fields outside the solenoid so the energy density of the fields are zero and there is a true vacuum, however, the vector potential changes the topology of the vacuum. We are familiar with the gauge idea that the vector potential can be written as the gradient of some other function χ so that ∇ × ∇χ = 0 identically. We can calculate χ by integrating the vector potential outside the solenoid with respect to the azimuthal coordinate Aθ = 1 r ∂χ ∂θ = BR2 2r or χ = 1 2BR2 θ which we see is not a single valued function of θ, that is χ(θ) 6= χ(θ + 2π). Functions that are not single valued can only exist in spaces that are not simply connected, or those that do not have trivial first homotopy group. We see that the manifold of the vacuum is thus the key to understanding the new physical result, and the topological defect gives us the intuition

• For the topology behind the Aharonov-Bohm effect, see section 10.5.3. in Geometry, Topology and Physics by Nakahara.

Example1
Example2: