Classical Aspects of the Aharonov-Bohm Effect

(−ih̄D)2 2m ψ = ih̄D0ψ, using the “altered” (covariant) derivative Dμ = ∂μ − ieAμ/h̄c, (1) which is gauge invariant only if the gauge transformation of the potentials, Aμ(xν) → Aμ + ∂μΩ(xν), is accompanied by a phase change of the wavefunction, ψ(xν) → e−ieΩ(xν )/h̄c ψ. Yang and Mills (1954) [4, 5] may have been the first to point out that Fock’s argument can be inverted such that a requirement of local phase invariance of the form ψ(xν) → e−ieΩ(xν )/h̄c ψ implies the existence of an interaction described by a potential Aμ (and charge e) which satisfies gauge invariance and modifies Schrödinger’s equation via the altered derivative Dμ. This led to a greater appreciation of the significance of potentials in the quantum realm. Separately, consideration of possible interference effects in electron microscopy [6] led Aharonov and Bohm [7, 8] to discuss an electron that moves only outside a long solenoid magnet (where B = 0 to a good approximation), and which accumulates a different phase in its wavefunction depending on which side of the magnet it passes. The resulting interference pattern, which depends on the (gauge-invariant) magnetic flux in the solenoid (that can be related to the vector potentialA in whatever gauge is used), has been observed in subsequent experiments [9, 10]. The quantum interference effect in the Aharonov-Bohm experiment is impressive, but there are already disconcerting issues in purely classical considerations thereof. It is often remarked that there is no classical effect on an electron that passes outside a long solenoid magnet, where Bsolenoid = 0. However, the current density that generates the solenoid field is affected by the magnetic field of the moving electron (even assuming that the electric charge density associated with the current density is zero). Problem: Deduce the force on a solenoid of radius a about the z-axis that carries azimuthal surface current density Kθ = I per unit length, when an electron of velocity v = v ŷ is at position (x, y, z) = (b, vt, 0), where v c and |b| a.