Quantum-mechanical inclusion of the source in the Aharonov-Bohm effects

Following semiclassical arguments by Vaidman [Phys. Rev. A 86, 040101(R) (2012)], we show that the phase shifts arising in the Aharonov-Bohm (AB) magnetic or electric effects can be treated as due to the electric force of a classical electron, respectively acting on quantized solenoid particles or quantized capacitor plates. This is in contrast to the usual approach that treats both effects as arising from non-field-producing potentials acting on the quantized electron. Moreover, we consider the problems of an interacting quantized electron and a quantized solenoid or a quantized capacitor to see what phase shift their joint wave function acquires. We show, in both cases, that the net phase shift is indeed the AB shift (for one might have expected twice the AB shift, given the above two mechanisms for each effect). The solution to the exact Schrödinger equation may be treated (approximately for the magnetic AB effect, which we show using a variational approach, exactly for the electric AB effect) as the product of two solutions of separate Schrödinger equations for each of the two quantized entities, but with an extra phase. The extra phase provides the negative of the AB shift, while the two separate Schrödinger equations without the extra phase each provide the AB phase shift so that the product wave function produces the net AB phase shift.

Figure 1

Electron packet trajectories in the $z=0$ plane and a cross section of the rotating cylindrical shells. The splitter boxes do not represent simply beam splitters but whatever optics is necessary to execute the electron behaviors explained in the text.

Figure 2

Illustrating one of the two superimposed cylindrical shells comprising the solenoid, a hollow ring within the shell containing charged pieces, and the wave packet of a piece.

Figure 3

Illustration of the notation.

Figure 4

Shown on top is the setup for electric AB effect, when the plates are together, before and after the electron nears the center of the plates. Here BS denotes beam splitter. (The left beam splitter's output differs from the right beam splitter's output, as illustrated.) On the bottom is an illustration of the plates with separation $\approx D$ while the electron moves for a time interval $T$. (Figures are not to scale; the plates and tubes should be greatly stretched in the $\pm x$ direction.)