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 Schr\"odinger 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 Schr\"odinger 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 Schr\"odinger equations without the extra phase each provide the AB phase shift so that the product wave function produces the net AB phase shift.