Quantum Mechanical Inclusion of the Source in the Aharonov-Bohm Effects

Abstract: Following semiclassical arguments by Vaidman we show, for the first time in a fully quantum mechanical way, that the phase shifts arising in the Aharonov Bohm (A-B) 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 which treats both effects as arising from non-field producing potentials acting on the quantized electron. Moreover, we consider the problems of interacting quantized electron and quantized solenoid or 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 A-B shift (for, one might have expected twice the A-B shift, given the above two mechanisms for each effect.) The solution to the exact Schrodinger equation may be treated (approximately for the magnetic A-B effect, which we show using a variational approach, exactly for the electric A-B effect) as the product of two solutions of separate Schrodinger equations for each of the two quantized entities, but with an extra phase. The extra phase provides the negative of the A-B shift, while the two separate Schrodinger equations without the extra phase each provide the A-B phase shift, so that the product wave function produces the net A-B phase shift.