  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.
