  We give a complete quantum analysis of the Aharonov-Bohm (AB) magnetic phase
shift involving three entities, the electron, the charges constituting the
solenoid current, and the vector potential. The usual calculation supposes that
the solenoid's vector potential may be well-approximated as classical. The AB
shift is then acquired by the quantized electron moving in this vector
potential. Recently, Vaidman presented a semi-classical calculation, later
confirmed by a fully quantum calculation of Pearle and Rizzi, where it is
supposed that the electron's vector potential may be well-approximated as
classical. The AB shift is then acquired by the quantized solenoid charges
moving in this vector potential. Here we present a third calculation, which
supposes that the electron and solenoid currents may be well-approximated as
classical sources. The AB phase shift is then shown to be acquired by the
quantized vector potential. We next show these are three equivalent alternative
ways of calculating the AB shift. We consider the exact problem where all three
entities are quantized. We approximate the wave function as the product of
three wave functions, a vector potential wave function, an electron wave
function and a solenoid wave function. We apply the variational principle for
the exact Schrodinger equation to this approximate form of solution. This leads
to three Schrodinger equations, one each for vector potential, electron and
solenoid, each with classical sources for the other two entities. However, each
Schrodinger equation contains an additional real c-number term, the time
derivative of an extra phase. We show that these extra phases are such that the
net phase of the total wave function produces the AB shift. Since none of the
three entities requires different treatment from any of the others, this leads
to three alternative views of the physical cause of the AB magnetic effect.
