  We study the Aharonov-Bohm effect under the conditions of the Tonomura et al.
experiments, that gave a strong evidence of the physical existence of the
Aharonov-Bohm effect, and we give the first rigorous proof that the classical
Ansatz of Aharonov and Bohm is a good approximation to the exact solution of
the Schroedinger equation. We provide a rigorous, quantitative, error bound for
the difference in norm between the exact solution and the approximate solution
given by the Aharonov-Bohm Ansatz. Our error bound is uniform in time. Using
the experimental data, we rigorously prove that the results of the Tonomura et
al. experiments, that were predicted by Aharonov and Bohm, actually follow from
quantum mechanics. Furthermore, our results show that it would be quite
interesting to perform experiments for intermediate size electron wave packets
(smaller than the ones used in the Tonomura et al. experiments, that were much
larger than the magnet) whose variance satisfies appropriate lower and upper
bounds that we provide. One could as well take a larger magnet. In this case,
the interaction of the electron wave packet with the magnet is negligible -the
probability that the electron wave packet interacts with the magnet is smaller
than $10^{-199}$- and, moreover, quantum mechanics predicts the results
observed by Tonomura et al. with an error bound smaller than $10^{-99}$, in
norm. Our error bound has a physical interpretation. For small variances it is
due to Heisenberg's uncertainty principle and for large variances to the
interaction with the magnet.
