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The Aharonov-Bohm Effect and Tonomura et al. Experiments. Rigorous Results

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.