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Abstract: 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 theAharonov-Bohm effect, and we give the first rigorous proof that the classicalAnsatz of Aharonov and Bohm is a good approximation to the exact solution ofthe Schroedinger equation. We provide a rigorous, quantitative, error bound forthe difference in norm between the exact solution and the approximate solutiongiven by the Aharonov-Bohm Ansatz. Our error bound is uniform in time. Usingthe experimental data, we rigorously prove that the results of the Tonomura etal. experiments, that were predicted by Aharonov and Bohm, actually follow fromquantum mechanics. Furthermore, our results show that it would be quiteinteresting to perform experiments for intermediate size electron wave packetssmaller than the ones used in the Tonomura et al. experiments, that were muchlarger than the magnet whose variance satisfies appropriate lower and upperbounds 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 -theprobability that the electron wave packet interacts with the magnet is smallerthan $10^{-199}$- and, moreover, quantum mechanics predicts the resultsobserved by Tonomura et al. with an error bound smaller than $10^{-99}$, innorm. Our error bound has a physical interpretation. For small variances it isdue to Heisenberg-s uncertainty principle and for large variances to theinteraction with the magnet.



Author: Miguel Ballesteros, Ricardo Weder

Source: https://arxiv.org/







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