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

We study the Aharonov-Bohm effect under the conditions of the Tonomura et al. experiments 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 Schrödinger equation. In remarkable experiments, Tonomura et al. enclosed a magnetic flux in the interior of a toroidal magnet and they superimposed behind the magnet an electron wave packet that traveled inside the hole of the magnet with a reference electron wave packet that traveled outside the magnet, and they measured the phase shift produced by the magnetic flux inside the magnet. These experiments gave a strong evidence of the physical existence of the Aharonov-Bohm effect. This effect is a fundamental issue in physics. It describes the physically important electromagnetic quantities in quantum mechanics, and its experimental verification constitutes a test of the theory of quantum mechanics itself. Under the assumption that the incoming free electron is a gaussian wave packet, we estimate the exact solution to the Schrödinger equation for all times. 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. We also prove that on the gaussian asymptotic state, the scattering operator is given by multiplication by e q ~c Φ̃ -where q is the charge of the electron, c is the speed of light, ~ is Planck’s constant, and Φ̃ is the magnetic flux in a transversal section of the magnetup to a quantitative error bound, that we provide. As suggested by Aharonov and Bohm and by Tonomura et al., we model the part of the electron wave packet that goes through the hole of the magnet. 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 10−199and, 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. If the variance in configuration space is small, the variance in momentum space is big, and then, the component of the momentum transversal to the axis of the magnet is large. In consequence, the opening angle of the electron wave packet is large, and there is a large interaction with the magnet. If the variance is large, the opening angle is small, but as the electron wave packet is big we have again a large interaction with the magnet. ∗PACS Classification (2008): 03.65Nk, 03.65.Ca, 03.65.Db, 03.65.Ta. Mathematics Subject Classification(2000): 81U40, 35P25, 35Q40, 35R30. †Research partially supported by CONACYT under Project P42553F. ‡Ricardo Weder is a Fellow of the Sistema Nacional de Investigadores. On Leave of absence from Departamento de Métodos Matemáticos y Numéricos. Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas. Universidad Nacional Autónoma de México. Apartado Postal 20-726, México DF 01000. 1 ar X iv :0 90 3. 26 09 v1 [ m at hph ] 1 5 M ar 2 00 9