The Aharonov-Bohm effect is a fundamental issue in physics. It describes the physically important electromagnetic quantities in quantum mechanics. Its experimental verification constitutes a test of the theory of quantum mechanics itself. The remarkable experiments of Tonomura ["Observation of Aharonov-Bohm effect by electron holography," Phys. Rev. Lett 48, 1443 (1982) and "Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave," Phys. Rev. Lett 56, 792 (1986)] are widely considered as the only experimental evidence of the physical existence of the Aharonov-Bohm effect. Here we give the first rigorous proof that the classical ansatz of Aharonov and Bohm of 1959 ["Significance of electromagnetic potentials in the quantum theory," Phys. Rev. 115, 485 (1959)], that was tested by Tonomura, is a good approximation to the exact solution to the Schrödinger equation. This also proves that the electron, that is, represented by the exact solution, is not accelerated, in agreement with the recent experiment of Caprez in 2007 ["Macroscopic test of the Aharonov-Bohm effect," Phys. Rev. Lett. 99, 210401 (2007)], that shows that the results of the Tonomura experiments can not be explained by the action of a force. 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 Aharonov-Bohm Ansatz. Our bound is uniform in time. We also prove that on the Gaussian asymptotic state the scattering operator is given by a constant phase shift, up to a quantitative error bound that we provide. Our results show that for intermediate size electron wave packets, smaller than the ones used in the Tonomura experiments, quantum mechanics predicts the results observed by Tonomura with an error bound smaller than 10-99. It would be quite interesting to perform experiments with electron wave packets of intermediate size. Furthermore, we provide a physical interpretation of our error bound. © 2009 American Institute of Physics.