Aharonov-Bohm Effect and High-Velocity Estimates of Solutions to the Schrödinger Equation

Abstract

The Aharonov-Bohm effect is a fundamental issue in physics that has been extensively studied in the literature and is discussed in most of the textbooks in quantum mechanics. The issues at stake are what are the fundamental electromagnetic quantities in quantum physics, if magnetic fields can act at a distance on charged particles and if the magnetic potentials have a real physical significance. The Aharonov-Bohm effect is a very controversial issue. From the experimental side the issues were settled by the remarkable experiments of Tonomura et al. (Phys Rev Lett 48:1443–1446, 1982; Phys Rev Lett 56:792–795, 1986) with toroidal magnets that gave a strong experimental evidence of the physical existence of the Aharonov-Bohm effect, and by the recent experiment of Caprez et al. (Phys Rev Lett 99:210401, 2007) that shows that the results of the Tonomura et al. experiments can not be explained by the action of a force. Aharonov and Bohm (Phys Rev 115:485-491, 1959) proposed an Ansatz for the solution to the Schrödinger equation in simply connected regions of space where there are no electromagnetic fields. It consists of multiplying the free evolution by the Dirac magnetic factor. The Aharonov-Bohm Ansatz predicts the results of the experiments of Tonomura et al. and of Caprez et al. Recently in Ballesteros and Weder (Math Phys 50:122108, 2009) we gave the first rigorous proof that the Aharonov-Bohm Ansatz is a good approximation to the exact solution for toroidal magnets under the conditions of the experiments of Tonomura et al. We provided a rigorous, simple, quantitative, error bound for the difference in norm between the exact solution and the Aharonov-Bohm Ansatz. In this paper we prove that these results do not depend on the particular geometry of the magnets and on the velocities of the incoming electrons used on the experiments, and on the gaussian shape of the wave packets used to obtain our quantitative error bound. We consider a general class of magnets that are a finite union of handlebodies. Each handlebody is diffeomorphic to a torus or a ball, and some of them can be patched though the boundary. We formulate the Aharonov-Bohm Ansatz that is appropriate to this general case and we prove that the exact solution to the Schrödinger equation is given by the Aharonov-Bohm Ansatz up to an error bound in norm that is uniform in time and that decays as a constant divided by v ^ρ, 0 < ρ < 1, with v the velocity. The results of Tonomura et al., of Caprez et al., our previous results and the results of this paper give a firm experimental and theoretical basis to the existence of the Aharonov-Bohm effect and to its quantum nature. Namely, that magnetic fields act at a distance on charged particles, and that this action at a distance is carried by the circulation of the magnetic potential which gives a real physical significance to magnetic potentials.