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International Journal of Modern Physics D

TOPOLOGICAL EFFECTS IN QUANTUM MECHANICS AND HIGH VELOCITY ESTIMATES

Ricardo Weder

Published May 19, 2011• 1 citation
Publisher

Abstract

We consider the problem of obtaining high-velocity estimates for finite energy solutions (wave packets) to Schrödinger equations for N-body systems. We discuss a time-dependent method that allows us to obtain precise estimates with error bounds that decay as a power of the velocity. We apply this method to the electric Aharonov–Bohm effect. We give the first rigorous proof that quantum mechanics predicts the existence of this effect. Our result follows from an estimate in norm, uniform in time, that proves that the Aharonov–Bohm Ansatz is a good approximation to the exact solution to the Schrödinger equation for high velocity.

DOI
10.1142/s0218271811019207

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We consider the problem of obtaining high-velocity estimates for finite energy solutions (wave packets) to Schrödinger equations for N-body systems. We discuss a time-dependent method that allows us to obtain precise estimates with error bounds that decay as a power of the velocity. We apply this method to the electric Aharonov–Bohm effect. We give the first rigorous proof that quantum mechanics predicts the existence of this effect. Our result follows from an estimate in norm, uniform in time, that proves that the Aharonov–Bohm Ansatz is a good approximation to the exact solution to the Schrödinger equation for high velocity.
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 et al. [“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 et al., 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 et al. in 2007 [“Macroscopic test of the Aharonov–Bohm effect,” 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. 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 et al. experiments, quantum mechanics predicts the results observed by Tonomura et al. 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.
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We give a direct proof of the magnetic Aharonov-Bohm effects without using the scattering theory and the theory of inverse boundary value problems. This proof can serve as a framework for a physical experiment to confirm the magnetic AB effect. We prove also the electric AB effect and we suggest a physical experiment to demonstrate the electric AB effect. In addition, we consider a combined electric and magnetic AB effect and we propose a new inverse problem for the time-dependent Schrödinger equations. Finally we study the gravitational AB effect.
The time-dependent Dirac equation is solved using the three-dimensional Finite Difference-Time Domain (FDTD) method. The dynamics of the electron wave packet in a vector potential is studied in the arrangements associated with the Aharonov-Bohm effect. The solution of the Dirac equation showed a change in the velocity of the electron wave packet even in a region where no fields of the unperturbed solenoid acted on the electron. The solution of the Dirac equation qualitatively agreed with the prediction of classical dynamics under the assumption that the dynamics was defined by the conservation of generalized or canonical momentum of the electron.
This paper introduces an unconditionally stable, accurate, and efficient algorithm to solve the time-dependent Schrödinger equation for a particle moving in an inhomogeneous magnetic field. The algorithm is used to simulate a fundamental thought experiment of quantum mechanics: the magnetic Aharonov–Bohm effect. The calculations demonstrate that the Aharonov–Bohm prediction of the phase shift holds with great precision, even when the dimensions of the interferometer are comparable to the de Broglie wavelength of the particle.