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Study on A. Tonomura's Experiment of Bi-prism of Electron Beam

Yoshifumi Ito

Published Jan 1, 2013• 4 citations
OpenAlex

Abstract

In this article, we study A. Tonomura's experiment of bi-prism of electron beam. Thereby, we verify the two facts that an electron is a particle and that the observation of the natural statistical phenomena is the sampling of the natural random variable defined on a certain physical system.

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•Jan 1, 2010•2 citations
The article discusses software development of the virtual lab “Tonomura's single-electron double slit experiment”. The second-year IT students' team of the Novosibirsk State Technical University NETI (NSTU NETI) developed the interactive simulator of the famous experiment demonstrating the corpuscular-wave properties of electrons. The relativistic energy-momentum relation for electrons and the superposition principle for wave functions were used in forming mathematical models. The original algorithm was developed to simulate random dynamics of the single electron interference pattern formation. The virtual lab is implemented on the basis of the Unity game engine (C\#) based on the principles of Clean Architecture. The virtual laboratory with the electron microscope and two monitors is visualized in the main window. The first virtual monitor is used to set up and display the user-specified values of the physical parameters. The second virtual monitor dynamically displays the interference pattern sequentially formed by individual electrons. The IT students' work was carried out in the process of organized STEM oriented project activities of computer modeling at the Department of General Physics of the NSTU NETI. The virtual lab can be used in the educational process when students learn quantum physics.
The article "A brief history of the double-slit experiment" (September 2002 p15; correction October p17) describes how Claus Jönsson of the University of Tübingen performed the first double-slit interference experiment with electrons in 1961. It then goes on to say: "The next milestone – an experiment in which there was just one electron in the apparatus at any one time – was reached by Akira Tonomura and co-workers at Hitachi in 1989 when they observed the build-up of the fringe pattern with a very weak electron source and an electron biprism (Am. J. Phys. 57 117–120)."
Several different statements and proof methods of uncertainty relation are presented. The agreement and disagreement among these statements are compared and analyzed. New developments of relevant quantum theory are described in association with some recent experiments such as the Tonomura’s Electron Diffraction experiment and Zeilinger’s Neutron Diffraction experiment. These experiments fully proved that a single particle has wave nature. Uncertainty relation turns out to be a natural consequence of the wave nature of particles, so it is suitable for a single particle.
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.
In this paper, the five arrangements of optical system are briefly compared with each other, by which the experiments of electron interference and holography have been carried out up to now. Using a Mollenstedt electron static bi-prism as the beam splitter, two operation modes making the experiment successfully in JEM-200CX type electron microscope are presented. Some parameters of the experimental and calculated results are given. Finally, on the basis of our own experience, we propose some points for attention in operating.