Skript / lecture notes - Universität Paderborn

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Prof. Dr. Wolf Gero Schmidt Universität Paderborn, Lehrstuhl für Theoretische Physik by a superconductor, Phys. Rev. A 34, 815 (1986). The Aharonov–Bohm effect illustrates the physicality of electromagnetic potentials, whereas previously it was possible to argue that only the electromagnetic fields were physical and that the electromagnetic potentials were purely mathematical constructs (ϕ and ⃗ A being non-unique, in addition to not appearing in the Lorentz Force formula). The Aharonov–Bohm effect shows that the local ⃗ E and ⃗ B fields do not contain full information about the electromagnetic field, and the electromagnetic potential, ϕ and ⃗ A, must be used instead. By Stokes’ theorem, the magnitude of the Aharonov–Bohm effect can be calculated using the ⃗ E and ⃗ B fields alone, or using the ϕ and ⃗ A potential alone. But when using just the ⃗ E and ⃗ B fields, the effect depends on the field values in a region from which the test particle is excluded, not only classically but also quantum mechanically. In contrast, when using just the electromagnetic potential ϕ and ⃗ A, the effect only depends on the potential in the region where the test particle is allowed. Therefore we can either abandon the principle of locality (which most physicists are reluctant to do) or we are forced to accept that the electromagnetic potential - composed of ϕ and ⃗ A - offers a more complete description of electromagnetism than the electric and magnetic fields can. In classical electromagnetism the two descriptions were equivalent. With the addition of quantum theory, though, the electromagnetic potentials ϕ and ⃗ A are seen as being more fundamental. The ⃗ E and ⃗ B fields can be derived from the potential, but the potential, by gauge freedom, cannot be derived from the ⃗ E and ⃗ B fields. In a broader sense, the Aharonov–Bohm effect illustrates that the Lagrangian approach to dynamics, based on energies, is not just a computational aid to the Newtonian approach, based on forces. Thus the Aharonov–Bohm effect validates the view that forces are an incomplete way to formulate physics, and potential energies must be used instead. Historical note: In their widely known 1959 paper on the ”Significance of Electromagnetic Potentials in the Quantum Theory,” Aharonov and Bohm made the observation that it is well known that in classical mechanics, the motion of a particle can be affected only by forces acting at the particle at its well-defined location. They noted that this is not the case in quantum mechanics where two different paths or histories of a single particle can interfere and potentials can have measurable effect outside of classically allowed regions. They illustrate this effect for electromagnetic potentials by two examples: the electric and the magnetic effects upon particles moving in regions where no force exists, but where differences of the scalar or vector potentials along two possible paths of the particle can physically manifest themselves by affecting the resulting interference pattern of the particle. They further note that in the magnetic case for an infinitely long and narrow solenoid, and for which they provide an exact solution of the scattering problem, that such a scenario can be experimentally tested. There is some controversy about the priority of this discovery, given the earlier (1949) work by Ehrenberg, and Siday on ”The Refractive Index in Electron Optics and the Principles of Dynamics.” In a crucial passage in the conclusion of this paper, they remark that, ”One might therefore expect wave-optical phenomena to arise which are due to the presence of a magnetic field, but not due to the magnetic field itself, i.e., which arise whilst the rays are in field-free regions only”. Moreover, it is known that Walter Franz, who was a student of Arnold 38
Prof. Dr. Wolf Gero Schmidt Universität Paderborn, Lehrstuhl für Theoretische Physik Sommerfeld, had predicted the ”unorthodox nature of interference processes” associated with the Aharonov–Bohm effect already earlier: Franz wrote an article 1939 published in ”Verhandlungen der Deutschen Physikalischen Gesellschaft”. In this article the subsequent A-B effect was preceded by 20 years, and this article can, therefore, be considered as the first publication of the phenomenon. Unfortunately, the significance of Franz’s work was not appreciated by the English language science community, as it was not translated into English. 39
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