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Physical Review Letters

Analysis of the Experiment to Determine the Spectrum of the Angular Momentum of a Charged-Boson, Magnetic-Flux-Tube Composite and the Aharonov-Bohm Effect

J.-Q. Liang

Published Aug 27, 1984• 19 citations
Publisher

Abstract

By means of the Feynman path-integral method the experiment proposed by Silverman is analyzed under the classical approximation. The results give a new verification of the Aharonov-Bohm effect. It is pointed out that the multiple-valued wave functions are reasonable for describing the experiment. Furthermore, it is verified that the eigenvalues of the angular momentum of a charged, spinless particle-solenoid composite are flux dependent.

DOI
10.1103/physrevlett.53.859

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Analysis of the Experiment to Determine the Spectrum of the Angular Momentum of a Charged-Boson, Magnetic-Flux-Tube Composite and the Aharonov-Bohm Effect

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By means of the Feynman path-integral method the experiment proposed by Silverman is analyzed under the classical approximation. The results give a new verification of the Aharonov-Bohm effect. It is pointed out that the multiple-valued wave functions are reasonable for describing the experiment. Furthermore, it is verified that the eigenvalues of the angular momentum of a charged, spinless particle-solenoid composite are flux dependent.
In the context of the Aharonov-Bohm effect, it is well-known that the local action of the vector potential or the non-local action of the magnetic field are properties attributable to quantum physics, which can explain the observed Aharonov-Bohm phase. Here we point out that, surprisingly, these properties are also attributable to classical physics and that this result follows from examining the electromagnetic angular momentum of a static electric charge outside an infinitely-long solenoid of finite radius carrying a uniform magnetic flux. This angular momentum may be expressed in three equivalent forms. The first form shows that there is an electromagnetic angular momentum even in regions for which the magnetic field vanishes but the potential does not, which stresses the physical nature of the vector potential in classical physics. The second form shows that there is an electromagnetic angular momentum produced by the non-local action of the magnetic field. The third form shows that the Aharonov-Bohm phase is proportional to the magnitude of the electromagnetic angular momentum, which sheds light on the physical meaning of this phase.
The intention of constructing a classically-electrodynamic theory of the Aharonov-Bohm effect (ABE) was a driving force to conduct the study presented in this paper. The ABE has been detailed by R. P. Feynman; it is a phenomenon in which an electrically charged particle (electron) passing by a very long and extremely thin magnetized filament is affected by a force, despite the fact that both the magnetic field and electric field are zero along the particle trajectory. The fact that nonzero vector potential exists around a magnetized filament turned out to be sufficient to give rise the quantum-mechanical explanation of this effect by Aharonov and Bohm in 1959. However, among the theorems of classical physics, the virtual work principle can be found, which represents one facet of the energy conservation law. This principle asserts that if the energy of a set of physical objects varies with the location of some object, then this object is subjected to a force, which can be calculated from the energy gradient at the point of the object location. This paper presents a classically-electrodynamic theory of the Aharonov-Bohm effect. The proposed theory uses the virtual work principle to determine the force acting on a point charge. The calculated value of the transverse momentum of an electron passing by an infinitely long solenoid is opposite in sign to that calculated by a quantum-mechanical method. The need for new, reasonably set up experiments is justified
The Aharonov-Bohm effect is investigated in the Feynman path-integral formulation of quantum mechanics. We consider an idealized situation with an electron moving in a magnetic-field-free region outside a solenoid whose radius to length ratio is very small. The nonvanishing vector potential term in the Lagrangian is written as an angular-velocity-dependent potential. In order to account for a singularity due to the presence of the solenoid itself, a periodic constraint is imposed on the path integral. The propagator can then be evaluated using the polar-coordinate methods of Peak and Inomata. It is found that the propagator has the general form $K({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}\ensuremath{'}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}};\ensuremath{\tau})=\ensuremath{\Sigma}{n}^{}{\ensuremath{\chi}}_{n}{K}_{n}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}\ensuremath{'}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}};\ensuremath{\tau})$ where the sum is taken over all classes of homotopic paths and the ${\ensuremath{\chi}}_{n}$ are a one-dimensional representation of the homotopy group. This is the form of the propagator as conjectured by Schulman.
The classical electromagnetic interaction for a charged particle with a long, constant-current solenoid is investigated in the context of classical electromagnetism. The conservation laws connected with energy, linear momentum, and angular momentum are first verified. Then the changes in the electromagnetic field quantities are evaluated explicitly, and the values are seen to have interesting connections with the solenoid vector potential. The calculations are of interest in connection with the Aharonov-Bohm effect involving the passage of electrons past a solenoid, and the classical expressions for energy and momentum are reminiscent of terms in the quantum-mechanical calculations.