What is the Boehm-Aharanov Effect

Gerhard W. Bruhn, Department of Mathematics at TU Darmstadt

On the role of the magnetic vector potential in the Aharonov-Bohm effect

The well-known attempt at diffraction of electrons at a double slit is considered, whereby an interference pattern of the frequency of impinging electrons is created on a screen behind the double slit diaphragm.



In the Aharonov-Bohm experiment, a magnetic flux is now additionally perpendicular to the plane of the drawing in the area between the two slots B. with the vector potential A. switched on.

(1)B = red A..

The flow can even be limited in such a way that there is no flow in the areas accessible to electrons, i.e. there B = 0 applies. Nevertheless, the electrons "feel" the magnetic field, and there is a shift in the interference pattern.

The phase difference of the probability waves hitting the screen is changed by this magnetic field

(2)/H q (∫γ1A ·dx − ∫γ2A ·dx )

postponed. Seen ostensibly, the phase shift depends on the vector potential A. from.

This has led superficially thinking viewers to believe that this is the first time in physics direct To recognize the effect of a vector potential. But this View is erroneous: Because the path integrals in (2) can be reversed by reversing the direction of the path γ2 rewrite in

(3) /H q ∫γ1−γ2A ·dx ,

where γ1−γ2 now called a closed path. But Stokes’s integral theorem can be used here

γ1γ2A. ·dx = ∫∫ΩredA. · dO,

where Ω is the one in the closed path γ1−γ2 means clamped area. So instead of (3) one obtains

(4)/H q ∫∫ΩredA. ·do = ∫∫Ω B.·dO.

This shows the phase shift of the Aharonov-Bohm effect from the river alone B. of the effective magnetic field. A dependence on that in the vector potential A. contained additional Information - A. is only except for one additive gradient definitely - cannot be observed:

Only the magnetic flux is effective for the phase shift B.

The electrons only "feel" the magnetic flux B., but not the one in its vector potential A. additional information contained.



R.P. Feynman, R. B. Leighton, M. Sands: The Feynman Lectures on Physics, Addison Wesley 1963, with German translation in Oldenbourg-Verlag 1973, part II-1, pp. 15-11 ff.

L. Motta, Aharonov-Bohm Effect, Eric Weisstein's World of Physics