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Bilješka o članku „područje valjanosti magnetskog Aharonov-Bohm efekta“

Comay, Eliahu

Published Jan 1, 1989• 0 citations
OpenAlex

Abstract

The present note contains some corrections to the previous published work on the magnetic Aharonov-Bohm effect.

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Područje valjanosti magnetskog Aharonov-Bohm efekta

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The magnetic Aharonov-Bohm effect is analyzed. Using basic properties of quantum mechanics, it is shown that the effect exists in a case where the energy spectrum of the source of the magnetic flux does not belong to a continuum. On the other hand, the effect disappears in an example of a source whose energy spectrum is continuous. It is shown that in all cases the system demonstrates the ordinary relations between classical physics and quantum mechanics.

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In a previous note (1), we argued that a potential V(x) may be written as Sum over k V(k) exp(ikx). Thus, -dV/dx is only an average force which may in fact be zero, yet -d/dx V(k) exp(ikx) is not zero leading to possible forward and backward motion/acceleration (due to each V(k) exp(ikx) which act at a different times). This gives rise to a momentum distribution P(p/x) =a(p) exp(ikx)/W(x) (W(x)=wavefunction) at each x. In this note, we wish to link these ideas to the Bohm Aharonov effect by considering two recent (2018, 2020) examples from the literature. In the first (2), the magnetic potential A(r) is written in a Fourier series in the same manner as V(k)exp(ikx). Thus, although grad A = B (magnetic field) = 0, grad A(k) exp(ikx) should not be zero and so there may be hits from “virtual photons”. (2) applies perturbation theory to find a correction to energy E and then considers exp(i Integral over t delta E dt) as in adiabatic theory. The second approach (3) applies the usual Hamiltonian containing the magnetic potential A, but imagines a little box at various xi positions such that A(xi) is a constant at each xi. In such a case, one may show directly that W(x)= W(x=0) exp( ie/hbar Integral xo to x1 A(x) dx) which yields the Ehrenberg–Siday–Aharonov–Bohm effect. In this approach, one essentially has a different constant potential at each point and hence a different energy. In a previous note, we argued that even though average kinetic energy may remain the same, a change in constant potential using Sum over k V(k) exp(ikx) gives rise to different physical effects which try to argue are related to the Bohm Aharonov effect. We also try to consider a more general case.

•Aug 24, 2020•0 citations
In a previous note (1), we argued that a potential V(x) may be written as Sum over k V(k) exp(ikx). Thus, -dV/dx is only an average force which may in fact be zero, yet -d/dx V(k) exp(ikx) is not zero leading to possible forward and backward motion/acceleration (due to each V(k) exp(ikx) which act at a different times). This gives rise to a momentum distribution P(p/x) =a(p) exp(ikx)/W(x) (W(x)=wavefunction) at each x. In this note, we wish to link these ideas to the Bohm Aharonov effect by considering two recent (2018, 2020) examples from the literature. In the first (2), the magnetic potential A(r) is written in a Fourier series in the same manner as V(k)exp(ikx). Thus, although grad A = B (magnetic field) = 0, grad A(k) exp(ikx) should not be zero and so there may be hits from “virtual photons”. (2) applies perturbation theory to find a correction to energy E and then considers exp(i Integral over t delta E dt) as in adiabatic theory. The second approach (3) applies the usual Hamiltonian containing the magnetic potential A, but imagines a little box at various xi positions such that A(xi) is a constant at each xi. In such a case, one may show directly that W(x)= W(x=0) exp( ie/hbar Integral xo to x1 A(x) dx) which yields the Ehrenberg–Siday–Aharonov–Bohm effect. In this approach, one essentially has a different constant potential at each point and hence a different energy. In a previous note, we argued that even though average kinetic energy may remain the same, a change in constant potential using Sum over k V(k) exp(ikx) gives rise to different physical effects which try to argue are related to the Bohm Aharonov effect. We also try to consider a more general case.