A useful and universal formula for the expectation value of the radial operator in the presence of the Aharonov-Bohm flux and the Coulomb Field is established. We find that the expectation value $< r^λ>$ $(-\infty \leq λ\leq \infty)$ is greatly affected due to the non-local effect of the magnetic flux although the Aharonov-Bohm flux does not have any dynamical significance in classical mechanics. In particular, the quantum fluctuation increases in the presence of the magnetic flux due to the Aharonov-Bohm effect. In addition, the Virial theory in quantum mechanics is also constructed for the spherically symmetric system under the Aharonov-Bohm effect.A useful and universal formula for the expectation value of the radial operator in the presence of the Aharonov-Bohm flux and the Coulomb Field is established. We find that the expectation value $< r^λ>$ $(-\infty \leq λ\leq \infty)$ is greatly affected due to the non-local effect of the magnetic flux although the Aharonov-Bohm flux does not have any dynamical significance in classical mechanics. In particular, the quantum fluctuation increases in the presence of the magnetic flux due to the Aharonov-Bohm effect. In addition, the Virial theory in quantum mechanics is also constructed for the spherically symmetric system under the Aharonov-Bohm effect.

arXiv:quant-ph/0210106Oct 14, 20020 citations

Aharonov-Bohm effect revisited

Donald H. Kobe

Annals of PhysicsDec 1, 197951 citations

Generalized Aharonov-Bohm Effect

Ghislain Golse, Hamza Jirari, H. Kröger +1 more

Springer eBooksJan 1, 19980 citations

Comment on ‘‘Asymmetry of the Aharonov-Bohm diffraction pattern and Ehrenfest’s theorem’’

Mark D. Semon, John R. Taylor

In a recent paper [D. H. Kobe, V. C. Aguilera-Navarro, and R. M. Ricotta, Phys. Rev. A 45, 6192 (1992)], Kobe et al. have calculated the expectation values of position and momentum in the Aharonov-Bohm effect and have found that both are shifted by the Aharonov-Bohm potentials. Although these shifts appear to contradict certain ``no-shift'' theorems proved by various authors, we show that this apparent contradiction is the result of different assumptions. Specifically, one of the assumptions on which the no-shift theorems are based does not hold for the parameters used by Kobe et al.; that is, with their parameters, there is no reason to expect the no-shift theorems to hold. On the other hand, with the parameters of actual two-slit experiments, all of the conditions of the no-shift theorems do hold and the no-shift theorems do apply.In a recent paper [D. H. Kobe, V. C. Aguilera-Navarro, and R. M. Ricotta, Phys. Rev. A 45, 6192 (1992)], Kobe et al. have calculated the expectation values of position and momentum in the Aharonov-Bohm effect and have found that both are shifted by the Aharonov-Bohm potentials. Although these shifts appear to contradict certain ``no-shift'' theorems proved by various authors, we show that this apparent contradiction is the result of different assumptions. Specifically, one of the assumptions on which the no-shift theorems are based does not hold for the parameters used by Kobe et al.; that is, with their parameters, there is no reason to expect the no-shift theorems to hold. On the other hand, with the parameters of actual two-slit experiments, all of the conditions of the no-shift theorems do hold and the no-shift theorems do apply.

Physical Review AAug 1, 19944 citations

A mathematical model for the Aharonov-Bohm effect

Christiane Martin

Letters in Mathematical PhysicsMar 1, 197625 citations

Revisiting Modular Momentum in the Aharonov-Bohm Effect: A Resolution of Aharonov’s Paradox

Gao, Shan

The Aharonov-Bohm (AB) effect demonstrates that electromagnetic potentials can influence quantum systems in field-free regions, challenging classical intuitions. In 2004, Aharonov and Kaufherr proposed that the AB phase shift arises from a nonlocal exchange of modular momentum, a gauge-invariant quantity, occurring at a definite time before wave packet interference. This framework, however, introduces a paradox: the expectation value of the modular momentum operator, $\langle \hat{M}_y \rangle$ , changes abruptly when the electron’s wave packets cross the solenoid, yet local, gauge-invariant properties of the electron, such as its velocity distribution, remain continuous. We resolve this paradox by showing that $\hat{M}_y=\exp(i \frac{m v_y L}{\hbar})$ and the velocity operator $\hat{v}_y$ have distinct spectral measures, and the abrupt change in $\langle \hat{M}_y \rangle$ reflects a global topological change in the electromagnetic environment, not a dynamical alteration in the electron’s state. Through a corrected, gauge-invariant formulation, we demonstrate that $\langle \hat{M}_y \rangle$ encodes the AB phase but does not explain the AB effect’s mechanism. This resolution reaffirms the conventional explanation of the AB effect while clarifying the nonlocal nature and experimental limitations of modular momentum.The Aharonov-Bohm (AB) effect demonstrates that electromagnetic potentials can influence quantum systems in field-free regions, challenging classical intuitions. In 2004, Aharonov and Kaufherr proposed that the AB phase shift arises from a nonlocal exchange of modular momentum, a gauge-invariant quantity, occurring at a definite time before wave packet interference. This framework, however, introduces a paradox: the expectation value of the modular momentum operator, $\langle \hat{M}_y \rangle$, changes abruptly when the electron’s wave packets cross the solenoid, yet local, gauge-invariant properties of the electron, such as its velocity distribution, remain continuous. We resolve this paradox by showing that $\hat{M}_y=\exp(i \frac{m v_y L}{\hbar})$ and the velocity operator $\hat{v}_y$ have distinct spectral measures, and the abrupt change in $\langle \hat{M}_y \rangle$ reflects a global topological change in the electromagnetic environment, not a dynamical alteration in the electron’s state. Through a corrected, gauge-invariant formulation, we demonstrate that $\langle \hat{M}_y \rangle$ encodes the AB phase but does not explain the AB effect’s mechanism. This resolution reaffirms the conventional explanation of the AB effect while clarifying the nonlocal nature and experimental limitations of modular momentum.

Sep 10, 20250 citations

A useful and universal formula for the expectation value of the radial operator in the presence of the Aharonov-Bohm flux and the Coulomb Field is established. We find that the expectation value

< r^λ>

(-\infty \leq λ\leq \infty)

is greatly affected due to the non-local effect of the magnetic flux although the Aharonov-Bohm flux does not have any dynamical significance in classical mechanics. In particular, the quantum fluctuation increases in the presence of the magnetic flux due to the Aharonov-Bohm effect. In addition, the Virial theory in quantum mechanics is also constructed for the spherically symmetric system under the Aharonov-Bohm effect.

\langle \hat{M}_y \rangle

, changes abruptly when the electron’s wave packets cross the solenoid, yet local, gauge-invariant properties of the electron, such as its velocity distribution, remain continuous. We resolve this paradox by showing that

\hat{M}_y=\exp(i \frac{m v_y L}{\hbar})

and the velocity operator

\hat{v}_y

have distinct spectral measures, and the abrupt change in

\langle \hat{M}_y \rangle

reflects a global topological change in the electromagnetic environment, not a dynamical alteration in the electron’s state. Through a corrected, gauge-invariant formulation, we demonstrate that

\langle \hat{M}_y \rangle

encodes the AB phase but does not explain the AB effect’s mechanism. This resolution reaffirms the conventional explanation of the AB effect while clarifying the nonlocal nature and experimental limitations of modular momentum.