The Aharonov-Bohm (AB) effect in non-commutative quantum mechanics (NCQM) is studied. First, by introducing a shift for the magnetic vector potential we give the Schr $\\ddot{o}$ dinger equations in the presence of a magnetic field on NC space and NC phase space, respectively. Then by solving the Schr $\\ddot{o}$ dinger equations, we obtain the Aharonov-Bohm (AB) phase on NC space and NC phase space, respectively.The Aharonov-Bohm (AB) effect in non-commutative quantum mechanics (NCQM) is studied. First, by introducing a shift for the magnetic vector potential we give the Schr$\\ddot{o}$dinger equations in the presence of a magnetic field on NC space and NC phase space, respectively. Then by solving the Schr$\\ddot{o}$dinger equations, we obtain the Aharonov-Bohm (AB) phase on NC space and NC phase space, respectively.

Aug 25, 200556 citations

On the role of potentials in the Aharonov–Bohm effect

Shan Gao

Physics Letters AOct 5, 20260 citations

Aharonov–Bohm quantum systems on a punctured 2-torus

Toshihiro Iwai, Yoshiro Yabu

This paper deals with Aharonov–Bohm (A-B) quantum systems on a punctured two-dimensional torus from the geometric and the operator theoretic point of view. First, flat connections on the U (1)-bundles over the punctured 2-torus are studied, which serve as vector potentials for A-B effect magnetic fields. It is proved that the moduli space of flat connections is identified with the ( N + 1)-dimensional torus T N +1 , if the punctured torus has N > 0 pinholes at which solenoids are assumed to penetrate the 2-torus. For a given point of T N +1 , an associated flat connection is constructed in terms of the Weierstrass zeta function on C together with an inhomogeneous linear function on R 2 . A-B quantum systems are defined in terms of position operators and momentum operators coupled with the A-B potentials. Necessary and sufficient conditions are given for two A-B quantum systems to be unitarily equivalent. Further, the A-B Hamiltonian is defined and analysed from the viewpoint of operator theory. The deficiency indices of the A-B Hamiltonian are determined to be ( N + M , N + M ), where M is the number of solenoids whose fluxes are not quantized. Finally, the eigenvalue problem is studied for the A-B Hamiltonian with all fluxes quantized to obtain eigenvalues together with eigenfunctions which are described in terms of the Weierstrass sigma functions.This paper deals with Aharonov–Bohm (A-B) quantum systems on a punctured two-dimensional torus from the geometric and the operator theoretic point of view. First, flat connections on the U (1)-bundles over the punctured 2-torus are studied, which serve as vector potentials for A-B effect magnetic fields. It is proved that the moduli space of flat connections is identified with the ( N + 1)-dimensional torus T N +1 , if the punctured torus has N > 0 pinholes at which solenoids are assumed to penetrate the 2-torus. For a given point of T N +1 , an associated flat connection is constructed in terms of the Weierstrass zeta function on C together with an inhomogeneous linear function on R 2 . A-B quantum systems are defined in terms of position operators and momentum operators coupled with the A-B potentials. Necessary and sufficient conditions are given for two A-B quantum systems to be unitarily equivalent. Further, the A-B Hamiltonian is defined and analysed from the viewpoint of operator theory. The deficiency indices of the A-B Hamiltonian are determined to be ( N + M , N + M ), where M is the number of solenoids whose fluxes are not quantized. Finally, the eigenvalue problem is studied for the A-B Hamiltonian with all fluxes quantized to obtain eigenvalues together with eigenfunctions which are described in terms of the Weierstrass sigma functions.

Journal of Physics A Mathematical and GeneralJan 11, 20063 citations

Geometry of the Aharonov–Bohm Effect

Stella Huérfano, M. A. López, M. Socolovsky

arXiv:math-ph/0701050International Journal of Theoretical PhysicsMay 29, 20071 citation

Aharonov–Bohm effect in phase space

José A. R. Cembranos, David García-López, Zoe Garcia del Toro

arXiv:2211.13199The European Physical Journal DFeb 1, 20243 citations

Aharonov–Bohm phase for an electromagnetic wave background

Max Bright, Douglas Singleton, Atsushi Yoshida

The canonical Aharonov–Bohm effect is usually studied with time-independent potentials. In this work, we investigate the Aharonov–Bohm phase acquired by a charged particle moving in time-dependent potentials. In particular, we focus on the case of a charged particle moving in the time-varying field of a plane electromagnetic wave. We work out the Aharonov–Bohm phase using both the potential (i.e. $\oint A_\mu \mathrm{d}x ^\mu$ ) and the field (i.e. $\frac{1}{2}\int F_{\mu \nu } \mathrm{d}\sigma ^{\mu \nu }$ ) forms of the Aharonov–Bohm phase. We give conditions in terms of the parameters of the system (frequency of the electromagnetic wave, the size of the space–time loop, amplitude of the electromagnetic wave) under which the time-varying Aharonov–Bohm effect could be observed.The canonical Aharonov–Bohm effect is usually studied with time-independent potentials. In this work, we investigate the Aharonov–Bohm phase acquired by a charged particle moving in time-dependent potentials. In particular, we focus on the case of a charged particle moving in the time-varying field of a plane electromagnetic wave. We work out the Aharonov–Bohm phase using both the potential (i.e. $$\oint A_\mu \mathrm{d}x ^\mu $$ ) and the field (i.e. $$\frac{1}{2}\int F_{\mu \nu } \mathrm{d}\sigma ^{\mu \nu }$$ ) forms of the Aharonov–Bohm phase. We give conditions in terms of the parameters of the system (frequency of the electromagnetic wave, the size of the space–time loop, amplitude of the electromagnetic wave) under which the time-varying Aharonov–Bohm effect could be observed.

arXiv:1509.01646The European Physical Journal CSep 1, 201510 citations

The Aharonov-Bohm (AB) effect in non-commutative quantum mechanics (NCQM) is studied. First, by introducing a shift for the magnetic vector potential we give the Schr

\\ddot{o}

dinger equations in the presence of a magnetic field on NC space and NC phase space, respectively. Then by solving the Schr

\\ddot{o}

dinger equations, we obtain the Aharonov-Bohm (AB) phase on NC space and NC phase space, respectively.

\oint A_\mu \mathrm{d}x ^\mu

) and the field (i.e.

\frac{1}{2}\int F_{\mu \nu } \mathrm{d}\sigma ^{\mu \nu }

) forms of the Aharonov–Bohm phase. We give conditions in terms of the parameters of the system (frequency of the electromagnetic wave, the size of the space–time loop, amplitude of the electromagnetic wave) under which the time-varying Aharonov–Bohm effect could be observed.