Magnetic flux and its topological effects in Aharonov-Bohm effect

Abstract

The Aharonov-Bohm effect is a physical phenomenon in which the quantum state of a charged particle acquires a phase shift that is directly proportional to the magnetic flux, $Φ$, due to a (classical) magnetic field, ${\mathbf B}$, which is confined in a spatial region from which the magnetic field cannot escape. Even though the charged particle is not allowed to interact with the magnetic field, it accumulates a phase shift that affects the interference pattern produced. Not surprisingly, this apparent nonlocality is puzzling and counter intuitive. In this work, we provide an explanation that explains the physics underlying this apparent nonlocality. We find that the role of the confined magnetic field is to impart a puncture in the configuration space, $\mathbb{R}^2$, of the charge. Therefore, the quantum state corresponding to the charged quantum particle acquires the phase shift due to its response to the modified topology of the configuration space, $\mathbb{R}^2-\{0\}$, corresponding to the charge.

Figures (1)

• Figure 1: A schematic overview of the Aharonov-Bohm setup, where a charged quantum particle is allowed to move on a two dimensional plane, $\mathbb{R}^2$ (the configuration space of the particle); the shaded region represents a subset of the configuration space, $\mathbb{R}^2$, through which the magnetic field, $\mathbf{B}$, passes through; and the magnetic field is made vanishing everywhere outside the shaded region. In such a setup, a charged quantum particle can take two possible paths, ABD and ACD, and the phase difference between the quantum states of the charged particle corresponding to the paths, ABD and ACD, turns out to be proportional to charge, $q$, of the particle, and the magntic flux, $\Phi$, through the shaded region.

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Definition 3