Quantum motion of a point particle in the presence of the Aharonov–Bohm potential in curved space

Abstract

The nonrelativistic quantum dynamics of a spinless charged particle in the presence of the Aharonov-Bohm potential in curved space is considered. We chose the surface as being a cone defined by a line element in polar coordinates. The geometry of this line element establishes that the motion of the particle can occur on the surface of a cone or an anti-cone. As a consequence of the nontrivial topology of the cone and also because of two-dimensional confinement, the geometric potential should be taken into account. At first, we establish the conditions for the particle describing a circular path in such a context. Because of the presence of the geometric potential, which contains a singular term, we use the self-adjoint extension method in order to describe the dynamics in all space including the singularity. Expressions are obtained for the bound state energies and wave functions.

Self-Adjoint Extension Approach to Motion of Spin-1/2 Particle in the Presence of External Magnetic Fields in the Spinning Cosmic String Spacetime

“…When we wish to study these models considering the spin of the particle, the following requirement arises: the term involving the δ function in the operator H ± cannot be neglected. It is known in the literature that the presence of such interaction guarantees the existence of bound states in Aharonov-Bohm-type systems [60,61,[63][64][65][66]. Then, we shall solve Equation (38) for bound states using the self-adjoint extension technique [35].…”

Geometric Potential and Dirac Quantization

“…In consequence, the commutators turn out to satisfy the following relation with , and the Hamiltonian operator turns out to be, The second and key finding of this Letter is: With the quantum condition being imposed, the curvature‐induced potential proves to be the geometric potential what has been expected for more than three and a half decades. Being in fact the trace of square of the extrinsic curvature tensor, the geometric potential is …”

“…This CPF has a distinct feature for no presence of any ambiguity. It is thus a powerful tool to examine various curvature‐induced consequences in two‐dimensional curved surfaces or curved wires . Experimental confirmations of the geometric potential include: an optical realization in 2010 and an observation of its effects in an uneven periodic caged peanut‐shaped nanostructure in 2012 .…”

Quantum effective potential from discarded degrees of freedom

“…2 The curved tube (1) is the entire configuration space of the system, so the covariant Schrödinger equation (11) can refer only to the intrinsic geometry of this manifold. Contrast this with a particle that actually exists in R 3 , but is constrained to a twodimensional surface Σ ⊂ R 3 by a steep potential well: here, the extrinsic curvature of Σ will also play a role [4][5][6][7][8].…”