On the spin- 1/2 Aharonov–Bohm problem in conical space: Bound states, scattering and helicity nonconservation

Abstract

In this work the bound state and scattering problems for a spin-1/2 particle undergone to an Aharonov-Bohm potential in a conical space in the nonrelativistic limit are considered. The presence of a δ-function singularity, which comes from the Zeeman spin interaction with the magnetic flux tube, is addressed by the self-adjoint extension method. One of the advantages of the present approach is the determination of the self-adjoint extension parameter in terms of physics of the problem. Expressions for the energy bound states, phase-shift and S matrix are determined in terms of the self-adjoint extension parameter, which is explicitly determined in terms of the parameters of the problem. The relation between the bound state and zero modes and the failure of helicity conservation in the scattering problem and its relation with the gyromagnetic ratio g are discussed. Also, as an application, we consider the spin-1/2 Aharonov-Bohm problem in conical space plus a two-dimensional isotropic harmonic oscillator.The Aharonov-Bohm (AB) effect [1] (first predicted by Ehrenberg and Siday [2]) is one of most weird results of quantum phenomena. The effect reveals that the electromagnetic potentials, rather than the electric and magnetic fields, are the fundamental quantities in quantum mechanics. The interest in this issue appears in the different contexts, such as solid-state physics [3], cosmic strings [4][5][6][7][8][9][10][11][12][13][14] κ-Poincaré-Hopf algebra [15, 16], δ-like singularities [17-19], supersymmetry [20, 21], condensed matter [22, 23], Lorentz symmetry violation [24], quantum chromodynamics [25], general relativity [26], nanophysics [27], quantum ring [28-30], black hole [31, 32] and noncommutative theories [33, 34].In the AB effect of spin-1/2 particles [7], besides the interaction with the magnetic potential, an additional two dimensional δ-function appears as the mathematical description of the Zeeman interaction between the spin and the magnetic flux tube [18,19]. This interaction is the basis of the spin-orbit coupling, which causes a splitting on the energy spectrum of atoms depending on the spin state. In Ref. [17] is argued that this δ-function contribution to the potential can not be neglected when the system has spin, having shown that changes in the amplitude and scattering cross section are implied in this case. The presence of a δ-function potential singularity, turns the problem more complicated to be solved. Such kind of point interaction potential can then be addressed by the self-adjoint extension approach [35]. The self-adjoint extension of symmetric operators [36] is a very powerful mathematical method and it can be applied to various systems in relativistic and nonrelativistic quantum mechanics, supersymmetric quantum mechanics and vortex-like models.This paper extends our previous report [37] on a general physical regularization method, both in details and depth. The method has the advantage of solving problems in relativistic and nonrelativistic quantum mechanics whose Hamiltonian is s...

Self-Adjoint Extension Approach for Singular Hamiltonians in (2 + 1) Dimensions

“…Therefore, for ν < 0, the bound state energy is given by [26,59] for the AB effect in curved space and is similar that one found in contact interactions of anyons [74]. It is also possible to express the S-matrix in terms of the bound state energy.…”

“…This is the scattering phase shift of the AB effect in the cosmic string spacetime [26,59]. It is important to mention that, for α = 1, it reduces to the phase shift for the usual AB effect in flat space δ AB m = π(|m| − |m + φ|)/2 [4].…”

Relativistic quantum dynamics of a neutral particle in external electric fields: An approach on effects of spin

“…The AC effect and consequently the AB effect also has been addressed to bound states. In the context of the method of operators in quantum mechanics such bound states are found by modeling the problem by boundary conditions at the origin [12][13][14][15][16][17][18][19][20][21][22][23] (See also Refs. [24][25][26][27] where bound states also are obtained for Aharonov-Bohm-like systems.…”

Quantum dynamics of scalar bosons in a cosmic string background

“…[33][34][35][36][37][38][39]. The Dirac oscillator embedded in a cosmic string background has inspired a great deal of research in last years [40][41][42][43][44][45][46]. A cosmic string is a linear defect that change the topology of the medium when viewed globally.…”