Abstract
In this work, we review two methods used to approach singular Hamiltonians in (2+1) dimensions. Both methods are based on the self-adjoint extension approach. It is very common to find singular Hamiltonians in quantum mechanics, especially in quantum systems in the presence of topological defects, which are usually modelled by point interactions. In general, it is possible to apply some kind of regularization procedure, as the vanishing of the wave function at the location of the singularity, ensuring that the wave function is square-integrable and then can be associated with a physical state. However, a study based on the self-adjoint extension approach can lead to more general boundary conditions that still gives acceptable physical states. We exemplify the methods by exploring the bound and scattering scenarios of a spin 1/2 charged particle with an anomalous magnetic moment in the Aharonov-Bohm potential in the conical space.
Highlights
Singular and pathological Hamiltonians are quite common in quantum mechanics and already have a long history [1]
The mathematical framework of quantum mechanics is that of linear operators in Hilbert spaces and the problems and paradoxes that could arise come from the use of simplified rules described in many textbooks
We shall obtain the Dirac equation to describe the motion of a spin-1/2 charged particle with mass M and anomalous magnetic moment μB interacting with an AB field in the cosmic string spacetime
Summary
Singular and pathological Hamiltonians are quite common in quantum mechanics and already have a long history [1]. The situation changes if we consider the delta function because the singularity is physically equivalent to an extraction of a single point from the plane R2, which leads to the loss of the self-adjointness of the Hamiltonian This has important consequences in the spectrum of the system [25]. The authors showed that the phase depends on the initial state of this atom and, in particular, there is no geometric phase acquired for the atom if the initial state is prepared in the excited state Another physical model of current interest that has several studies in cosmic string spacetime is the Dirac oscillator [39]. In reference [42], the self-adjoint extension method was used to study the effects of spin on the dynamics of a two-dimensional Dirac oscillator in the magnetic cosmic string background.
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