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Chinese Physics C

ON THE HERMITIAN OF THE HAMILTONIAN OF RADIAL EQUATION

Zhanwen Ma, Aiguo Dai

Published Jan 1, 1988• 0 citations
OpenAlex

Abstract

The Hamiltonian of a radial equation is defined on a half-line, and there is a close relation between its hermitian and the boundary condition of the wave functions at the origin. If the wave functions are nonvanishing and convergent at the origin, the Hamiltonian has a one-parameter family of self-adjoint extensions which are related with the vanishness of the radial probability current at the origin. In this paper the problem on the hermitian of the Hamiltonian of a radial equation is studied systematically. Some methods for determining the parameter for the fermion moving in the magnetic monopole field are discussed.

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The Hamiltonian of a radial equation is defined on a half-line, and there is a close relation between its hermitian and the boundary condition of the wave functions at the origin. If the wave functions are nonvanishing and convergent at the origin, the Hamiltonian has a one-parameter family of self-adjoint extensions which are related with the vanishness of the radial probability current at the origin. In this paper the problem on the hermitian of the Hamiltonian of a radial equation is studied systematically. Some methods for determining the parameter for the fermion moving in the magnetic monopole field are discussed.
There is much discussion in the mathematical physics literature as well as in quantum mechanics textbooks on spherically symmetric potentials. Nevertheless, there is no consensus about the behavior of the radial function at the origin, particularly for singular potentials. A careful derivation of the radial Schrödinger equation leads to the appearance of a delta function term when the Laplace operator is written in spherical coordinates. As a result, regardless of the behavior of the potential, an additional constraint is imposed on the radial wave function in the form of a vanishing boundary condition at the origin.
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, an analysis 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.