We discuss the Hamiltonian for a nonrelativistic electron with spin in the presence of an abelian magnetic monopole and note that it is not self-adjoint in the lowest two angular momentum modes. We then use von Neumann's theory of self-adjoint extensions to construct a self-adjoint operator with the same functional form. In general, this operator will have eigenstates in which the lowest two angular momentum modes mix, thereby removing conservation of angular momentum. However, consistency with the solutions of the Dirac equation limits the possibilities such that conservation of angular momentum is restored. Because the same effect occurs for a spinless particle with a sufficiently attractive inverse square potential, we also study this system. We use this simpler Hamiltonian to compare the eigenfunctions corresponding to a particular self-adjoint extension with the eigenfunctions satisfying a boundary condition consistent with probability conservation.
1997 Academic Press
I. INTRODUCTION
In this article, we first examine the Pauli Equation for an electron in the field of a magnetic monopole. This equation has appeared in the literature before, and it is well known that an extension is needed to make the Hamiltonian self-adjoint in the j=0 sector [1]. What seems to have gone unnoticed is that, for eg= 1 2 , the domain to be extended includes the j=1 sector as well. With the inclusion of this sector, the structure of the extensions becomes richer, and the number of parameters required to describe them jumps from 1 to 16. While the parameters may be chosen to be consistent with angular momentum conservation, this is not required: a pure incoming s-wave can come out with a p-wave component, even though the functional form of the Hamiltonian is spherically symmetric. However, if we require our states to match the states of the Dirac equation in the nonrelativistic limit, we will only have 1 free parameter, and angular momentum will be conserved. To better article no. PH965638 11 0003-4916ร97 25.00