  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.
