Electron Waves in the Magnetic Dipole Field of a Neutron

Abstract

The relativistic wave functions for an electron moving in the field of a central magnetic dipole that has spin $\frac{1}{2}\ensuremath{\hbar}$ are studied. Spin inertia and retardation of the central field are neglected. In general, there are then four, first-order, coupled differential equations for the radial dependence of the electron waves; and these are equivalent to two coupled Schroedinger equations having a positive definite potential energy operator. In the particular case that the total angular momentum of the system is zero, however, the equations are equivalent to a single Schroedinger equation in which the potential energy is positive at all radii only for the electron-neutron interaction. The positron-neutron potential becomes strongly negative in a small region near a separation of the order of $\frac{{e}^{2}}{M{c}^{2}}$, where $M$ is the mass of the neutron. The attraction is just sufficient to form a bound ${P}_{\frac{1}{2}}$-state of the positron with zero binding energy. The reality of such neutron-positron state is open to question because of the simplifications that have been made and also because the magnetic dipole moment of the neutron is most probably of extended origin. The single equation obtained when the total angular momentum vanishes is the Mathieu differential equation which may be solved in terms of an integral function by a method developed by Lindemann. Asymptotic forms of these solutions, as well as of the fourth-order equation for total angular momentum unity are presented. The results indicate agreement with those obtained in various applications of perturbation methods. In particular, the ambiguity of the wave functions discussed previously by Bloch is apparent in the solutions for unit angular momentum.