Zero-range potentials for Dirac particles: Scattering and related continuum problems

Abstract

The zero-range potential model, widely used in nonrelativistic quantum mechanics, is extended to continuum problems involving Dirac particles. A bispinor wave function of a Dirac particle scattered from a system of zero-range potentials is sought in the form of an incident wave superposed with waves emerging from points where targets are located. Interactions between the particle and individual targets are described by imposing certain limiting conditions, relating linearly upper and lower components of the wave function at target locations. This yields an inhomogeneous algebraic system for superposition coefficients appearing in the expression for the wave function. After preliminary considerations, admitting a quite general form of the incident wave, the case of the monochromatic plane-wave scattering is considered in detail. Expressions for $2\ifmmode\times\else\texttimes\fi{}2$ and $4\ifmmode\times\else\texttimes\fi{}4$ matrix scattering amplitudes and scattering kernels, as well as for various kinds of differential and total cross sections, are given. An eigenchannel formalism for the model is developed in the manner analogous to that presented in the author's recent work [R. Szmytkowski, Ann. Phys. (N.Y.) 311, 503 (2004)] on scattering from short-range potentials. Eigenchannel representations of the scattering wave function, of a ``final-state'' wave function for photodetachment, as well as of outgoing and ingoing matrix Green functions, are derived. Formulas for matrix scattering amplitudes, scattering kernels, and cross sections expressed in terms of eigenphase shifts and eigenchannel spinor harmonics are presented. A possibility to formulate the zero-range potential model for Dirac particles in an alternative way is also discussed.