Bound States in a Debye-Hückel Potential

Abstract

This is a study of some of the properties of the discrete energy levels of an electron in an exponentially shielded Coulomb potential, which is known in plasma physics as the Debye-Hückel potential and in nuclear physics as the Yukawa potential. A system with this potential possesses a finite number of bound states and these states are not degenerate with respect to the orbital angular momentum. First-order perturbation theory is used to obtain simple, analytical expressions for estimating the energy levels; the approximations appear to be quite accurate for large values of the Debye length. The perturbation calculations lead to an estimate of the number of bound states as a function of the Debye length. The matrix elements of the Hamiltonian for this potential are calculated in the representation of the hydrogen atom wave functions. The Hamiltonian matrix may serve as a convenient starting point for several other methods of computing the energy levels.