Laguerre polynomials in a relativistic quantum-statistical model

Abstract

The main aim of this work is to find analytical expressions for the eigenvalues and eigenfunctions corresponding to Dirac equation, for hot and dense matter. It is shown that the Laguerre polynomials depending on the effective charge are solutions for self-consistent fields. To determine the screening constant and affective charge we introduce and minimize a functional. These formulas are accurate for machine calculation of bound-bound, bound-free and free-free transitions, including large values of principal quantum numbers. Hence these expressions would be in accordance with quantum-statistical results based on more sophisticated calculations. Comparison with the solutions of the Schrödinger equation for different substancies are discussed.