Towards the Theory of the Yukawa Potential

Abstract

Using three different approaches, Perturbation Theory (PT), the Lagrange Mesh Method (Lag-Mesh) and the Variational Method (VM), we study the low-lying states of the Yukawa potential $V(r)=-(λ/r)e^{-αr}\,$. First orders in PT in powers of $α$ are calculated in the framework of the Non-Linerization Procedure. It is found that the Padé approximants to PT series together with the Lag-Mesh provide highly accurate values of the energy and the positions of the radial nodes of the wave function. The most accurate results, at present, of the critical screening parameters ($α_c$) for some low-lying states and the first coefficients in the expansion of the energy at $α_c$ are presented. A locally-accurate and compact approximation for the eigenfunctions of the low-lying states for any $r\in [ 0,\infty)$ is discovered. This approximation used as a trial function in VM eventually leads to energies as precise as those of PT and Lag-Mesh. Finally, a compact analytical expression for the energy as a function of $α$, that reproduce at least $6$ decimal digits in the entire physical range of $α$, is found.