Towards the Theory of the Yukawa Potential

TL;DR

This paper develops and compares three methods—Perturbation Theory, Lagrange Mesh, and Variational Method—to accurately analyze the low-lying states of the Yukawa potential, providing new approximations and analytical expressions for energy and wavefunctions.

Contribution

It introduces a new compact approximation for eigenfunctions, compares multiple methods for accuracy, and derives an analytical energy expression valid across the entire physical range.

Findings

Padé approximants and Lag-Mesh yield highly accurate energies.

Critical screening parameters are precisely determined.

An analytical energy function reproduces results to six decimal places.

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 $V(r)=-(\lambda/r)e^{-\alpha r}\,$. First orders in PT in powers of $\alpha$ are calculated in the framework of the Non-Linerization Procedure. It is found that the Pad\'e 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 ($\alpha_c$) for some low-lying states and the first coefficients in the expansion of the energy at $\alpha_c$ 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 $\alpha$, that reproduce at least $6$ decimal digits in the entire physical range of $\alpha$, is found.