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 $V(r)=-(\lambda /r)e^{-\alpha r}$. First orders in PT in powers of $\alpha$ are calculated in the framework of the Non-Linerization Procedure. It is found that the 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 (${\alpha }_c$) 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.

arXiv comment28 pages, 4 figures

References in corpus (6)

• Cores in Dwarf Galaxies from Dark Matter with a Yukawa Potential
• Ground state of a confined Yukawa plasma
• Anharmonic oscillator and double-well potential: approximating eigenfunctions
• {H${}_2^{+}$, HeH and H${}_2$}: approximating potential curves, calculating rovibrational states
• Accurate solution of the Dirac equation on Lagrange meshes
• Fluid Modes of a Spherically Confined Yukawa Plasma

Cited by in corpus (4)

• Radial Anharmonic Oscillator: Perturbation Theory, New Semiclassical Expansion, Approximating Eigenfunctions. I. Generalities, Cubic Anharmonicity Case
• Solving the One-Dimensional Time-Independent Schrödinger Equation with High Accuracy: The LagrangeMesh Mathematica Package
• Solving Schrödinger Equation with Scattering Matrices. Bound states of Lennard-Jones Potential
• Radial Anharmonic Oscillator: Perturbation Theory, New Semiclassical Expansion, Approximating Eigenfunctions. II. Quartic and Sextic Anharmonicity Cases