Coulomb plus power-law potentials in quantum mechanics

TL;DR

This paper analyzes the discrete energy spectrum of a quantum Hamiltonian with Coulomb plus power-law potential using envelope theory and trial functions, providing accurate bounds and numerical comparisons for specific cases.

Contribution

It introduces a semiclassical approximation and envelope theory to estimate eigenvalues of Coulomb plus power-law potentials, with detailed bounds and numerical validation.

Findings

Derived approximate eigenvalues using envelope theory.

Obtained accurate upper bounds with trial functions.

Validated results with direct numerical calculations.

Abstract

We study the discrete spectrum of the Hamiltonian H = -Delta + V(r) for the Coulomb plus power-law potential V(r)=-1/r+ beta sgn(q)r^q, where beta > 0, q > -2 and q \ne 0. We show by envelope theory that the discrete eigenvalues E_{n\ell} of H may be approximated by the semiclassical expression E_{n\ell}(q) \approx min_{r>0}\{1/r^2-1/(mu r)+ sgn(q) beta(nu r)^q}. Values of mu and nu are prescribed which yield upper and lower bounds. Accurate upper bounds are also obtained by use of a trial function of the form, psi(r)= r^{\ell+1}e^{-(xr)^{q}}. We give detailed results for V(r) = -1/r + beta r^q, q = 0.5, 1, 2 for n=1, \ell=0,1,2, along with comparison eigenvalues found by direct numerical methods.