Coulomb plus power-law potentials in quantum mechanics

Abstract

We study the discrete spectrum of the Hamiltonian H = −∆ + V (r) for the Coulomb plus power-law potential V (r) = −1/r + β sgn(q)r q , where β > 0, q > −2 and q = 0 . We show by envelope theory that the discrete eigenvalues E nℓ of H may be approximated by the semiclassical expression E nℓ (q) ≈ min r>0 {1/r 2 − 1/(µr) + sgn(q)β(νr) q }. Values of µ and ν are prescribed which yield upper and lower bounds. Accurate upper bounds are also obtained by use of a trial function of the form, ψ(r) = r ℓ+1 e −(xr) q . We give detailed results for V (r) = −1/r + βr q , q = 0.5, 1, 2 for n = 1, ℓ = 0, 1, 2, along with comparison eigenvalues found by direct numerical methods.

Discrete spectra for confined and unconfined −a/r + br2 potentials in d-dimensions

“…But the bound is weak. For potentials such as V (r) that satisfy For the unconstrained case R = ∞, however, envelope methods [29][30][31][32][33]35] allow one to construct analytical upper and lower energy bounds with general forms similar to (7). In this case we shall write E d nℓ = E(a, b).…”

Solution of the radial Schrödinger equation for the potential family using the asymptotic iteration method

“…The energy eigenvalues are obtained from the roots of the quantization condition, given by the termination condition of the method in equation (4). The quantization condition of the method together with equation (5) can also be written as follows…”

“…and it is solved analytically by using various methods [2,4]. Furthermore, this potential becomes the spiked harmonic oscillator potential [4] when A = 0, B = 1 and κ = 2.…”

“…Furthermore, this potential becomes the spiked harmonic oscillator potential [4] when A = 0, B = 1 and κ = 2.…”

“…[1,2,3,4,5,6,7,8,9,10,11,12,13] and references therein). Thus, it would be interesting and important to solve the non-relativistic radial Schrödinger equation for a new solvable potential family for any n and l quantum states.…”