[Paper Review] A perturbative treatment for the bound states of the Hellmann potential

Sameer M. Ikhdair, R. Sever|ArXiv.org|Mar 23, 2006

Quantum Mechanics and Non-Hermitian Physics3 references4 citations

TL;DR

This paper presents a perturbative approach to calculate bound-state energy levels of the Hellmann potential, a superposition of Coulomb and Yukawa potentials, using a novel formalism that decomposes the radial Schrödinger equation into exactly solvable and perturbed parts. The method yields accurate energy eigenvalues for weak screening and low quantum numbers, with results deteriorating as screening strength and Yukawa potential magnitude increase due to insufficiently small expansion parameters.

ABSTRACT

A new approximation formalism is applied to study the bound states of the Hellmann potential, which represents the superposition of the attractive Coulomb potential $-a/r$ and the Yukawa potential $b\exp (-δr)/r$ of arbitrary strength $b$ and screening parameter $δ$. Although the analytic expressions for the energy eigenvalues $E_{n,l ext{}}$ yield quite accurate results for a wide range of $n,\ell $ in the limit of very weak screening, the results become gradually worse as the strength $b$ and the screening coefficient $δ$ increase. This is because that the expansion parameter is not sufficiently small enough to guarantee the convergence of the expansion series for the energy levels.

Motivation & Objective

• To develop and apply a new perturbative formalism for calculating bound-state energy levels of the Hellmann potential.
• To investigate the validity and accuracy of the perturbative approach across varying strengths of the Yukawa potential (b) and screening parameters (δ).
• To compare the method’s results with existing variational and large-N expansion techniques, especially for hydrogenic-like states.
• To analyze the level ordering and energy shifts relative to hydrogenic states due to the finite-range Yukawa interaction.
• To assess the method’s performance for both attractive (b < 0) and repulsive (b > 0) Yukawa contributions across different quantum numbers (n, ℓ).

Proposed method

• The radial Schrödinger equation is decomposed into an exactly solvable part (Coulomb plus centrifugal term) and a perturbed part (Yukawa potential).
• The formalism uses logarithmic derivatives of unperturbed and perturbed wave functions to derive a perturbative expansion for energy shifts.
• Energy eigenvalues are calculated up to third-order perturbation using the derived expressions for ΔWn and Wn.
• The method relies on the decomposition of the radial wave function into a known unperturbed part χn(r) and a modulating function un(r).
• The approach is applied to the Hellmann potential V(r) = -a/r + b exp(-δr)/r, with a and b as strength parameters and δ as the screening parameter.
• The formalism is tested numerically for various n, ℓ, b, and δ values, comparing results with high-precision numerical data and other theoretical works.

Experimental results

Research questions

• RQ1 How accurate is the perturbative formalism for calculating bound-state energies of the Hellmann potential across different values of b and δ?
• RQ2 How do energy levels shift relative to hydrogenic levels (EnH) due to the finite-range Yukawa potential, and how does this depend on the sign and magnitude of b?
• RQ3 What is the dependence of energy level ordering on ℓ and n for both attractive and repulsive Yukawa potentials?
• RQ4 How does the convergence of the perturbation series degrade as b and δ increase, and what are the limits of applicability of the method?
• RQ5 To what extent does the method reproduce known results for the static screened Coulomb potential (SSCP) when a=0 and b=-αZ?

Key findings

• The perturbative method yields highly accurate energy eigenvalues for weak screening (small δ) and low quantum numbers, with results approaching hydrogenic levels as n and ℓ increase.
• For increasing b and δ, the accuracy of the perturbative expansion degrades significantly due to the expansion parameter becoming too large for convergence.
• Energy levels shift downward (relative to hydrogenic levels) for b < 0 (attractive Yukawa) and upward for b > 0 (repulsive Yukawa), with the magnitude of shift decreasing with increasing ℓ.
• The method correctly reproduces the level ordering theorems: for b < 0, E_nℓ > E_nℓ' for ℓ < ℓ', and for b > 0, E_nℓ < E_nℓ' for ℓ < ℓ', consistent with Grosse and Martin's theorem.
• For large b, the relative error in energy calculations decreases, and the results compare favorably with high-precision numerical calculations, especially for ℓ = 0.
• Non-monotonic level ordering is observed: for certain b and δ, higher n and lower ℓ states can have lower energy than lower n and higher ℓ states, both for attractive and repulsive Yukawa potentials.