[Paper Review] A perturbative treatment for the exponential-cosine-screened Coulomb potential

Sameer M. Ikhdair|ArXiv.org|Sep 1, 2005

Quantum Mechanics and Non-Hermitian Physics4 references4 citations

TL;DR

This paper presents a novel perturbative approach to solve the Schrödinger equation for the exponential-cosine-screened Coulomb (ECSC) potential, using a decomposition into an exactly solvable part and a perturbed component. The method yields analytical expressions for bound-state energies and wave functions up to second-order corrections, showing high accuracy for low to moderate screening parameters and validating results against existing numerical and analytical methods.

ABSTRACT

An alternative approximation scheme has been used in solving the Schroedinger equation for the exponential-cosine-screened Coulomb potential. The bound state energies for various eigenstates and the corresponding wave functions are obtained analytically up to the second perturbation term.

Motivation & Objective

• To develop a new perturbative formalism for solving the Schrödinger equation with the exponential-cosine-screened Coulomb (ECSC) potential, which lacks exact solutions.
• To overcome limitations of standard perturbation theory by introducing a decomposition of the radial equation into an exactly solvable part and a perturbed part.
• To derive analytical expressions for bound-state energies and radial wave functions up to second-order corrections for various quantum states.
• To validate the method’s accuracy by comparing computed energy eigenvalues with existing numerical and analytical results across different screening parameters and atomic numbers.
• To explore the behavior of the ECSC potential in the low- and high-screening regimes, particularly for light to heavy atoms.

Proposed method

• The radial Schrödinger equation is decomposed into an exactly solvable part $ V_0(r) + \frac{\hbar^2 \ell(\ell+1)}{2m r^2} $ and a perturbation $ \Delta V(r) $, with the full wave function expressed as a product of unperturbed $ \chi_n(r) $ and a modulating function $ u_n(r) $.
• Logarithmic derivatives $ W_n(r) $ and $ \Delta W_n(r) $ are introduced to transform the second-order differential equation into a form suitable for perturbative treatment.
• The perturbation equation (Eq. 7) is solved analytically by assuming a power-series ansatz for $ \Delta W_n(r) $, enabling derivation of energy corrections $ \Delta \varepsilon_n = E_n^{(1)} + E_n^{(2)} + \cdots $.
• The method uses a modified perturbation scheme that avoids nodal singularities and allows consistent analytical treatment for both ground and excited states.
• Energy corrections are computed up to second order using the derived expressions for $ W_n^{(1)}(r) $ and $ W_n^{(2)}(r) $, with explicit formulas provided for $ E_n^{(1)} $ and $ E_n^{(2)} $.
• The approach is applied numerically in atomic units ($ \hbar = m = A = 1 $) and extended to physical parameters ($ A = Z $, $ \delta = \sqrt{2}G $) for comparison with literature.

Experimental results

Research questions

• RQ1 Can a new perturbative formalism provide accurate analytical solutions for the bound-state energies and wave functions of the ECSC potential?
• RQ2 How do the first- and second-order energy corrections behave across different quantum numbers $ n $ and $ \ell $, and how do they depend on the screening parameter $ \delta $?
• RQ3 To what extent does the method maintain accuracy for increasing $ A $ and $ \delta $, especially in the high-screening and high-potential-strength regimes?
• RQ4 How do the computed energy levels compare with existing numerical and analytical results from the literature for $ 1s $, $ 2s $, $ 2p $, $ 3s $, $ 3p $, and $ 3d $ states?
• RQ5 Can the method be generalized to yield analytical expressions for higher-order corrections without numerical instability or divergence?

Key findings

• The first-order energy correction for the $ n=2, \ell $ state is analytically derived as $ E_2^{(1)} = -\frac{\hbar^4 (\ell+3)^2 (\ell+2)(2\ell+23)}{6A m^2} \delta^3 $, valid for all $ \ell $.
• The second-order energy correction for the $ n=2, \ell $ state is given by $ E_2^{(2)} = \frac{\hbar^6 (\ell+2)(\ell+3)^2(2\ell+5)(2\ell^2 + 45\ell + 153)}{24A^2 m^3} \delta^4 - \frac{\hbar^{10} (\ell+2)(\ell+3)^5 (16\ell^4 + 474\ell^3 + 3879\ell^2 + 12118\ell + 12873)}{72A^4 m^5} \delta^6 $, showing explicit dependence on quantum number and screening.
• Numerical results for $ 1s $ and $ 2s $ states in atomic units ($ \hbar = m = A = 1 $) agree with prior works (Refs. [9,22,23]) up to $ \delta^6 $, confirming consistency and accuracy.
• For $ A = Z = 4, 8, 16, 24 $, the method yields good agreement with other methods at low $ \delta $, but accuracy degrades as $ \delta $ and $ A $ increase, indicating limitations in strong-screening regimes.
• The method successfully avoids nodal singularities in the perturbation expansion, enabling consistent analytical treatment for excited states including $ 2p $, $ 3s $, $ 3p $, and $ 3d $.
• The formalism provides analytical wave functions via the modulating function $ u_n(r) $, derived from the perturbed logarithmic derivative $ \Delta W_n(r) $, enabling full analytical description of the radial wave function.