⚛️ quantum physics

Bound state solutions of the Schrödinger equation for the atomic systems interacting with the radial screened Coulomb potential: analytical approximation methods

This paper investigates the bound state properties of hydrogen-like atoms and Positronium interacting with a radial screened Coulomb potential by deriving approximate energy eigenvalues through three complementary analytical methods, which are validated against high-precision numerical data to demonstrate their accuracy and utility for estimating errors in plasma-embedded atomic systems.

Original authors: Fatma Zohra Khaled, Mustafa Moumni, Mokhtar Falek

Published 2026-07-22

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CC BY 4.0

Original authors: Fatma Zohra Khaled, Mustafa Moumni, Mokhtar Falek

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible ocean. Usually, when we think about atoms—the tiny building blocks of everything around us—we picture them floating in a perfect vacuum, like a lonely island in a calm sea. In this quiet state, the electron dances around the nucleus in a predictable rhythm, governed by a simple, well-known rule called the Coulomb potential. It's like a magnet pulling a paperclip; the closer they get, the stronger the pull.

But what happens when that ocean isn't empty? What if the atom is submerged in a thick, busy crowd, like a party where everyone is jostling for space? In the real world, atoms often live in "plasma"—a super-hot, electric soup found in stars, lightning, and fusion reactors. In this crowded environment, the other charged particles act like a protective shield, or a "screen," that weakens the atom's internal pull. It's as if the crowd is holding up a wall between the magnet and the paperclip, making the attraction feel weaker and changing the dance steps of the electron. Scientists need to know exactly how these steps change to understand how stars shine or how to build clean energy. The big question is: how do we calculate the new energy levels of an atom when it's being squeezed and shielded by its neighbors?

This paper tackles that exact problem by looking at a specific type of "shield" called the Radial Screened Coulomb Potential (RSCP). Think of the RSCP not as a wall that blocks the pull from far away, but as a special kind of fog that is thickest right next to the nucleus and thins out as you move away. This is different from the more common "Yukawa" model, which acts like a fog that is thick far away and clear up close. The authors wanted to find a way to predict the energy of an electron in this specific "foggy" environment without needing to run massive, time-consuming computer simulations every single time.

To solve this, the researchers acted like master architects trying to build a model of a complex, foggy house. They didn't try to measure every single molecule of fog; instead, they used three clever "blueprint" strategies to approximate the answer.

First, they tried a Coulomb Reference. Imagine trying to guess the shape of a house in a fog by looking at a clear, sunny house next door and just guessing how the fog changes it. This method is simple, but it tends to underestimate how "stuck" the electron is, making the energy look a bit too high (too close to zero).

Second, they used a Kratzer Reference. This is like realizing the fog isn't just a wall; it actually pushes back a little bit near the center, changing the shape of the foundation. By adjusting their blueprint to account for this push, they got a much better picture. This method was surprisingly accurate, getting the energy right within about 0.63% for the first ten energy levels when the screening was moderate.

Third, and most impressively, they used a Variational Method. This is like taking that improved blueprint and adding a "dial" that lets them stretch or shrink the house until it fits the fog perfectly. By tweaking this dial, they found the absolute best fit. This approach was the winner, achieving errors as low as 0.40% for the ground state and even better for higher energy states.

The paper also checked their work against the "gold standard" of high-precision computer simulations (called GPS data) and found their math held up remarkably well. They even showed that their formulas work for other atom-like systems, such as Positronium (a weird atom made of an electron and its antimatter twin, a positron). They found that while the math works for both, the regular hydrogen atom feels the "fog" more strongly than the Positronium does because of how its internal forces interact with the screen.

One interesting thing the authors ruled out is the idea of a "critical point" where the atom suddenly falls apart. Some older, simpler models suggested that if the fog got too thick, the electron would just pop off. However, the authors' more accurate calculations show that the electron stays bound, just less tightly, no matter how thick the fog gets (as long as it's finite). The "falling apart" idea was just an illusion caused by using a too-simple blueprint.

In short, this paper provides a new, fast, and highly accurate set of mathematical tools for scientists to predict how atoms behave when they are crowded in plasma. Instead of waiting for a supercomputer to crunch the numbers for every new scenario, researchers can now use these elegant formulas to get answers that are almost as good as the most complex simulations, but in a fraction of the time. It's a reminder that sometimes, the best way to understand a complex, foggy world is to build a clever, adjustable model that fits just right.

Technical Summary: Bound State Solutions of the Schrödinger Equation for Atomic Systems Interacting with the Radial Screened Coulomb Potential

Problem Statement
The paper addresses the challenge of accurately determining the bound state energy eigenvalues of hydrogen-like atoms embedded in plasma environments. Standard models often rely on the Yukawa (Debye-Hückel) potential, which screens interactions at large distances. However, the authors focus on the Radial Screened Coulomb Potential (RSCP), proposed by Stachura, defined as $V_{RSCP}(r) = -(Ze^2/r)e^{-c/r}$ . Unlike the Yukawa potential, the RSCP is non-singular at the origin ( $r \to 0$ ) and retains the correct unscreened Coulomb tail at large distances ( $r \to \infty$ ). This potential is physically motivated by scenarios involving finite nuclear size effects, muonic atoms, and core polarization where screening depends on proximity to the nucleus rather than distance from it. While high-precision numerical methods (such as the Generalized Pseudospectral or GPS method) exist, there is a need for closed-form analytical approximations that offer rapid parametric scaling and direct insight into the functional dependence on the screening parameter $c$ .

Methodology
The authors employ three complementary analytical approaches to derive approximate energy eigenvalues for arbitrary quantum numbers $n$ and $\ell$ . All methods utilize the exact, unexpanded RSCP Hamiltonian, avoiding the truncation of the potential itself; approximations arise solely from the choice of trial wavefunctions.

Expectation Value Approach with Reference States:
The authors decompose the RSCP Hamiltonian into a reference Hamiltonian ( $H_{ref}$ ) and a residual potential ( $\Delta V$ ). Two reference potentials are derived from the Taylor expansion of the RSCP:
- Coulomb Reference ( $j=0$ ): Uses the pure Coulomb potential $-1/r$ as the reference. The trial wavefunctions are standard hydrogenic eigenfunctions. This method neglects the $c/r^2$ term in the expansion, leading to an underestimation of binding energy.
- Kratzer Reference ( $j=1$ ): Uses the potential $-1/r + c/r^2$ as the reference. This incorporates the leading-order screening correction by modifying the effective angular momentum quantum number to $\nu_l = -1/2 + \sqrt{(l+1/2)^2 + 2c}$ . The trial wavefunctions are Kratzer eigenfunctions. This method accounts for the repulsive $c/r^2$ term, leading to a slight overestimation of binding energy.
- In both cases, the energy is calculated as $E \approx E_{ref} + \langle \Delta V \rangle_{ref}$ , where the expectation values of the full RSCP term $e^{-c/r}/r$ are evaluated analytically using modified Bessel functions of the second kind ( $K_\nu$ ).
Variational Method with Scaled Kratzer Basis:
Building on the Kratzer reference, the authors introduce a variational scaling parameter $\beta$ into the trial wavefunction to optimize its spatial extent. The energy functional $E_{var}(\beta) = \langle \psi_{trial} | H_{RSCP} | \psi_{trial} \rangle$ is minimized numerically with respect to $\beta$ . This approach systematically improves upon the fixed-reference methods by adjusting the wavefunction to better match the exact potential.
Hellmann-Feynman Theorem Approach:
The authors utilize the Hellmann-Feynman theorem, $\partial E / \partial c = \langle \partial V / \partial c \rangle$ , to derive energies by integrating the expectation value of $e^{-c/r}/r^2$ from the unscreened limit ( $c=0$ ) to a finite $c$ . This is performed using both Coulomb and Kratzer basis functions. The Coulomb-based integration is shown to be mathematically equivalent to the Coulomb reference expectation value method, while the Kratzer-based integration (HF-Kr) provides a distinct route to the energy.

Key Contributions

Analytical Formulas: The paper derives closed-form analytical expressions for energy eigenvalues involving modified Bessel functions and Gamma functions, valid for arbitrary $n$ and $\ell$ .
Complementary Bias Analysis: The study explicitly characterizes the systematic biases of the reference methods: the Coulomb reference underestimates binding (energies too close to zero), while the Kratzer reference overestimates binding (energies too negative). This complementary behavior allows for robust error estimation.
Extension to Positronium: The formalism is generalized to Positronium (Ps), accounting for the reduced mass $\mu_{Ps} = 1/2$ . The authors demonstrate that the mass-dependent screening response leads to different energy shifts compared to hydrogen.
Benchmarking: The analytical results are rigorously benchmarked against high-precision GPS data from Xu et al. and previous results from Stachura and Hancock.

Results

Accuracy: At a screening parameter $c=0.1$ , the Kratzer reference method achieves relative errors below 0.63% for the first ten s-states. The variational method further improves this, reducing errors to approximately 0.40% for the 1s state and 0.06% for the 10s state.
Convergence: The error scaling analysis reveals that the variational method provides the most robust accuracy across all states. The Coulomb reference errors decrease as $1/n$ , while the Kratzer-based methods maintain relatively constant errors across $n$ .
Critical Screening: The authors note that the Coulomb reference method incorrectly predicts a "critical screening" (energy crossing zero) for the 1s state at $c \approx 0.6$ . In contrast, the Kratzer-based methods and exact GPS data confirm that bound states persist for all finite $c$ , as the effective potential remains attractive.
Positronium: For Positronium at $c=0.1$ , the variational method yields a relative error of 0.106% for the 1s state. The study finds that the fractional energy shift for hydrogen (24.44%) is larger than for Positronium (14.74%) at the same $c$ , attributed to the larger effective centrifugal barrier modification in hydrogen.

Significance and Claims
The paper claims to provide a unified analytical framework that balances computational efficiency with high accuracy. The authors emphasize that while numerical methods like GPS provide benchmark-quality data, their analytical expressions offer distinct advantages:

Rapid Parametric Scaling: They enable quick studies of how energy levels shift with the screening parameter $c$ without re-running complex numerical simulations.
Physical Insight: The closed-form expressions provide direct insight into the functional dependence of the energy on the screening parameter and the role of the effective angular momentum.
Diagnostic Utility: The methods serve as efficient tools for plasma diagnostics where repeated calculations are required.

The authors position their work as a complementary approach to standard Debye-Hückel models, specifically targeting regimes where short-range screening and internal atomic structure effects are significant. They acknowledge that for very strong screening ( $c \ge 1$ ), deviations from reference values increase, suggesting that higher-order corrections or resummation techniques may be necessary in that regime. The work does not propose new experimental setups but validates the utility of the RSCP model for interpreting precision spectroscopy in exotic atoms and dense plasmas.

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