⚛️ quantum physics

On Solutions of the Killingbeck Potential and Clarifying Comments on a Related Analytical Approach

This paper presents exact analytical solutions for the Schrödinger equation with the Killingbeck potential using a series expansion method to describe quarkonium and confined hydrogen-like atoms, while also providing a constructive critique to correct mathematical errors in a recent study by Obu et al. regarding similar systems.

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

Published 2026-07-23

📖 4 min read🧠 Deep dive

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

The Cosmic Dance of Invisible Particles

Imagine the universe as a giant, invisible playground where tiny particles like electrons and quarks are constantly running, jumping, and spinning. But they aren't running freely; they are trapped in a "dance floor" created by invisible forces. In the world of quantum mechanics, scientists use something called a "potential" to describe the shape of this dance floor. Think of a potential like the terrain of a roller coaster: some parts dip down (attracting the particle), some parts go up (pushing it away), and some parts curve smoothly.

To predict exactly how these particles move and what energy they have, physicists use a famous recipe called the Schrödinger equation. It's like a complex math machine that takes the shape of the roller coaster (the potential) and spits out the allowed paths and speeds (the energy levels) for the particle. However, some roller coasters are so weirdly shaped that the math machine breaks down, and scientists have to find clever tricks to solve it. One of these tricky shapes is the "Killingbeck potential," a hybrid track that mixes three different types of forces: a smooth curve (like a spring), a straight slope (like a ramp), and a sharp dip (like a deep hole). This specific mix is crucial for understanding heavy particles like quarks (which make up protons) and atoms trapped in hot plasma, where forces behave in unusual ways.

The Paper's Mission: Fixing the Map and Solving the Puzzle

In this paper, the authors, Fatma Zohra Khaled, Mustafa Moumni, and Mokhtar Falek, take on the challenge of solving the Schrödinger equation for this specific "Killingbeck" roller coaster. They don't just want to guess the answer; they want to find the exact mathematical formula that describes the energy and movement of particles in this environment. To do this, they use a technique called the "series expansion method," which is essentially building the solution piece by piece, like stacking Lego bricks to form a tower. They also use a more advanced mathematical tool called the "Heun equation," which acts like a specialized translator that converts the messy physics problem into a language where the solution is easier to read.

But there's a twist. The authors noticed that a recent study by other scientists (Obu et al.) tried to solve a very similar problem but made a critical mistake in their logic. Imagine trying to solve a puzzle where you assume that because two pieces look different, they must be unrelated. The authors of this paper point out that Obu et al. confused the "independence" of the puzzle pieces (the mathematical terms) with the independence of the instructions used to build them. This led to a wrong conclusion about how the energy levels are calculated. The authors of this paper step in to correct this "mathematical misstatement," showing that the pieces are actually linked in a specific way that creates a chain of rules (recurrence relations) rather than standing alone.

Once they fixed the logic, the authors successfully derived the exact formulas for the energy levels of particles in the Killingbeck potential. They found that the energy isn't just one thing; it's a hybrid mix. It combines the energy of a spring (harmonic oscillator) and the energy of a deep hole (Coulomb potential), with a special term that accounts for the linear slope. They showed that when the "screening" effects (which happen in plasma environments) are weak, this complex potential simplifies into the familiar models scientists already know.

However, the authors are careful to note that their method of stopping the infinite series of Lego bricks (truncating the series) to get a final answer isn't as simple as just saying "stop here." They discovered that a common shortcut used in previous works doesn't always guarantee the series actually stops; sometimes, the math keeps going even when you think it should stop. By using the Heun equation approach, they provided a more rigorous way to ensure the solution is finite and correct. They tested their new formulas by checking what happens in extreme cases: when the linear slope disappears, the result perfectly matches the standard harmonic oscillator, and when the spring force disappears, it matches the standard hydrogen atom. This confirms their math is solid.

In short, this paper does two main things: it fixes a logical error in a recent study that confused how mathematical terms relate to each other, and it provides a complete, exact map of the energy levels for particles moving in the complex Killingbeck potential. They prove that this potential is a powerful tool for describing systems where particles are both confined and screened, offering a clearer picture of the quantum dance floor than before.

Technical Summary: Solutions of the Killingbeck Potential and Clarification of Related Analytical Approaches

Problem Statement
The paper addresses the need for accurate analytical solutions to the radial Schrödinger equation for the Killingbeck potential, a hybrid model combining harmonic ( $Cr^2$ ), linear ($Br$), and Coulombic ( $-A/r$ ) terms. This potential is significant for modeling quantum systems ranging from heavy quarkonium (meson physics) to hydrogen-like atoms in plasma environments, where both confinement and screening effects are present. The authors identify two specific issues in the existing literature:

A conceptual misstatement regarding the linear independence of power series coefficients in a recent study by Obu et al. (2023).
A potential mathematical flaw in the series truncation conditions used by Guvendi and Mustafa (2021) and others, which may not guarantee the termination of the series required for bound-state solutions.

Methodology
The authors employ two primary analytical frameworks to solve the radial Schrödinger equation:

Series Expansion Method (SEM): The authors derive the Killingbeck potential as a third-order Taylor expansion of a screened Yukawa-type potential. They propose a wavefunction ansatz $R(r) = e^{-(\alpha r^2 + \beta r)}F(r)$ and substitute a general polynomial series $F(r) = \sum a_k r^{k+s}$ into the differential equation. Unlike previous works that arbitrarily selected specific polynomial forms, the authors derive the starting power $s$ and the recurrence relations rigorously from the boundary conditions at the origin.
Biconfluent Heun Functions (BHE): To address the limitations of the standard truncation condition, the authors transform the radial equation into the Biconfluent Heun equation. This allows for a more robust analysis of the series termination conditions required to obtain physical, normalizable wavefunctions.

Key Contributions and Corrections

Correction of Obu et al. (2023): The authors clarify a mathematical error in Obu et al.'s treatment of linear independence. Obu et al. incorrectly stated that the independence of coefficients implies they are separately zero. The authors correct this by stating that the linear independence of the basis functions $r^n$ requires that the coefficient of each power vanishes separately, leading to recurrence relations rather than independent constraints on the coefficients themselves. This correction resolves a contradiction regarding the principal quantum number found in the previous work.
Refinement of Truncation Conditions: The paper critiques the condition used by Guvendi and Mustafa (and others) where setting a specific coefficient to zero (e.g., $a_{n_r+2}=0$ ) is assumed to truncate the series. The authors demonstrate that this condition does not guarantee $a_{n_r+3}=0$ , meaning the series may not actually terminate. They propose that the Biconfluent Heun formalism provides the necessary rigorous conditions (simultaneous vanishing of specific coefficients) to ensure true polynomial solutions.

Results

Exact Energy Eigenvalues: The authors derive exact analytical expressions for the energy eigenvalues ( $E_{n_r, l}$ ) of the Killingbeck potential. These expressions are shown to interpolate between the harmonic oscillator and Coulomb limits.
Wavefunctions: Explicit forms for the radial wavefunctions are obtained in terms of the Biconfluent Heun functions, multiplied by the asymptotic decay factors and the angular momentum term $r^L$ .
Validation via Limits: The derived energy formulas are tested against known physical limits:
- In the limit of vanishing linear and repulsive terms, the spectrum reduces to that of the harmonic oscillator.
- In the Coulomb limit (strong screening parameter $\alpha_I \to \infty$ ), the results converge to the standard hydrogenic energy levels.
- For the specific case of the first excited state ( $n_r=1$ ), the authors identify two mathematical solutions for the energy. By analyzing the Coulomb limit, they determine that only one of these solutions ( $E^-_{1,l}$ ) correctly corresponds to the physical $n_r=1$ state, thereby generalizing and correcting the results found in previous literature.

Significance
The paper claims that its primary significance lies in providing mathematically rigorous and exact analytical solutions for the Killingbeck potential, a model of high utility in hadronic and plasma physics. By correcting the conceptual errors in the treatment of power series independence and refining the truncation conditions for series solutions, the authors aim to ensure the mathematical integrity of future spectral calculations. The work serves to clarify the relationship between screened potentials and the Killingbeck model, offering a unified framework for systems where both short-range and long-range interactions are critical. The authors emphasize that these corrections are essential for maintaining the pedagogical value and physical accuracy of analytical approaches in quantum mechanics.

The Cosmic Dance of Invisible Particles

The Paper's Mission: Fixing the Map and Solving the Puzzle

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