Bound states of the hydrogen-like atomic systems in plasma environments

Technical Summary: Bound States of Hydrogen-like Atomic Systems in Plasma Environments

Problem Statement
The paper addresses the challenge of solving the non-relativistic Schrödinger equation for hydrogen-like atomic systems immersed in a plasma environment. The interaction is modeled using the Screened Coulomb Potential (SCP), also known as the Yukawa or Debye-Hückel potential, $V(r) = -(Ze^2/r)e^{-kr}$, where $k$ is the screening parameter. While the physical importance of this system is well-established in astrophysics, semiconductors, and fusion research, the Schrödinger equation for the full Yukawa potential cannot be solved exactly in terms of elementary functions. Existing numerical and variational methods provide accurate results but often lack explicit analytic expressions for wavefunctions and energies as functions of the screening parameter $k$. Furthermore, previous analytic attempts, such as those based on the bi-confluent Heun (BCH) equation for truncated potentials, suffer from a critical limitation: they fail to reduce smoothly to the standard Coulomb eigenfunctions as the screening parameter $k \to 0$.

Methodology
The authors employ a multi-stage analytical approach:

1. Potential Approximation: The Yukawa potential is expanded to the third order in $k$, resulting in a cubic-truncated potential of the Killingbeck type: $V(r) \approx Ze^2(-1/r + k - k^2r/2 + k^3r^2/6)$.
2. Exact BCH Solutions: The radial Schrödinger equation for this truncated potential is transformed into a bi-confluent Heun (BCH) equation. The authors derive exact eigenfunctions and eigenenergies for radial quantum numbers $n_r = 0$ and $n_r = 1$. However, they identify that these exact solutions impose rigid truncation conditions that prevent a smooth transition to the Coulomb limit ($k=0$).
3. Inspired BCH Ansatz: To overcome the Coulomb limit deficiency, the authors propose an alternative "inspired" approach. They retain the functional form of the BCH ansatz (involving exponential and polynomial terms) but relax the rigid truncation conditions. Instead, the parameters and polynomial coefficients are treated as analytic functions of $k$. These are determined order-by-order by enforcing the Schrödinger equation and ensuring the recovery of Coulomb solutions when $k=0$. This yields corrected eigenfunctions expressed in terms of Laguerre polynomials, valid up to $O(k^3)$.
4. Energy Calculation: Using these corrected wavefunctions, the energy spectrum is computed via two independent analytic methods:
   • Direct Expectation Value: Evaluating the expectation value of the full Yukawa Hamiltonian (not just the truncated expansion), yielding closed-form expressions involving hypergeometric functions.
   • Hellmann-Feynman Theorem: Utilizing the theorem to derive an integral representation of the energy, which is then expanded to match the perturbative series.
5. Thermodynamics: The partition function is derived for the $n_r=0$ states using the exact BCH energy expressions. The Euler-Maclaurin summation formula is applied to evaluate the sum over angular momentum states, allowing for the derivation of analytic expressions for free energy, mean energy, entropy, and heat capacity.

Key Contributions and Results

• Analytic Wavefunctions: The paper provides explicit, systematic analytic formulas for wavefunctions that are valid for weak to moderate screening and reduce exactly to Coulomb eigenfunctions when $k=0$. These are constructed up to $O(k^3)$ and can be extended to higher orders.
• Energy Spectrum Accuracy: Numerical comparisons with reference variational results (Paul and Ho) show that the corrected BCH wavefunctions yield energy eigenvalues with relative errors below 0.01% for most states when screening lengths $\lambda_D \gtrsim 20$ and third-order corrections are included. This represents a significant improvement over the exact BCH polynomial solutions, which show deviations of up to 8.46% at similar screening levels.
• Methodological Consistency: The energy values obtained via the direct expectation value method and the Hellmann-Feynman theorem are virtually identical (differences $\le 10^{-5}$ Ry), serving as a strong internal consistency check that validates the quality of the corrected wavefunctions.
• Thermodynamic Properties: Analytic expressions for thermodynamic quantities (free energy, mean energy, entropy, specific heat) are derived. These expressions correctly converge to the standard Coulomb limits as the screening parameter vanishes ($\lambda \to \infty$).
• Radial Distributions: The radial probability distributions computed from the analytic wavefunctions correctly show broadening as screening increases, reflecting reduced nuclear attraction.

Significance
The paper establishes a powerful analytic framework for studying confined quantum systems with screened interactions. Its primary significance lies in providing explicit analytic formulas for both wavefunctions and energies, which are often unavailable in purely numerical or variational approaches. The authors highlight several advantages of their method:

• Systematic Improvement: The approach allows for the inclusion of higher-order terms to improve accuracy systematically.
• Smooth Limits: The construction ensures a smooth transition to the Coulomb limit, a feature lacking in exact BCH polynomial treatments.
• Utility for Variational Studies: The derived wavefunctions, particularly the inspired form involving Laguerre polynomials, serve as flexible trial functions for variational calculations with one or more parameters.
• Closed-Form Representations: The methods yield closed-form expressions (involving hypergeometric functions) that facilitate immediate numerical evaluation and further analytic work, such as calculating transition probabilities and expectation values.

The work is presented as a foundational analytic tool that can be extended to other exponential-type potentials and relativistic equations, offering a robust alternative to numerical methods for plasma-screened atomic systems.