A new comparison theorem on Hellmann potential.

Area of Science:

• Quantum mechanics and atomic physics.

Background:

• Screened Coulomb potentials are crucial for modeling atomic and molecular systems.
• The relative ordering of energy levels in these potentials is important for understanding spectral properties.

Purpose of the Study:

• To establish the hierarchical ordering of the spherical Hellmann potential relative to standard screened Coulomb potentials.
• To extend existing comparison theorems for screened Coulomb potentials to include the Hellmann potential.

Main Methods:

• Utilized basic inequalities to determine the relative ordering of potentials.
• Expanded a known comparison theorem for screened Coulomb potentials.
• Performed analytical derivations and supported them with numerical calculations.

Main Results:

• The spherical Hellmann potential is definitively placed within the hierarchy of screened Coulomb potentials at all radial distances.
• The energy levels of nℓ-states for the Hellmann potential are comparatively ordered with respect to screened Coulomb potentials.
• Analytical findings are validated through numerical computations.

Conclusions:

• The study provides a comprehensive framework for comparing the Hellmann potential with screened Coulomb potentials.
• The expanded comparison theorem offers new insights into the energy level ordering of quantum systems.
• This work contributes to a deeper understanding of inter-particle interactions in various physical contexts.