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The European Physical Journal Plus

Approximate solutions of Schrodinger equation and expectation values of Inversely Quadratic Hellmann-Kratzer (IQHK) potential

Chibueze Paul Onyenegecha, Chioma J. Okereke, Ifeanyi Jude Njoku, C. A. Madu, R. U. Ndubuisi, Ugochukwu Kizito Nwajeri

Published Jan 1, 2022• 12 citations
Publisher
DOI
10.1140/epjp/s13360-021-02315-w

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Phase Space Representation for the Inversely Quadratic Hellmann-Kratzer Potential

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Abstract In this work, we develop a phase space formulation of quantum mechanics to investigate the inversely quadratic Hellmann-Kratzer (IQHK) potential, with the aim of simultaneously constructing the Wigner distribution and the corresponding characteristic functions. By employing Weyl transformations, we establish two independent computational approaches, each yielding explicit and generalized analytical expressions for higher-order moments and momentum. In addition, we derive a generalized analytical formulation of the momentum operator and establish a corresponding generalized expression of Heisenberg’s uncertainty principle, explicitly dependent on the quantum numbers n and L. This constitutes a novel and complementary contribution to the study of the IQHK potential. Moreover, our approach enables the identification of a dimensionless principal parameter β in phase-space transformations, which further refines the generalized analytical expressions for the energy levels En,L. These energy levels are numerically evaluated and exhibit excellent agreement with previously published results for all special cases of the IQHK potential across various diatomic molecular systems. Furthermore, we verify that our results are consistent with the canonical Heisenberg-Born-Jordan-Dirac commutation relations and Heisenberg’s uncertainty principle, thereby confirming the robustness and reliability of the proposed methodology within this framework.

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Abstract: In this study, the Schrödinger equation was solved with a superposition of the Hulthen potential and generalized inverse quadratic Yukawa potential model using the Nikiforov-Uvarov (NU) method. For completeness, we also calculated the wave function. To validate our results, the numerical bound state energy eigenvalues was computed for various principal n and angular momentum l quantum numbers. With the aid of the Hellmann-Feynman theorem, the expressions for the expectation values of the square of inverse of position, r^(-2), inverse of position, r^(-1), kinetic energy, T ̂ and square of momentum, p ̂ are calculated. By adjusting the potential parameters, special cases of the potential were considered, resulting in Generalized Inverse Quadratic Yukawa potential, Hulthen potential, Coulomb potential, Kratzer potential, Inversely Quadratic Yukawa potential and Coulomb plus inverse square potential, respectively. Their energy eigenvalue expressions and numerical computations agreed with the literature. Keywords: Schrödinger equation, Hulthen potential (HP), Generalized inverse quadratic Yukawa potential (GIQYP), Nikiforov-Uvarov method. PACS: 03.65.−w, 03.65.Ca, 03.65.Ge.

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

Fatma Zohra Khaled, Mustafa Moumni, Mokhtar Falek

We investigate the bound state properties of the hydrogen-like atoms in the radial screened Coulomb potential (RSCP). using three complementary analytical approaches - expectation values with Coulomb and Kratzer reference states, variational optimization with a scaled Kratzer basis, and the Hellmann-Feynman theorem - we derive approximate energy eigenvalues as function of the screening parameter c. Benchmarked against high-precision generalized pseudospectral data, the expectation value-approach with the Kratzer basis achieves relative errors of 0.63% for the first ten s-states at $c=0.1$, while the variational method improves this further. The formalism extends naturally to Positronium, demonstrating its generality for arbitrary reduced-mass systems. The complementary biases of the methods provide robust error estimation for plasma-embedded atoms.

quant-pharXiv:2607.19197•Jul 22, 2026
Abstract In this work, we develop a phase space formulation of quantum mechanics to investigate the inversely quadratic Hellmann-Kratzer (IQHK) potential, with the aim of simultaneously constructing the Wigner distribution and the corresponding characteristic functions. By employing Weyl transformations, we establish two independent computational approaches, each yielding explicit and generalized analytical expressions for higher-order moments and momentum. In addition, we derive a generalized analytical formulation of the momentum operator and establish a corresponding generalized expression of Heisenberg’s uncertainty principle, explicitly dependent on the quantum numbers n and L. This constitutes a novel and complementary contribution to the study of the IQHK potential. Moreover, our approach enables the identification of a dimensionless principal parameter β in phase-space transformations, which further refines the generalized analytical expressions for the energy levels En,L. These energy levels are numerically evaluated and exhibit excellent agreement with previously published results for all special cases of the IQHK potential across various diatomic molecular systems. Furthermore, we verify that our results are consistent with the canonical Heisenberg-Born-Jordan-Dirac commutation relations and Heisenberg’s uncertainty principle, thereby confirming the robustness and reliability of the proposed methodology within this framework.
Abstract: In this study, the Schrödinger equation was solved with a superposition of the Hulthen potential and generalized inverse quadratic Yukawa potential model using the Nikiforov-Uvarov (NU) method. For completeness, we also calculated the wave function. To validate our results, the numerical bound state energy eigenvalues was computed for various principal n and angular momentum l quantum numbers. With the aid of the Hellmann-Feynman theorem, the expressions for the expectation values of the square of inverse of position, r^(-2), inverse of position, r^(-1), kinetic energy, T ̂ and square of momentum, p ̂ are calculated. By adjusting the potential parameters, special cases of the potential were considered, resulting in Generalized Inverse Quadratic Yukawa potential, Hulthen potential, Coulomb potential, Kratzer potential, Inversely Quadratic Yukawa potential and Coulomb plus inverse square potential, respectively. Their energy eigenvalue expressions and numerical computations agreed with the literature. Keywords: Schrödinger equation, Hulthen potential (HP), Generalized inverse quadratic Yukawa potential (GIQYP), Nikiforov-Uvarov method. PACS: 03.65.−w, 03.65.Ca, 03.65.Ge.
We investigate the bound state properties of the hydrogen-like atoms in the radial screened Coulomb potential (RSCP). using three complementary analytical approaches - expectation values with Coulomb and Kratzer reference states, variational optimization with a scaled Kratzer basis, and the Hellmann-Feynman theorem - we derive approximate energy eigenvalues as function of the screening parameter c. Benchmarked against high-precision generalized pseudospectral data, the expectation value-approach with the Kratzer basis achieves relative errors of 0.63% for the first ten s-states at c=0.1c=0.1c=0.1, while the variational method improves this further. The formalism extends naturally to Positronium, demonstrating its generality for arbitrary reduced-mass systems. The complementary biases of the methods provide robust error estimation for plasma-embedded atoms.