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Physica Scripta

Schrödinger equation and displacement operator for pseudo-harmonic and cornell potentials

M’hamed Hadj Moussa

Published Dec 1, 2025• 0 citations
Publisher

Abstract

Abstract This work investigates analytical solutions of the Schrödinger equation within the framework of the displacement-operator formalism. Three distinct potential models are considered: a hybrid potential combining the Cornell and pseudo-harmonic forms, V 1 ( x ); the pseudo-harmonic potential, V 2 ( x ); and the squared Cornell potential, V 3 ( x ). The proposed approach enables the derivation of expressions for both the energy spectrum, E n , and the corresponding wave functions, ϕ . In each case, the solutions for ϕ are expressed in terms of the confluent Heun function, specifically HeunC C 1 , C 2 , C 3 , C 4 , C 5 , 1 + γ x . Furthermore, we present graphical illustrations of the positive energy spectrum, E n + , accompanied by a detailed discussion of their physical implications. Finally, the thermodynamic properties associated with these potential models are analyzed and interpreted, highlighting their relevance to various quantum systems.

DOI
10.1088/1402-4896/ae26ed

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