Schrödinger equation and displacement operator for pseudo-harmonic and cornell potentials
M’hamed Hadj Moussa
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.
Physica Scripta•Dec 1, 2025•0 citations