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International Journal of the Physical Sciences

Exact solutions of N-Dimensional Schrdinger equation for a potential containing coulomb and quadratic terms

HASSAN HASSANABADI, M. Hamzavi, S. Zarrinkamar, ALI AKBAR RAJABI

Published Feb 4, 2011• 9 citations
Publisher

Abstract

We encounter Coulomb and quadratic terms in the Hamiltonian relation in many branches of physics. Here we consider the N-dimensional space Schrodinger equation in the presence of Coulomb and quadratic terms and analytically calculate the eigenfunctions and the corresponding eigenenergies.   Key words: Schrodinger equation, N-dimension, Coulomb and quadratic potentials. PACS numbers: 02.30Em, 03.65.Fd and 03.65.Ge.

DOI
10.5897/ijps10.555

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We encounter Coulomb and quadratic terms in the Hamiltonian relation in many branches of physics. Here we consider the N-dimensional space Schrodinger equation in the presence of Coulomb and quadratic terms and analytically calculate the eigenfunctions and the corresponding eigenenergies.
 

Key words: Schrodinger equation, N-dimension, Coulomb and quadratic potentials.

PACS numbers: 02.30Em, 03.65.Fd and 03.65.Ge.
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.
Exactly solvable models play an extremely important role in many fields of quantum physics. In this study, the Schrödinger equation is applied for a solution of a two--dimensional (2D) problem for two particles interacting via Kratzer, and modified Kratzer potentials. We found the exact bound state solutions of the two--dimensional Schrödinger equation with Kratzer--type potentials and present analytical expressions for the eigenvalues and eigenfunctions. The eigenfunctions are given in terms of the associated Laguerre polynomials.