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arXiv:quant-ph/0702141 · Open Physics

Exact solutions of the radial Schrödinger equation for some physical potentials

Sameer Ikhdair, Ramazan Sever

Published Jan 1, 2007• 49 citations
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Abstract

Abstract By using an ansatz for the eigenfunction, we have obtained the exact analytical solutions of the radial Schrödinger equation for the pseudoharmonic and the Kratzer potentials in two dimensions. The bound-state solutions are easily calculated from this eigenfunction ansatz. The corresponding normalized wavefunctions are also obtained.

DOI
10.2478/s11534-007-0022-9

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Abstract By using an ansatz for the eigenfunction, we have obtained the exact analytical solutions of the radial Schrödinger equation for the pseudoharmonic and the Kratzer potentials in two dimensions. The bound-state solutions are easily calculated from this eigenfunction ansatz. The corresponding normalized wavefunctions are also obtained.
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We consider the radial Schrödinger equation with the pseudo-Gaussian potential. By making an ansatz to the solution of the eigenvalue equation for the associate Hamiltonian, we arrive at the general exact eigenfunction. The values of energy levels for the bound states are calculated along with their corresponding normalized wave-functions. The case of positive energy levels, known as meta-stable states, is also discussed and the magnitude of transmission coefficient through the potential barrier is evaluated.
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.