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Journal of Mathematical Physics

Exact solution of the Schrödinger equation with noncentral parabolic potentials

Asok Guha, S. Mukherjee

Published Apr 1, 1987• 46 citations
Publisher

Abstract

The Schrödinger equation with a class of parabolic potentials has been studied. A constant of motion related to angular momentum has been calculated. It has been shown that the scattering states for these potentials can be defined only in some special cases. The Hartmann potential and the Aharonov–Bohm potential are studied as special cases.

DOI
10.1063/1.527573

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The Schrödinger equation with a class of parabolic potentials has been studied. A constant of motion related to angular momentum has been calculated. It has been shown that the scattering states for these potentials can be defined only in some special cases. The Hartmann potential and the Aharonov–Bohm potential are studied as special cases.
We have studied the path integral solution of a system of particle moving in a certain class of noncentral potential without using the Kustannheimo–Stiefel transformation. The Hamiltonian of the system has been converted to a separable Hamiltonian of Liouville type in parabolic coordinates and has further reduced to a Hamiltonian corresponding to two two-dimensional simple harmonic oscillators. The energy spectrum for this system is calculated analytically. The Hartmann ring-shaped potential and compound Coulomb plus the Aharanov–Bohm potential have also been studied as special cases.
The general solutions of the Schrödinger equation for a non-central potential are obtained by using the Nikiforov–Uvarov method. The Schrödinger equation with general non-central potential is separated into radial and angular parts, and energy eigenvalues and eigenfunctions are derived analytically. By making special selections, the non-central potential is reduced to Coulomb and Hartmann ring-shaped potentials, and the obtained results are compared with the solutions of Coulomb and Hartmann potentials given in the literature.
The general solutions of Schrodinger equation for non central potential are obtained by using Nikiforov Uvarov method. The Schrodinger equation with general non central potential is separated into radial and angular parts and energy eigenvalues and eigenfunctions for these potentials are derived analytically. Non central potential is reduced to Coulomb and Hartmann potential by making special selections, and the obtained solutions are compared with the solutions of Coulomb and Hartmann ring shaped potentials given in literature.
We have studied the path integral solution of a system of particle moving in certain class of non-central potential without using Kustannheimo-Stiefel transformation. The Hamiltonian of the system has been converted to a separable Hamiltonian of Liouville type in parabolic coordinates and has further reduced to a Hamiltonian corresponding to two 2- dimensional simple harmonic oscillators. The energy spectrum for this system is calculated analytically. Hartmann ring-shaped potential and compound Coulomb plus Aharanov- Bohm potential have also been studied as special cases.