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Advances in Mathematical Physics
Volume 2016 (2016), Article ID 3693572, 6 pages
Research Article

Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System

1Department of Material Sciences, Faculty of Science, University of M’sila, 28000 M’sila, Algeria
2Faculty of Technology, University of Ferhat Abbas Sétif-1, 19000 Sétif, Algeria
3Laboratory of Optoelectronics and Compounds (LOC), Department of Physics, Faculty of Science, University of Ferhat Abbas Sétif-1, 19000 Sétif, Algeria
4Department of Radiologic Technology, Daegu Health College, Daegu 41453, Republic of Korea

Received 27 October 2015; Accepted 28 March 2016

Academic Editor: Yao-Zhong Zhang

Copyright © 2016 Salim Medjber et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The Schrödinger solutions for a three-dimensional central potential system whose Hamiltonian is composed of a time-dependent harmonic plus an inverse harmonic potential are investigated. Because of the time-dependence of parameters, we cannot solve the Schrödinger solutions relying only on the conventional method of separation of variables. To overcome this difficulty, special mathematical methods, which are the invariant operator method, the unitary transformation method, and the Nikiforov-Uvarov method, are used when we derive solutions of the Schrödinger equation for the system. In particular, the Nikiforov-Uvarov method with an appropriate coordinate transformation enabled us to reduce the eigenvalue equation of the invariant operator, which is a second-order differential equation, to a hypergeometric-type equation that is convenient to treat. Through this procedure, we derived exact Schrödinger solutions (wave functions) of the system. It is confirmed that the wave functions are represented in terms of time-dependent radial functions, spherical harmonics, and general time-varying global phases. Such wave functions are useful for studying various quantum properties of the system. As an example, the uncertainty relations for position and momentum are derived by taking advantage of the wave functions.