Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System
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 conve...
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| Format: | Article |
| Language: | English |
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Wiley
2016-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2016/3693572 |
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| author | Salim Medjber Hacene Bekkar Salah Menouar Jeong Ryeol Choi |
| author_facet | Salim Medjber Hacene Bekkar Salah Menouar Jeong Ryeol Choi |
| author_sort | Salim Medjber |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-78b5f594b8ee4d27bbb945a274537b92 |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-78b5f594b8ee4d27bbb945a274537b922025-02-03T06:44:02ZengWileyAdvances in Mathematical Physics1687-91201687-91392016-01-01201610.1155/2016/36935723693572Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential SystemSalim Medjber0Hacene Bekkar1Salah Menouar2Jeong Ryeol Choi3Department of Material Sciences, Faculty of Science, University of M’sila, 28000 M’sila, AlgeriaFaculty of Technology, University of Ferhat Abbas Sétif-1, 19000 Sétif, AlgeriaLaboratory of Optoelectronics and Compounds (LOC), Department of Physics, Faculty of Science, University of Ferhat Abbas Sétif-1, 19000 Sétif, AlgeriaDepartment of Radiologic Technology, Daegu Health College, Daegu 41453, Republic of KoreaThe 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.http://dx.doi.org/10.1155/2016/3693572 |
| spellingShingle | Salim Medjber Hacene Bekkar Salah Menouar Jeong Ryeol Choi Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System Advances in Mathematical Physics |
| title | Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System |
| title_full | Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System |
| title_fullStr | Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System |
| title_full_unstemmed | Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System |
| title_short | Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System |
| title_sort | quantization of a 3d nonstationary harmonic plus an inverse harmonic potential system |
| url | http://dx.doi.org/10.1155/2016/3693572 |
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