Dunkl-Schrodinger Equation in Higher Dimension
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912542521032704 |
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| author | Hamil, B. Lütfüoğlu, B. C. Merad, M. |
| author_facet | Hamil, B. Lütfüoğlu, B. C. Merad, M. |
| contents | This paper presents analytical solutions for eigenvalues and eigenfunctions of the Schrödinger equation in higher dimensions, incorporating the Dunkl operator. Two fundamental quantum mechanical problems are examined in their exact forms: the d-dimensional harmonic oscillator and the Coulomb potential. In order to obtain analytical solutions to these problems, both Cartesian and polar coordinate systems were employed. Firstly, the Dunkl-Schrödinger equation is derived in d-dimensional Cartesian coordinates, and then for the isotropic harmonic potential interaction, its solutions are given. Subsequently, using polar coordinates the angular and radial parts of the Dunkl-Schrödinger equation are obtained. It is demonstrated that the system permits the separation of variables in both coordinate systems, with the resulting separated solutions expressed through Laguerre and Jacobi polynomials. Then, the radial Dunkl-Schrödinger equation is solved using the isotropic harmonic, pseudoharmonic, and Coulomb potentials. The eigenstates and eigenvalues are obtained for each case and the behavior of the energy eigenvalue functions are illustrated graphically with the reduced probability densities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12653 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dunkl-Schrodinger Equation in Higher Dimension Hamil, B. Lütfüoğlu, B. C. Merad, M. Quantum Physics Mathematical Physics This paper presents analytical solutions for eigenvalues and eigenfunctions of the Schrödinger equation in higher dimensions, incorporating the Dunkl operator. Two fundamental quantum mechanical problems are examined in their exact forms: the d-dimensional harmonic oscillator and the Coulomb potential. In order to obtain analytical solutions to these problems, both Cartesian and polar coordinate systems were employed. Firstly, the Dunkl-Schrödinger equation is derived in d-dimensional Cartesian coordinates, and then for the isotropic harmonic potential interaction, its solutions are given. Subsequently, using polar coordinates the angular and radial parts of the Dunkl-Schrödinger equation are obtained. It is demonstrated that the system permits the separation of variables in both coordinate systems, with the resulting separated solutions expressed through Laguerre and Jacobi polynomials. Then, the radial Dunkl-Schrödinger equation is solved using the isotropic harmonic, pseudoharmonic, and Coulomb potentials. The eigenstates and eigenvalues are obtained for each case and the behavior of the energy eigenvalue functions are illustrated graphically with the reduced probability densities. |
| title | Dunkl-Schrodinger Equation in Higher Dimension |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2409.12653 |