Cartesian Operator Factorization Method for Hydrogen
We generalize Schrödinger’s factorization method for Hydrogen from the conventional separation into angular and radial coordinates to a Cartesian-based factorization. Unique to this approach is the fact that the Hamiltonian is represented as a sum over factorizations in terms of coupled operators th...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-01-01
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| Online Access: | https://www.mdpi.com/2218-2004/10/1/14 |
| _version_ | 1797472813179731968 |
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| author | Xinliang Lyu Christina Daniel James K. Freericks |
| author_facet | Xinliang Lyu Christina Daniel James K. Freericks |
| author_sort | Xinliang Lyu |
| collection | DOAJ |
| description | We generalize Schrödinger’s factorization method for Hydrogen from the conventional separation into angular and radial coordinates to a Cartesian-based factorization. Unique to this approach is the fact that the Hamiltonian is represented as a sum over factorizations in terms of coupled operators that depend on the coordinates and momenta in each Cartesian direction. We determine the eigenstates and energies, the wavefunctions in both coordinate and momentum space, and we also illustrate how this technique can be employed to develop the conventional confluent hypergeometric equation approach. The methodology developed here could potentially be employed for other Hamiltonians that can be represented as the sum over coupled Schrödinger factorizations. |
| first_indexed | 2024-03-09T20:06:29Z |
| format | Article |
| id | doaj.art-1e23e382f3004298ba526f8e2909832b |
| institution | Directory Open Access Journal |
| issn | 2218-2004 |
| language | English |
| last_indexed | 2024-03-09T20:06:29Z |
| publishDate | 2022-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Atoms |
| spelling | doaj.art-1e23e382f3004298ba526f8e2909832b2023-11-24T00:28:08ZengMDPI AGAtoms2218-20042022-01-011011410.3390/atoms10010014Cartesian Operator Factorization Method for HydrogenXinliang Lyu0Christina Daniel1James K. Freericks2College of Civil Engineering, Tongji University, 1239 Siping Road, Shanghai 200092, ChinaDepartment of Physics, Georgetown University, 37th and O Sts. NW, Washington, DC 20057, USADepartment of Physics, Georgetown University, 37th and O Sts. NW, Washington, DC 20057, USAWe generalize Schrödinger’s factorization method for Hydrogen from the conventional separation into angular and radial coordinates to a Cartesian-based factorization. Unique to this approach is the fact that the Hamiltonian is represented as a sum over factorizations in terms of coupled operators that depend on the coordinates and momenta in each Cartesian direction. We determine the eigenstates and energies, the wavefunctions in both coordinate and momentum space, and we also illustrate how this technique can be employed to develop the conventional confluent hypergeometric equation approach. The methodology developed here could potentially be employed for other Hamiltonians that can be represented as the sum over coupled Schrödinger factorizations.https://www.mdpi.com/2218-2004/10/1/14bound states of hydrogenSchrödinger factorization methodreverse Bessel polynomialsseparation of variablesoperator methods |
| spellingShingle | Xinliang Lyu Christina Daniel James K. Freericks Cartesian Operator Factorization Method for Hydrogen Atoms bound states of hydrogen Schrödinger factorization method reverse Bessel polynomials separation of variables operator methods |
| title | Cartesian Operator Factorization Method for Hydrogen |
| title_full | Cartesian Operator Factorization Method for Hydrogen |
| title_fullStr | Cartesian Operator Factorization Method for Hydrogen |
| title_full_unstemmed | Cartesian Operator Factorization Method for Hydrogen |
| title_short | Cartesian Operator Factorization Method for Hydrogen |
| title_sort | cartesian operator factorization method for hydrogen |
| topic | bound states of hydrogen Schrödinger factorization method reverse Bessel polynomials separation of variables operator methods |
| url | https://www.mdpi.com/2218-2004/10/1/14 |
| work_keys_str_mv | AT xinlianglyu cartesianoperatorfactorizationmethodforhydrogen AT christinadaniel cartesianoperatorfactorizationmethodforhydrogen AT jameskfreericks cartesianoperatorfactorizationmethodforhydrogen |