Multicomplex wave functions for linear and nonlinear Schrödinger equations
We consider a multicomplex Schrödinger equation with general scalar potential, a generalization of both the standard Schrödinger equation and the bicomplex Schrödinger equation of Rochon and Tremblay, for wave functions mapping onto (Formula presented.). We determine the equivalent real-valued syste...
| Main Authors: | , |
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| Format: | Journal article |
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Springer
2016
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| _version_ | 1797078734271938560 |
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| author | Theaker, K van Gorder, R |
| author_facet | Theaker, K van Gorder, R |
| author_sort | Theaker, K |
| collection | OXFORD |
| description | We consider a multicomplex Schrödinger equation with general scalar potential, a generalization of both the standard Schrödinger equation and the bicomplex Schrödinger equation of Rochon and Tremblay, for wave functions mapping onto (Formula presented.). We determine the equivalent real-valued system in recursive form, and derive the relevant continuity equations in order to demonstrate that conservation of probability (a hallmark of standard quantum mechanics) holds in the multicomplex generalization. From here, we obtain the real modulus and demonstrate the generalized multicomplex version of Born’s formula for the probability densities. We then turn our attention to possible generalizations of the multicomplex Schrödinger equation, such as the case where the scalar potential is replaced with a multicomplex-valued potential, or the case where the potential involves the real modulus of the wave function, resulting in a multicomplex nonlinear Schrödinger equation. Finally, in order to demonstrate the solution methods for such equations, we obtain several particular solutions to the multicomplex Schrödinger equation. We interpret the generalized results in the context of the standard results from quantum mechanics. |
| first_indexed | 2024-03-07T00:35:59Z |
| format | Journal article |
| id | oxford-uuid:816deaf4-a4ca-4c32-b188-b28735492c12 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T00:35:59Z |
| publishDate | 2016 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:816deaf4-a4ca-4c32-b188-b28735492c122022-03-26T21:30:09ZMulticomplex wave functions for linear and nonlinear Schrödinger equationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:816deaf4-a4ca-4c32-b188-b28735492c12Symplectic Elements at OxfordSpringer2016Theaker, Kvan Gorder, RWe consider a multicomplex Schrödinger equation with general scalar potential, a generalization of both the standard Schrödinger equation and the bicomplex Schrödinger equation of Rochon and Tremblay, for wave functions mapping onto (Formula presented.). We determine the equivalent real-valued system in recursive form, and derive the relevant continuity equations in order to demonstrate that conservation of probability (a hallmark of standard quantum mechanics) holds in the multicomplex generalization. From here, we obtain the real modulus and demonstrate the generalized multicomplex version of Born’s formula for the probability densities. We then turn our attention to possible generalizations of the multicomplex Schrödinger equation, such as the case where the scalar potential is replaced with a multicomplex-valued potential, or the case where the potential involves the real modulus of the wave function, resulting in a multicomplex nonlinear Schrödinger equation. Finally, in order to demonstrate the solution methods for such equations, we obtain several particular solutions to the multicomplex Schrödinger equation. We interpret the generalized results in the context of the standard results from quantum mechanics. |
| spellingShingle | Theaker, K van Gorder, R Multicomplex wave functions for linear and nonlinear Schrödinger equations |
| title | Multicomplex wave functions for linear and nonlinear Schrödinger equations |
| title_full | Multicomplex wave functions for linear and nonlinear Schrödinger equations |
| title_fullStr | Multicomplex wave functions for linear and nonlinear Schrödinger equations |
| title_full_unstemmed | Multicomplex wave functions for linear and nonlinear Schrödinger equations |
| title_short | Multicomplex wave functions for linear and nonlinear Schrödinger equations |
| title_sort | multicomplex wave functions for linear and nonlinear schrodinger equations |
| work_keys_str_mv | AT theakerk multicomplexwavefunctionsforlinearandnonlinearschrodingerequations AT vangorderr multicomplexwavefunctionsforlinearandnonlinearschrodingerequations |