The dynamical evolution of geometric uncertainty principle for spin 1/2 system
Geometric Quantum Mechanics is a formulation that demonstrates how quantum theory may be casted in the language of Hamiltonian phase-space dynamics. In this framework, the states are referring to points in complex projective Hilbert space, the observables are real valued functions on the space and t...
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| Format: | Article |
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Union of Researchers of Macedonia
2021
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| _version_ | 1825937564208463872 |
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| author | Umair, H. Zainuddin, H. Chan, K. T. Said Husain, Sh. K. |
| author_facet | Umair, H. Zainuddin, H. Chan, K. T. Said Husain, Sh. K. |
| author_sort | Umair, H. |
| collection | UPM |
| description | Geometric Quantum Mechanics is a formulation that demonstrates how quantum theory may be casted in the language of Hamiltonian phase-space dynamics. In this framework, the states are referring to points in complex projective Hilbert space, the observables are real valued functions on the space and the Hamiltonian flow is defined by Schr{\"o}dinger equation. Recently, the effort to cast uncertainty principle in terms of geometrical language appeared to become the subject of intense study in geometric quantum mechanics. One has shown that the stronger version of uncertainty relation i.e. the Robertson-Schr{\"o}dinger uncertainty relation can be expressed in terms of the symplectic form and Riemannian metric. In this paper, we investigate the dynamical behavior of the uncertainty relation for spin $\frac{1}{2}$ system based on this formulation. We show that the Robertson-Schr{\"o}dinger uncertainty principle is not invariant under Hamiltonian flow. This is due to the fact that during evolution process, unlike symplectic area, the Riemannian metric is not invariant under the flow. |
| first_indexed | 2024-03-06T11:01:25Z |
| format | Article |
| id | upm.eprints-95375 |
| institution | Universiti Putra Malaysia |
| last_indexed | 2024-03-06T11:01:25Z |
| publishDate | 2021 |
| publisher | Union of Researchers of Macedonia |
| record_format | dspace |
| spelling | upm.eprints-953752023-04-12T04:40:15Z http://psasir.upm.edu.my/id/eprint/95375/ The dynamical evolution of geometric uncertainty principle for spin 1/2 system Umair, H. Zainuddin, H. Chan, K. T. Said Husain, Sh. K. Geometric Quantum Mechanics is a formulation that demonstrates how quantum theory may be casted in the language of Hamiltonian phase-space dynamics. In this framework, the states are referring to points in complex projective Hilbert space, the observables are real valued functions on the space and the Hamiltonian flow is defined by Schr{\"o}dinger equation. Recently, the effort to cast uncertainty principle in terms of geometrical language appeared to become the subject of intense study in geometric quantum mechanics. One has shown that the stronger version of uncertainty relation i.e. the Robertson-Schr{\"o}dinger uncertainty relation can be expressed in terms of the symplectic form and Riemannian metric. In this paper, we investigate the dynamical behavior of the uncertainty relation for spin $\frac{1}{2}$ system based on this formulation. We show that the Robertson-Schr{\"o}dinger uncertainty principle is not invariant under Hamiltonian flow. This is due to the fact that during evolution process, unlike symplectic area, the Riemannian metric is not invariant under the flow. Union of Researchers of Macedonia 2021-09-30 Article PeerReviewed Umair, H. and Zainuddin, H. and Chan, K. T. and Said Husain, Sh. K. (2021) The dynamical evolution of geometric uncertainty principle for spin 1/2 system. Advances in Mathematics: Scientific Journal, 10 (9). 3241 - 3251. ISSN 1857-8365; ESSN: 1857-8438 https://www.research-publication.com/amsj/all-issues/vol-10/iss-09 10.37418/amsj.10.9.13 |
| spellingShingle | Umair, H. Zainuddin, H. Chan, K. T. Said Husain, Sh. K. The dynamical evolution of geometric uncertainty principle for spin 1/2 system |
| title | The dynamical evolution of geometric uncertainty principle for spin 1/2 system |
| title_full | The dynamical evolution of geometric uncertainty principle for spin 1/2 system |
| title_fullStr | The dynamical evolution of geometric uncertainty principle for spin 1/2 system |
| title_full_unstemmed | The dynamical evolution of geometric uncertainty principle for spin 1/2 system |
| title_short | The dynamical evolution of geometric uncertainty principle for spin 1/2 system |
| title_sort | dynamical evolution of geometric uncertainty principle for spin 1 2 system |
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