Three-Hilbert-Space Formulation of Quantum Mechanics
In paper [Znojil M., Phys. Rev. D 78 (2008), 085003, 5 pages, arXiv:0809.2874] the two-Hilbert-space (2HS, a.k.a. cryptohermitian) formulation of Quantum Mechanics has been revisited. In the present continuation of this study (with the spaces in question denoted as H(auxiliary) and H(standard)) we s...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2009 |
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2009
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/149281 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Three-Hilbert-Space Formulation of Quantum Mechanics / M. Znojil // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 25 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-149281 |
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Znojil, M. 2019-02-19T19:46:57Z 2019-02-19T19:46:57Z 2009 Three-Hilbert-Space Formulation of Quantum Mechanics / M. Znojil // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 25 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 81Q10; 47B50 https://nasplib.isofts.kiev.ua/handle/123456789/149281 In paper [Znojil M., Phys. Rev. D 78 (2008), 085003, 5 pages, arXiv:0809.2874] the two-Hilbert-space (2HS, a.k.a. cryptohermitian) formulation of Quantum Mechanics has been revisited. In the present continuation of this study (with the spaces in question denoted as H(auxiliary) and H(standard)) we spot a weak point of the 2HS formalism which lies in the double role played by H(auxiliary). As long as this confluence of roles may (and did!) lead to confusion in the literature, we propose an amended, three-Hilbert-space (3HS) reformulation of the same theory. As a byproduct of our analysis of the formalism we offer an amendment of the Dirac's bra-ket notation and we also show how its use clarifies the concept of covariance in time-dependent cases. Via an elementary example we finally explain why in certain quantum systems the generator H(gen) of the time-evolution of the wave functions may differ from their Hamiltonian H. This paper is a contribution to the Proceedings of the VIIth Workshop “Quantum Physics with NonHermitian Operators” (June 29 – July 11, 2008, Benasque, Spain). In various stages of development the work has been supported by Institutional Research Plan AV0Z10480505, by the MSMT “Doppler Institute” project Nr. LC06002, by GA ˇ CR, grant Nr. 202/07/1307 and by the hospitality of Universidad de Santiago de Chile. Last but not least, three anonymous referees should be acknowledged for their constructive commentaries. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Three-Hilbert-Space Formulation of Quantum Mechanics Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Three-Hilbert-Space Formulation of Quantum Mechanics |
| spellingShingle |
Three-Hilbert-Space Formulation of Quantum Mechanics Znojil, M. |
| title_short |
Three-Hilbert-Space Formulation of Quantum Mechanics |
| title_full |
Three-Hilbert-Space Formulation of Quantum Mechanics |
| title_fullStr |
Three-Hilbert-Space Formulation of Quantum Mechanics |
| title_full_unstemmed |
Three-Hilbert-Space Formulation of Quantum Mechanics |
| title_sort |
three-hilbert-space formulation of quantum mechanics |
| author |
Znojil, M. |
| author_facet |
Znojil, M. |
| publishDate |
2009 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
In paper [Znojil M., Phys. Rev. D 78 (2008), 085003, 5 pages, arXiv:0809.2874] the two-Hilbert-space (2HS, a.k.a. cryptohermitian) formulation of Quantum Mechanics has been revisited. In the present continuation of this study (with the spaces in question denoted as H(auxiliary) and H(standard)) we spot a weak point of the 2HS formalism which lies in the double role played by H(auxiliary). As long as this confluence of roles may (and did!) lead to confusion in the literature, we propose an amended, three-Hilbert-space (3HS) reformulation of the same theory. As a byproduct of our analysis of the formalism we offer an amendment of the Dirac's bra-ket notation and we also show how its use clarifies the concept of covariance in time-dependent cases. Via an elementary example we finally explain why in certain quantum systems the generator H(gen) of the time-evolution of the wave functions may differ from their Hamiltonian H.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/149281 |
| citation_txt |
Three-Hilbert-Space Formulation of Quantum Mechanics / M. Znojil // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 25 назв. — англ. |
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