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Three-Hilbert-Space Formulation of Quantum Mechanics
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2009, том 5
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Three-Hilbert-Space Formulation of Quantum Mechanics
Інший ID:
2000 Mathematics Subject Classification: 81Q10; 47B50
Посилання:
Three-Hilbert-Space Formulation of Quantum Mechanics / M. Znojil // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 25 назв. — англ.
Підтримка:
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.
Дата:
2009
Переглядів:
477
Завантажень:
256
Анотація:
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.
