Three-Hilbert-Space Formulation of Quantum Mechanics

0329440 - ÚJF 2010 RIV UA eng J - Journal Article
Znojil, Miloslav
Three-Hilbert-Space Formulation of Quantum Mechanics.
Symmetry, Integrability and Geometry: Methods and Applications. Roč. 5, - (2009), 001/1-001/19. ISSN 1815-0659.
[7th Workshop on Quantum Physics with Non-Hermitian Operators. Benasque, 29.06.2007-11.07.2007]
R&D Projects: GA ČR GA202/07/1307; GA MŠMT LC06002
Institutional research plan: CEZ:AV0Z10480505
Keywords : formulation of Quantum Mechanics * cryptohermitian operators of observables * triplet of the representations of the Hilbert space of states
Subject RIV: BE - Theoretical Physics
Impact factor: 0.789, year: 2009

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.
Permanent Link: http://hdl.handle.net/11104/0175477