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General solution of quantum mechanical equations of motion with time-dependent Hamiltonians: A Lie algebraic approach
Abstrakt
The unitary operators U(t), describing the quantum time evolution of systems with a time-dependent Hamiltonian, can be constructed in an explicit manner using the method of time-dependent invariants. We clarify the role of Lie-algebraic techniques in this context and elaborate the theory for SU(2) and SU(1,1). In these cases we give explicit formulae for obtaining general solutions from special ones. We show that the constructions known as Magnus expansion and Wei-Norman expansion correspond with different representations of the rotation group. A simpler construction is obtained when representing rotations in terms Euler angles.Progress can be made if one succeds in finding a nontrivial special solution of the equations of motion. Then the general solution can be derived by means of the Lie theory. The problem of evaluating the evolution of the system is translated from noncommutative integration in the sense of Dyson into an ordinary commutative integration.The two main applications of our method are reviewed, namely the Bloch equations and harmonic oscllator with time-dependent frequency. Even in these well-known examples some new results are otained.
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- Wersja publikacji
- Accepted albo Published Version
- DOI:
- Cyfrowy identyfikator dokumentu elektronicznego (otwiera się w nowej karcie) 10.1016/S0034-4877(10)80007-2
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Informacje szczegółowe
- Kategoria:
- Publikacja w czasopiśmie
- Typ:
- artykuł w czasopiśmie wyróżnionym w JCR
- Opublikowano w:
-
REPORTS ON MATHEMATICAL PHYSICS
nr 65,
strony 77 - 108,
ISSN: 0034-4877 - Język:
- angielski
- Rok wydania:
- 2010
- Opis bibliograficzny:
- Kuna M., Naudts J.: General solution of quantum mechanical equations of motion with time-dependent Hamiltonians: A Lie algebraic approach// REPORTS ON MATHEMATICAL PHYSICS. -Vol. 65, nr. Iss. 1 (2010), s.77-108
- Weryfikacja:
- Politechnika Gdańska
wyświetlono 117 razy