General solution of quantum mechanical equations of motion with time-dependent Hamiltonians: A Lie algebraic approach - Publikacja - MOST Wiedzy

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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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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

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