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General solution of quantum mechanical equations of motion with time-dependent Hamiltonians: A Lie algebraic approach
Abstract
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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- Category:
- Articles
- Type:
- artykuł w czasopiśmie wyróżnionym w JCR
- Published in:
-
REPORTS ON MATHEMATICAL PHYSICS
no. 65,
pages 77 - 108,
ISSN: 0034-4877 - Language:
- English
- Publication year:
- 2010
- Bibliographic description:
- 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
- Verified by:
- Gdańsk University of Technology
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