Title
General solutions of quantum mechanical equations of motion with time-dependent Hamiltonians: a Lie algebraic approach General solutions of quantum mechanical equations of motion with time-dependent Hamiltonians: a Lie algebraic approach
Author
Faculty/Department
Faculty of Sciences. Physics
Publication type
article
Publication
Subject
Mathematics
Physics
Source (journal)
Reports on mathematical physics
Volume/pages
65(2010) , p. 77-107
ISSN
0034-4877
ISI
000275029600004
Carrier
E
Target language
English (eng)
Full text (Publishers DOI)
Affiliation
University of Antwerp
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 of Euler angles. Progress can be made if one succeeds 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 a 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 the harmonic oscillator with time-dependent frequency. Even in these well-known examples some new results are obtained.
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