Title 



General solutions of quantum mechanical equations of motion with timedependent Hamiltonians: a Lie algebraic approach
 
Author 



 
Abstract 



The unitary operators U(t), describing the quantum time evolution of systems with a timedependent Hamiltonian, can be constructed in an explicit manner using the method of timedependent invariants. We clarify the role of Liealgebraic 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 WeiNorman 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 timedependent frequency. Even in these wellknown examples some new results are obtained.   
Language 



English
 
Source (journal) 



Reports on mathematical physics  
Publication 



2010
 
Volume/pages 



65(2010), p. 77107
 
ISI 



000275029600004
 
Full text (Publisher's DOI) 


  
