 
Summary: Parameter differentiation and quantum state decomposition for time varying
Schršodinger equations
Claudio Altafini
SISSAISAS
International School for Advanced Studies
via Beirut 24, 34014 Trieste, Italy
For the unitary operator, solution of the Schršodinger equation corresponding to a timevarying
Hamiltonian, the relation between the Magnus and the product of exponentials expansions can be
expressed in terms of a system of first order differential equations in the parameters of the two
expansions. A method is proposed to compute such differential equations explicitly and in a closed
form.
I. INTRODUCTION
For energy preserving finite dimensional closed quantum systems, the time evolution is wellknown to
be completely trivial. In fact, the linear and hermitian hamiltonian H admits the spectral decomposition
H =
N
i=1 ii i with i the energy eigenvalues corresponding to the stationary eigenstates i , and
the solution of the associated Schršodinger equation in the i(t) is simply obtained by adding a phase
factor proportional to the corresponding energy eigenvalue: i(t) = exp (i it/ ) i . Even without
resorting to the spectral resolution and to the change of basis that diagonalizes the Hamiltonian, the time
