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Summary: Parameter differentiation and quantum state decomposition for time varying
Schršodinger equations
Claudio Altafini
SISSA-ISAS
International School for Advanced Studies
via Beirut 2-4, 34014 Trieste, Italy
For the unitary operator, solution of the Schršodinger equation corresponding to a time-varying
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 well-known to
be completely trivial. In fact, the linear and hermitian hamiltonian H admits the spectral decomposition
H =
N
i=1 i|i 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
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