  The solutions of the time independent Schrodinger equation for non-Hermitian
(NH) Hamiltonians have been extensively studied and calculated in many
different fields of physics by using L^2 methods that originally have been
developed for the calculations of bound states. The existing non-Hermitian
formalism breaks down when dealing with wavepackets(WP). An open question is
how time dependent expectation values can be calculated when the Hamiltonian is
NH ? Using the F-product formalism, which was recently proposed, [J. Phys.
Chem., 107, 7181 (2003)] we calculate the time dependent expectation values of
different observable quantities for a simple well known study test case model
Hamiltonian. We carry out a comparison between these results with those
obtained from conventional(i.e., Hermitian) quantum mechanics (QM)
calculations. The remarkable agreement between these results emphasizes the
fact that in the NH-QM, unlike standard QM, there is no need to split the
entire space into two regions; i.e., the interaction region and its
surrounding. Our results open a door for a type of WP propagation calculations
within the NH-QM formalism that until now were impossible.
