Let's consider, I have an equation U(t2-t1) χ= σzχ. Here U represents the time evolution operator and χ is the spin state. Now if take log of both side, left side gives the Hamiltonian while the right side is not defined (log(-1) is not allowed). What can I do in this case to find the Hamiltonian?
Dear A. Amit Vikram ,
σz is also unitary. So it can be a time evolution operator. Consider a situation where the spin state evolves such a way that after every Delta t=t2-t1 it changes to a state from 'A' to 'σzA'. Hence it is realistic.
From
U(Delta t) = \sigma_z
we get
U(2 Delta t) = 1
U(3 Delta t) = \sigma_z
U(4 Delta t) = 1 ...
is this what you had in mind ? Hopefully not, since nobody in his right mind would think up such a blunder.
I'm sure that your task, whatever it is, asks for considering
U(t1-t2) = exp(-i(t1-t2) \sigma_z)
which, due to the fact that \sigma_z is a diagonal matrix, really simple and should not be outside your reach.
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