Dear Aleksei Bykov,

Look at Modern Quantum Mechanics by Sakurai ch.1 for discussion on  Infinitesimal evolution operator.

Dear Aleksei Bykov,

Sorry, it is ch. 2.1.  I am not talking about contradiction with your point. I am  asking if I can use the form  the equation 2.1.21  i.e. U(t0+dt,t0)= 1-iH dt/h  in this case or not ?

Debajyoti> if the Hamiltonian is dependent on the time interval

then it is no longer an infinitesimal generator. However, if you want to make some numerical approximation of the evolution, then it makes perfect sense to replace H(t) dt by H(t,Dt) Dt (where Dt is notation for a finite time interval).

Maybe your question have a different meaning within the realm of nonstandard analysis.

Thanks Kåre Olaussen .

Dear Kåre Olaussen,

Thanks for making it clear. Say I have a time evolution operator such that

U(t0+dt,t0) =σz . How can I find out the Hamiltonian in this case? As σz is unitary, we can use it as evolution operator.

Debajyoti> U(t0+dt,t0) =σz

If dt is infinitesimal you cannot find any Hamiltonian, because σz is not close to the identity. If dt is finite (Dt in my notation) you could choose  H = k(1-σz), where k = pi/(2*Dt).

Dear Kåre Olaussen,

But, $ln(0)=2ni\pi$ and $ ln(-1)=(2n+1) i\pi. $So  we can use any value of 'n', right?

Yes, one can (with n integer).