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I was looking into the progression of quantum states with respect to time. From what I understood the progression of a state ## \left|\psi(t)\right> ## is given by:

$$ \left|\psi(t)\right> = U(t)\left|\psi(0)\right> $$

I'm not sure if that's right. But it's okay if I haven't got that yet.. That was just to give some context to my actual question.

What is this representation of the linear operator ## U(t) ## at ## t = \epsilon ##, where ## \epsilon ## is an infinitesimal?

$$ U(\epsilon) = I - i\epsilon H $$

Where ## i ## is the imaginary unit, ## I ## is the identity matrix and I think ## H ## is the hamiltonian.

It also apparently has more terms of order ## \epsilon^2 ## and so on. What "series" is this? Is it some first order approximation of ## U(t) ##? What should I look into to understand where those terms are coming from?

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# Representation of linear operator using series ?

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