Let $|\psi\rangle$ be an element of a Hilbert space $\mathcal{H}$ and $U$ a unitary operator on $\mathcal{H}$. I am concerned with the actual physical manifestation of such a unitary operator in the sense portrayed in the following parable.

I am in an inertial reference frame, and $|\psi\rangle$ is a quantum system at rest relative to me. Thus, I make the natural choice to parametrize the time evolution of $|\psi\rangle$ with my proper time. Now, I am given an apparatus that takes as input a quantum state like $|\psi\rangle$ and outputs $U|\psi\rangle$. Being the good note-taking scientist that I am, I write in my notebook that at time $t_0$ (I glanced at my watch), the state of the quantum system before me is $|\psi\rangle$. I then, at time $t_{\text{in}} > t_0$, feed the state $|\psi\rangle$ into the apparatus. At time $t_{\text{out}}$, the output state, $U|\psi\rangle$, emerges. Finally, knowing that the apparatus performs a $U$ operation, I write that the final state is $U|\psi\rangle$ at time $t_f > t_{\text{out}}$. My question is if $t_{\text{out}} > t_{\text{in}}$ necessarily? In other words, do actual unitary operations on quantum systems ever happen instantaneously? Is this even possible to answer?

If indeed $t_{\text{out}} > t_{\text{in}}$ for all $U$, does this imply that all unitary operations that actually happen in nature manifest from a Hamiltonian (or set of Hamiltonians) that generates an appropriate time evolution operator and implements the desired unitary $U$?


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