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I'm curious about what happen to a system when the configuration of the system changes. If we have a system in a state $|\psi_{\textrm{in}}\rangle$ and we change the configuration of the system, the new state is going to be $U(t)|\psi_{\textrm{in}}\rangle=|\psi_{\textrm{final}}\rangle$, where $U(t)$ is the time evolution operator.

I'm curious about what is this time evolution operator. Where can I find a derivation of this operator? How does this operator work?

Also, if we have $\phi$ a possible state of the system after we change the configuration, I want to know if it's correct to say that the probability is going to be $|\langle\phi|U(t)|\psi_{\textrm{in}}\rangle|^2$ because $U(t)|\psi_{\textrm{in}}\rangle=|\psi_{\textrm{final}}\rangle$?

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One way to look at this is through the Schrodinger's equation:

$i\hbar|\dot\psi(t)\rangle = H|\psi(t)\rangle$

Then a general solution to this equation is:

$|\psi(t)\rangle = e^{-iHt/\hbar} |\psi(0) \rangle$

(Notice that $H$ is an operator here instead of a scalar. $H$ also has to be time-independent, as is usually the case for introductory quantum mechanical problems. But ordinary laws of differentiation works if you expand $e^{-iHt/\hbar}$ term by term. For the sake of intuition, there is no need to worry about mathematical details too much now)

so if you look at this equation you realize that the time evolution operator $U(t) = e^{-iHt/\hbar} $ !! This is sometimes also called a propagator since it propagates a state in time.

The probabilities you wrote are correct.

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    $\begingroup$ The time evolution operator is of that form only if the Hamiltonian does not depend on time. Otherwise it is a Dyson series. Also, it is not exactly the propagator for the wave function. $\endgroup$ – gented Oct 3 '15 at 20:51
  • $\begingroup$ Thanks for pointing out the time-independence condition Gennaro! Maybe our definitions of propagators are a little different... my propagator is applied to a ket vector, while your propagator is applied to wavefunctions (which are projections of kets onto certain bases). $\endgroup$ – Zhengyan Shi Oct 3 '15 at 21:13
  • $\begingroup$ A ket is a wavefunction under the identification of the Hilbert space with an $L^2$-space. The operator $U$ can just be seen as the solution to the Schrödinger equation, i.e. the equation that introduces a dynamics on the Hilbert space. This is the quantum version of the Hamilton equation bringing a dynamics inside the phase space (this point of view is strengthened in the geometric approach to quantum mechanics). $\endgroup$ – Phoenix87 Oct 3 '15 at 22:22

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