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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 $U(t)|\psi_{\textrm{in}}\rangle=|\psi_{\textrm{final}}\rangle$?

Qmechanic
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WernerSpin
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1 Answers1

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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.

Zhengyan Shi
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