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## Homework Statement

i know that

[itex] H |n> = \varepsilon_n |n>[/itex]

[itex] H |i> = \varepsilon_i |i>[/itex]

and i want to estimate

[itex] < n | e^{i H t / \hbar} V(t) e^{-i H t / \hbar} |i> [/itex]

## Homework Equations

[itex] <\psi | \varphi> = \int \psi^*(q) \varphi(q) dq [/itex]

## The Attempt at a Solution

i don't understand well how the operator interacts with the wave function.... i'm pretty sure that the right solution is

[itex] < n | e^{i H t / \hbar} V(t) e^{-i H t / \hbar} |i> = \int{ \left(n e^{- i H t / \hbar}\right)^* V e^{-i \varepsilon_i t / \hbar}} i = \int n^* e^{i \varepsilon_n t / \hbar} V e^{-i \varepsilon_f t / \hbar} i= <n | V(t) | i> e^{i \frac{\varepsilon_n - \varepsilon_i}{\hbar}t} [/itex]

but i'm not sure about "why is this the right way to do this" ... for example i dont know why [itex] H |n> = \varepsilon_n |n> [/itex] implies [itex] e^{iHt / \hbar} |n> = e^{i \varepsilon_n t / \hbar } |n> [/itex]... or why this way to do is wrong:

[itex] < n | e^{i H t / \hbar} V(t) e^{-i H t / \hbar} |i> = \int \left(n e^{i H t / \hbar}\right)^* e^{i H t / \hbar} V(t) i = \int n^* V(t) i = <n|V(t)|i> [/itex]

(i didn't put the "dq" in the integral because i dont know in wich variable i should integrate)

if someone can explain me why this is the right way to do this expression i'll be happy.

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