Time evolution of a state. (a missing t)

[naima]

Hi PF there is one thing that i cannot understand here.
Please look at eqn 1068
I try to compute the first term (without $V^\dagger$)
I get something like
$c_f (t) =-i/\hbar exp [i(\omega + \omega_{fi})t/2] \frac{sin(\omega + \omega_{fi})t/2}{(\omega + \omega_{fi})/2} \}$

Unlike eqn 1071 a "t" is lacking under the sine and i cannot write it as a sinc function.

Could you help me? I do not see my mistake.
(I can write my intermediate formulas)

 

[naima]

We begin with a $\int_0^t xxx... dt' term$
With a change of variables t"" = t' -t/2 we get
$\int_{-t/2}^{t/2} exp[i \Omega (t"" + t/2)] dt""$
$= exp[i \Omega t/2]\int_{-t/2}^{t/2} exp[i \Omega t""] dt""$
In the integral the odd i sin function gives no contribution so we have
$= exp[i \Omega t/2]\int_{-t/2}^{t/2}cos (\Omega t"") dt""$
so we finish with a $sin (\Omega t/2) / (\Omega/2)$
But the real formula has a $sin (\Omega t/2) / (\Omega t/2)$
Where does this t come from?

 

[naima]

Sorry I did not see that in the link we had $c_n(t) = it/\hbar ...$
I have no more problem with the formula