If anyone can help me, I'm very grateful.
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The passage \(\displaystyle U ^{\dagger}(t,0)\) to exponential: n  1 came from comutation \(\displaystyle [a, a{^\dagger}] = 1\) or applying the in \(\displaystyle {\psi_n}>\) with n became n  1?A few quick question for you and about the notation.
1) \(\displaystyle  \psi _n >\) is the oscillator state with n oscillators, correct?
2) That being the case, do you know that \(\displaystyle a  \psi _n > = \sqrt{n} ~  \psi _{n  1} >\) and \(\displaystyle a^{\dagger}  \psi _{n + 1} > = \sqrt{n + 1} ~  \psi _{n + 1} >\) ?
3) Do you know that the number operator \(\displaystyle N  \psi _n > = a^{\dagger} a  \psi _n > = n  \psi _n > \) ?
Okay. We have to calculate
\(\displaystyle \overline{a} (t)  \psi _n > = U^{\dagger} (t, 0) ~ a ~ U(t, 0)  \psi _n >\).
(The tilde doesn't come out so well, so I'm using the overline.)
I'm going to do this the "long way." I think it gives a better idea of how to do this correctly. So step by step:
\(\displaystyle U^{\dagger} (t, 0) ~ a ~ e^{iHt/ \hbar}  \psi _n > = U^{\dagger} (t, 0) ~ a ~ e^{i \hbar \omega (n + 1/2) t/ \hbar }  \psi _n > = e^{i \hbar \omega (n + 1/2) t/ \hbar } U^{\dagger} (t, 0) ~ a  \psi _n > \)
\(\displaystyle = e^{i \hbar \omega (n + 1/2) t/ \hbar } U^{\dagger} (t, 0) \sqrt{n}  \psi _{n  1} > = \sqrt{n} ~ e^{i \hbar \omega (n + 1/2) t/ \hbar } U^{\dagger} (t, 0)  \psi _{n  1} >\)
\(\displaystyle = \sqrt{n} ~ e^{i \hbar \omega (n + 1/2) t/ \hbar } e^{i \hbar \omega ((n  1) + 1/2) t/ \hbar}  \psi _{n 1} >\)
\(\displaystyle = \sqrt{n} ~ e^{i \hbar \omega t/ \hbar }  \psi _{n  1} >\)
Now to pretty it up a bit:
\(\displaystyle \overline{a} (t)  \psi _n > = U^{\dagger} (t, 0) ~ a ~ U(t, 0)  \psi _n > = = \sqrt{n} ~ e^{i \hbar \omega t/ \hbar }  \psi _{n  1} >\)
\(\displaystyle \overline{a} (t)  \psi _n > = e^{i \hbar \omega t/ \hbar } \sqrt{n}  \psi _{n  1} > = e^{i \hbar \omega t/ \hbar } a ~  \psi _n >\)
You can similarly show that
\(\displaystyle \overline{a ^{\dagger}} (t)  \psi _n > = e^{i \hbar \omega t/ \hbar } a ^{\dagger} ~  \psi _n >\)
b) and c) can be done in a similar fashion. See if you can finish these. If you have problems, just let us know.
For d) this is just an application of U(t) on the wavefunction. So calculate \(\displaystyle U(t)  \psi _n >\). (Hint: What is \(\displaystyle  \psi _n (t = 0) >\)?
Let's let e) and f) wait until you have a better idea about these.
Dan