Finding the State of a Quantum System with Given Hamiltonian and Observable

[LizardWizard]

Homework Statement
We are given the Hamiltonian H and an observable A
##H=\begin{pmatrix}
2 & 0 & 0\\0 & 0 & 1\\0 & 1 & 0
\end{pmatrix}\hbar\omega
A=
\begin{pmatrix}
1 & 0 & 0\\0 & 1 & 0\\0 & 0 & -1
\end{pmatrix}a
##
We are also told that at ##t=0## we have that a measurement of A gives us -a, and then we are asked to determine the generic state of the system at time t

The attempt at a solution

For starters, the solution the professor gives is $$|\psi(t)\rangle=
\begin{pmatrix}
0\\-isin(\omega t)\\cos(\omega t)
\end{pmatrix}
$$

Now while I don't know how to arrive to this solution in particular, I know this result most likely follows from $$\psi(t)=e^{-iEt/\hbar}$$
since we already have that ##E=\hbar\omega##
But how exactly do I arrive at the solution provided? This is somewhat different from all the exercises I've seen so far so I don't really know the calculations involved.

 

[LizardWizard]

I thought I'd add some extra information. In another exercise we have
##H=\begin{pmatrix}
0 & 1 & 0\\1 & 0 & 0\\0 & 0 & 1
\end{pmatrix}\hbar\omega
A=
\begin{pmatrix}
-1 & 0 & 0\\0 & 1 & 0\\0 & 0 & -1
\end{pmatrix}a
##
and solution
$$|\psi(t)\rangle=
\begin{pmatrix}
-isin(\omega t)\\cos(\omega t)\\0
\end{pmatrix}
$$
Perhaps this helps illustrate my problem with this. The solution itself makes sense to me but the mechanism behind it elude me.

By the way this is my first post so please don't hesistate to give feedback.

 

[blue_leaf77]

$$\psi(t)=e^{-iEt/\hbar}$$

It should be ##\psi(t)=e^{-iHt/\hbar}\psi(0)##, with ##\psi(0) = |u_3\rangle## (##|u_3\rangle## is the eigenket of ##A## correponding to eigenvalue ##-a##) as given by the problem. Next, make use of the completeness relation for the eigenkets of ##H##,
$$
\sum_{n=1}^3 |e_n\rangle \langle e_n| = 1
$$
with ##|e_n\rangle## the eigenket of ##H##, and place it somewhere in ##e^{-iHt/\hbar}\psi(0)## such that the exponential operator can act on ##|e_n\rangle##.

 

[LizardWizard]

If the solution lies in having the eigenvectors of H act on ##\psi(t)## then I have, for the first example $$|e_1>=[1 0 0]$$$$|e_2>=[0 -1 1]$$$$|e_1>=[0 1 1]$$
from these ##|e_2>## would lead to the solution I am given, but would the other eigenvectors also lead to valid solutions? The same happens with the second example.

 

[blue_leaf77]

from these |e2>|e2>|e_2> would lead to the solution I am given, but would the other eigenvectors also lead to valid solutions?

I don't know what you mean by ##|e_2\rangle## being the only eigenvector of ##H## which contributes to the answer. Using the completeness relation above, I have checked that ##|e_3\rangle## also contributes.
By the way, the eigenvectors you have there must first be normalized, otherwise, the completeness relation won't work.

 

[LizardWizard]

I see, could you psot your solution then perhaps?

 

[blue_leaf77]

In PF, the helper is only allowed to guide the OP towards the correct answer without providing the final answer.
What about your own try, what did you get after properly utilizing the completeness relation in the equation ##\psi(t)=e^{-iHt/\hbar}|u_3\rangle##?

 

[LizardWizard]

Ok, I understand. I'll go try it one more time now and see if it works out.