Quantum mechanics problem? I had a test on Quantum mechanics a few days ago, and there was a problem which I had no clue how to solve. Could you please explain me?
The problem is:

Let's look at the $\hat H=E_0[|1 \rangle \langle 2| + |2 \rangle \langle1|]$ two-state quantum system, where $E_0$ is a constant, and $\langle i|j \rangle=\delta_{ij}$ $(i,j=1,2)$.
  \begin{equation}
\hat O= \Omega_0 [3 |1 \rangle \langle1|- |2 \rangle \langle2|] 
\end{equation} 
  is an observable quantity, and its expectation value at $t=0$ is: $\langle \hat O \rangle =-\Omega_0$, where $\Omega_o$ is a constant. 
  What is the $|\psi(0) \rangle$ state of the system at $t=0$, and what is the minimum $t>0$ time, that is needed for the system to be in the state: $|\psi(t) \rangle =|1 \rangle$?

I never came across a problem like this, I tried to construct the time evolution operator, $\hat U$, but I couldn't, and I have no idea how to start. 
 A: Part 1
The state vector can be written in terms of the two states at time $t$ as
$$
\left|\psi\left(t\right)\right> = c_1\left(t\right) \left|1\right> + c_2\left(t\right) \left|2\right>
$$
and at time $t=0$ as
$$
\left|\psi\left(0\right)\right> = c_1\left(0\right) \left|1\right> + c_2\left(0\right) \left|2\right>.
$$
We know
$$
\begin{align}
-\Omega_0 = \left<\hat{O}\right> &= \left<\psi\left(0\right)\right| \hat{O} \left|\psi\left(0\right)\right> \\
&= \Omega_0 \left(c^*_1\left(0\right) \left<1\right| + c^*_2\left(0\right) \left<2\right|\right) \left(3 \left|1\right>\left<1\right|-\left|2\right>\left<2\right|\right) \left(c_1\left(0\right) \left|1\right> + c_2\left(0\right) \left|2\right>
\right) \\
&= \Omega_0 \left(c^*_1\left(0\right) \left<1\right| + c^*_2\left(0\right) \left<2\right|\right)\left(3c_1\left(0\right) \left|1\right> - c_2\left(0\right) \left|2\right>
\right) \\
&= \Omega_0 \left(3 \left|c_1\left(0\right)\right|^2  - \left|c_2\left(0\right)\right|^2 \right),
\end{align}
$$
so
$$
3 \left|c_1\left(0\right)\right|^2  - \left|c_2\left(0\right)\right|^2 = -1.
$$
Since the state vector must be normalized,
$$
\left|c_1\left(0\right)\right|^2 + \left|c_2\left(0\right)\right|^2 = 1.
$$
You can finish this part.
Part 2
The Schrödinger equation tells us
$$
i \hbar \frac{d}{dt} \left|\psi\left(t\right)\right> = \hat{H} \left|\psi\left(t\right)\right>,
$$
or
$$
i \hbar  \left({\dot{c}}_1\left(t\right) \left|1\right> + {\dot{c}}_2\left(t\right) \left|2\right>\right) = E_0 \left(\left|1\right>\left<2\right| + \left|2\right>\left<1\right|\right) \left(c_1\left(t\right) \left|1\right> + c_2\left(t\right) \left|2\right>\right).
$$
I'll let you take it from here.
