Time evolution of an eigenstate and probabilities Suppose we have some Hamiltonian $H$ with at least one normalized eigenstate $v$ with real eigenvalue $\lambda$.
The time evolution operator is given by
$$ U(t,0) = e^{- i \frac{H}{\hbar}t} \ .$$
Now let $\psi_0 = v$ and let $\psi(t) = U(t, 0) \psi_0$. Now if my computation is correct we have
\begin{align}
\left| \left< \psi(t) | \psi_0 \right> \right|^2
& = \left| \left< \psi_0 | \psi(t) \right> \right|^2 = \left| \left< \psi_0 | U(t,0)\psi_0 \right> \right|^2 = \left| \left< \psi_0 | e^{- i \frac{\lambda}{\hbar}t} \psi_0 \right> \right|^2
\\ &  = \left| e^{ i \frac{\lambda}{\hbar}t} \left< \psi_0 | \psi_0 \right> \right|^2 = \left|\left< \psi_0 | \psi_0 \right> \right|^2 = 1 \ . 
\end{align}
Now in the context of my exercise this makes no sense as I am asked what the "survival probability" of some state $\psi(t)$ is and the exercise in question asked me to calculate the above probability, but I seem to get that the probability is $1$ for all times, so the interpretation would be that the state remains the same for all time, or something like that.
I have a feeling I am misunderstanding something in the above calculation. 
 A: Your observation is totally correct and nothing is wrong with that.
Eigenstates of the (time independant) Hamiltonian are called stationary states. Such system will stay in a state who's amplitude is constant. Only the relative phase will change:
$$ \psi(t) = U(t,0) \psi_0 = e^{-iHt/\hbar}\psi_o = e^{-i\lambda t/\hbar}\psi_0 $$
So a system with definite energy will not change it's energy, commonly known as energy conservation.
Also, if a system is in a constant superposition of states $\psi_0,~\psi_1,~\psi_2,\dots$ with different energy eigenvalues $\lambda_0,~\lambda_1,~\lambda_2,\dots$, the system's probability to be in one of those states is constant:
$$ \left| \langle\psi_0|U(t,0)\sum_i\alpha_i|\psi_i\rangle \right|^2 = |\alpha_i|^2 \left| \langle\psi_0|\psi_0\rangle \right|^2 = const. $$
However, the propablity to stay in that superposition may oscillate:
$$ \left| \left(\sum_j\alpha_j^*\langle\psi_j|\right)U(t,0)\sum_i\alpha_i|\psi_i\rangle \right|^2 = \left| \sum_i |\alpha_i|^2 e^{-i\lambda_i t/\hbar}\right|^2 \ne const. $$
