Probability to find a particle in a particlar state $\psi_{n}$ I have a problem to understand the probabilities in QM. In particular, if I have a particle in state $\psi_{n}$, then we change the system and we ask for the probability to find the particle in a state $\varphi_{n}$. Where $\psi_{n}$ are the stationary states of the system 1 and $\varphi_{n}$ of the system 2. 
When I talk about changing the system, it's for example if we have a harmonic oscillator system with a constant electric field and then we remove the electric field. I understand it have something to do with $|c_{n}|^2$, where $c_{n}$ is:
\begin{equation}
c_{n}=\langle{\psi_{n}|\varphi_{n}}\rangle=\int_{-\infty}^{+\infty}\psi_{n}^{*}\;\varphi_{n}\;dx
\end{equation}
I understand the mathematics of the situation. My problem is really to understand the physics of this situation. If I could simply list my questions, it would be like that:


*

*What are the probabilities of being in a stationary state?

*What are the probabilities of being in a state (it could be a linear combination of several stationary states)?

*What are the probabilities of being in a state after the system changes?

*How can I use the equations to help me to understand the probabilities in the system or systems?

 A: A quantum system is described by a set of self-adjoint operators $(A_1\ldots A_n, H)$ and a Hilbert space $\mathcal{H}$. The mentioned operators represent the observables that you can experimentally measure and their eigenvalues the possible outcomes. Among them there is a special one, the Hamiltonian $H$, describing the time evolution of the system. A state is any element $|\psi\rangle \in \mathcal{H}$.

What are the probabilities of being in a state (it could be a linear combination of several stationary states)?

Assumptions provide that any state can be expanded onto a basis of eigenvectors of any of the observables, namely $|\psi\rangle = \sum_i c_i |a_i\rangle$. This means that after having performed a measurement of the observable $A$ your state can become any of its eigenvectors (namely any of its possible outcomes), collapsing into them with frequencies of $|c_i|^2$ if performing infinite measurements.

What are the probabilities of being in a stationary state?

It is not clear what you mean by stationary state. Any system changes in time, thus nothing is stationary. What can happen is that after a measurement of the energy your initial state becomes an eigenstate of the Hamiltonian (by definition of measurement); if so, and if the time evolution operator is diagonal onto the eigenstates of the Hamiltonian, then a subsequent measurement of the energy will give back the same state because the eigenvectors remain such.

What are the probabilities of being in a state after the system changes?

Given the initial state $|\psi_{\textrm{in}}\rangle$ and the time evolution operator $U(t)$, the state after "the change" (as evolution) will be $U(t)|\psi_{\textrm{in}}\rangle = |\psi_{\textrm{final}}\rangle$. The probability of this being any other state $|\phi\rangle$ after infinite measurements is $|\langle\phi|U(t)|\psi_{\textrm{in}}\rangle|^2$.
A: I think this is saying much the same as Gennaro, but I'd phrase things slightly differently.
In config $\mathbf{1}$ your system has some eigenfunctions $\psi$, and you start with the system in one of these eigenfunctions $\psi_n$.
When you change to config $\mathbf{2}$ the function $\psi_n$ is no longer an eigenfunction because your system now has a set of different eigenfunctions $\varphi$. However we can express the function $\psi_n$ as a linear combination of the new eigenfunctions:
$$ \psi_n = \sum_i c_i \varphi_i $$
So your system is now in a superposition of lots of different eigenfunctions. In effect you have just set $\psi_n$ to be the initial state of your system. You could have chosen any function as the initial state - $\psi_n$ just happens to be the initial state inherited from the pre-change state.
To extract information you just proceed as usual. For example if you want the energy you simply use:
$$ \langle E \rangle = \langle \sum_i c_i \varphi_i | \mathcal{H} | \sum_j c_i \varphi_j \rangle $$
and because the $\varphi$ functions are orthonormal this expands to:
$$ \langle E \rangle = \sum_i c_i^2 E_i $$
As you say in the question the constants $c_i$ are calculated using:
$$\begin{align}
 c_i &= \langle \varphi_i | \psi \rangle \\
     &= \langle \varphi_i | \sum_j c_j \varphi_j \rangle \\
     &= \sum_j c_j \langle \varphi_i | \varphi_j \rangle 
\end{align} $$
