Meaning of matrix elements What does $⟨ϕ|Q|ψ⟩$ physically mean, where $|ψ⟩$ and $|ϕ⟩$ are states and $Q$ is a linear operator? I know what its mathematical meaning is but I am looking for an interpretation: What does $Q$ do to $|ψ⟩$ and then the result to $⟨ϕ|$?
More specifically, what happens physically in the Stern-Gerlach experiment when you have the following expression:
$⟨S_z;+|S_x|S_z;+⟩$ 
, where $S_x$ is the operator of the measurement including an inhomogeneous magnetic field directed along the $x$-axis and $|S_z;+⟩$ is the spin-state of the particle along the $z$-axis.
 A: In brief: in general, none.
There might be operators $O$ for which $\langle\phi|O|\psi\rangle$ has some physical meaning. These are generally unitary operators (which preserve the normalization of $|\psi\rangle$, so that $O|\psi\rangle$ is still a physical, normalized state if $|\psi\rangle$ is). For instance, if $O=U(t)$, where $U$ is the time evolution operator, then $\langle\phi|U(t)|\psi\rangle$ is the amplitude for the process $\psi\to \phi$ to happen after a time interval $t$. If $O=T_{a}$, where $T$ is the spatial translation operator, then $\langle\phi|T_{a}|\psi\rangle$ is the overlap between the state $|\psi\rangle$ translated of an amount $\Delta x=a$ and the state $|\phi\rangle$. For rotations it works similarly, etc.
Moreover, $\langle\phi|O|\psi\rangle$ might have a direct (albeit approximate) interpretation in contexts such as the time-dependent perturbation theory, where in the case of a time-independent perturbation, $H=H_{0}+O$, $\langle\phi|O|\psi\rangle$ is proportional to the amplitude for going from state $\psi$ to state $\phi$ in the presence of the perturbation (and to first order in the perturbative series).
As for your example, $\langle S_{z};+|S_{x}|S_{z};+\rangle$ is just the mean value of $S_{x}$ in the up state of $S_{z}$, meaning that if you measured $S_{x}$ in the state $|S_{z};+\rangle$ in an infinite amount of different experiments and then averaged the outcomes you would get $\langle S_{z};+|S_{x}|S_{z};+\rangle$. This is because, for any hermitian operator $Q$,
$$
\langle\psi|Q|\psi\rangle=\sum_{q}\ q\ |\langle q|\psi\rangle|^{2}
$$
where $|q\rangle$ is the eigenstate of $Q$ with eigenvalue $q$ (notice that in the above $\phi=\psi$), and $|\langle q|\psi\rangle|^{2}$ does have a physical interpretation: it is the probability of finding $\psi$ in the state $|q\rangle$, hence the probability for the outcome of the measurement to be $q$.
Note: of course, not having a direct physical meaning does not mean that those matrix elements do not impact the physics of the system. Their impact, however, depends on how they enter into the equations.
A: $\langle \phi|Q|\psi\rangle$ is a mathematical statement that gives the inner (dot) product between the states $|\phi\rangle$ and $|Q\psi\rangle$. Here $Q$ is a linear operator that takes $|\psi\rangle$ to some other state $|Q\psi\rangle$. 
What it means is what inner products usually mean. It is a measure of overlap between the two states $|\phi\rangle$ and $|Q\psi\rangle$. How much of $|Q\psi\rangle$ is in $|\phi\rangle$, loosely speaking. 
Now for the physical interpretation of this we need to make use of completeness and the Born postulate. Say $|q_i\rangle$s are the eigenstates of the operator $Q$ with eigenvalues $q_i$, then we can rewrite:
$$|\psi\rangle=\sum_i |q_i\rangle\langle q_i|\psi\rangle=\sum_i \alpha_i|q_i\rangle$$
where $\alpha_i$s are the overlap between $|q_i\rangle$ and $|\psi\rangle$. 
Using this in our expression we get:
$$\langle \phi|Q|\psi\rangle=\langle\phi|Q \sum_i \alpha_i|q_i\rangle\\
=\langle\phi|\sum_i \alpha_iQ|q_i\rangle\\
=\langle\phi|\sum_i \alpha_iq_i|q_i\rangle
$$ 
This here as of now doesn’t really make much sense. But consider the case when $|\phi\rangle$ is an eigenstate $|q_j\rangle$ of $Q$, then our expression simplifies to:
$$\langle q_j|Q|\psi\rangle=\langle q_j|\sum_i \alpha_iq_i|q_i\rangle\\
=\sum_i\alpha_iq_i\langle q_j|q_i\rangle=\sum_i\alpha_i q_i\delta_{i,j}=\alpha_j q_j
$$
This physically means that when we measure $Q$ on a state $|\psi\rangle$, we get the outcome $q_i$ with a probability of $|\alpha_i|^2$, assuming the states are normalised. 
Now consider the general case where $|\phi\rangle=\sum_n \beta_n|q_n\rangle$. Now the inner product looks like:
$$
\langle \phi|Q|\psi\rangle=\sum_{i,n}\langle q_n|\beta^*_n \alpha_iq_i|q_i\rangle\\
=\sum_{i,n}\beta^*_n\alpha_iq_i\langle q_n|q_i\rangle=\sum_{i,n}\beta^*_n\alpha_i q_i\delta_{i,n}=\sum_n\beta^*_n \alpha_n q_n
$$
You can easily extend this argument to your system at hand. 
