Nomenclature in QM textbook by Landau and Lifshitz According to Landau and Lifshitz ["Quantum Mechanics non-relativistic theory", page 6], the probability of various results of any measurement is given, in general by the following expression:
$$\iint \Psi(q)\Psi^\star (q') \phi(q,q') \,dq\,dq' $$ 
Where the function $\phi$ depends on the nature of the measurement, and $q$ is the set of coordinates of the quantum system. I have no idea why this is the case. But what makes me anxious is that the authors did not explain anywhere in the text what does $q'$ stand for. Is this a derivative of $q$? (I don't think so). So what is $q'$?    
 A: L&L's expression
$$\iint \Psi(q)\Psi^\star (q') \phi(q,q') \,dq\,dq' $$ 
is just the "familiar" expression (in case you have some other exposure to QM)
$$\int \Psi(q)^\star ~ \hat \phi ~\Psi (q)   \,dq ~ , $$ 
a sandwich between wavefunctions of a Hermitean  operator $\hat \phi$ representing the observable. (For instance, if it were the momentum operator $\hat p$, this would be the expectation value of the momentum measurement.) q' is but another coordinate "like" q.
Later on, in eqns (3.10, 3.11) they explain their tasteful convention of representing operators through integral kernels,
$$
\hat f \Psi ~(q)= \int dq' ~ f(q,q') \Psi(q') ~,\\
f(q,q')=\sum_n f_n ~ \Psi_n(q')^\star ~~ \Psi_n(q) . 
$$ 
Perhaps you are more familiar with the Dirac bracket notation,
$$
\langle \psi| \hat \phi| \psi\rangle= \iint dq dq' ~ \langle \psi|q'\rangle \langle q'|\hat \phi|q\rangle \langle q|\psi\rangle ~,
$$
that is
$$
\hat \phi =  \iint dq dq' ~  |q'\rangle   \phi (q,q')\langle q|  ~.
$$
For example,
$$
\hat p = \iint dq dq' ~  |q'\rangle    i\hbar (\partial_q \delta (q'-q))\langle q| = \int dq   ~  |q \rangle \frac{\hbar}{i} \partial_q  \langle q| ,
$$
after integration by parts and collapsing the δ function, so that, 
$$
\langle \Psi|\hat p |\Psi \rangle= -i\hbar \int dq   ~ \Psi(q)^\star ~\partial_q \Psi(q).
$$
