While I've covered a basic course in Quantum Mechanics, I'm self-studying Landau & Lifshitz's book to help me understand what's going on.
Unfortunately, I'm stuck on the very first equation in the book.
Landau & Lifshitz write:
The basis of the mathematical formalism of quantum mechanics lies in the proposition that the state of a system can be described by a definite (in general complex) function $\Psi(q)$ of the coordines
so far so good...
The square of the modulus of this function determines the probability that a measurement performed on the system will find the values of the coordinates to be in the element $dq$ of configuration space. The function $\Psi$ is called the wave function of the system
A knowledge of the wave function allows us, in principle, to calculate the probability of the various results of any measurement (not necessarily of the coordinates) also. All these probabilities are determined by expressions bilinear in $\Psi$ and $\Psi^*$. The most general form of such an expression is
$\int \int \Psi(q)\Psi^*(q')\phi(q,q')dq dq'$
where the function $\phi(q,q')$ depends on the nature and the result of the measurement, and the integration is extended over all configuration space. The probability $\Psi\Psi^*$ of various values of the coordinates is itself an expression of this type.
OK, there's a lot here I don't understand. I can accept most of the statements on faith, but what I don't understand is what the hell $q'$ is for. Why can't the equation just use q throughout?