Help me understand the first equation in Landau & Lifshitz's Quantum Mechanics

Question

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

again, fine...

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?

Answer 1

This is trying to motivate matrices for observables--- the reason that they use q and q' is because they want to write down the most general quadratic bilinear function of $\psi$. Imagine you have only three possible positions 1,2,3, then there are three values of $\psi$, which you can call ABC and the most general bilinear function is the sum of products of pairs of these:

$$\phi(1,1)A^2 + \phi(1,2)AB + \phi (1,3) AC + ...$$

The general form is a matrix of coefficients $\phi(i,j)$ and the quadratic function is

$$ \sum_{ij} \phi(i,j)\psi^*(i)\psi(j)$$

And for continuous space you replace the sum by an integral using limits. This is what they are saying.

This is a poor motivation in my opinion. The idea that quantum mechanics only yeilds expectation values is false. The reason we use matrices for observables can be better motivated like this: two states where a physical observable is different must be orthogonal, by measuring the observable, you learn that every state can be written as a superposition of different states with different values for the observable, therefore you can define the operator to be the outer product of the definite-value vectors times the value of the observable (which becomes the eigenvalue) and this defines the observable. Then Landau and Lifschitz's thing becomes the expected value in a natural way.