Something has been bothering me for a while. I encounter this kind of statement everywhere:
While a single particle is described by a wave function $\Psi({\vec r};t)$, a system of two particles, call them 1 and 2, is described by a wave function $ \Psi({\vec r}_1,{\vec r}_2;t)$ depending on both particle positions. Leon van Dommelon
and
A free quantum particle is described by the Schrödinger equation:
$\frac{\hbar^2}{2m} \nabla^2 \ \psi(\mathbf{r}, t) = i\hbar\frac{\partial}{\partial t} \psi (\mathbf{r}, t)$ where $\psi$ is the wavefunction of the particle at position r and time t. Wikipedia
Many other examples of this kind of phrasing are on Physics StackexchangeStack Exchange as well.
The problem is that it seems to me that the wavefunction is not a function of particle positions at all, for positions are not defined until the system is collapsed with a measurement. It would be weird if the wavefunction was a function of random eigenvalues. Instead, it looks like the wavefunction is a function of coordinates in Euclidean position space $\mathbb{R}^{3n}$ that maps a list of $x_1, y_1, z_1, x_2, y_2, z_2 ...$$x_1, y_1, z_1, x_2, y_2, z_2\ldots$ to a complex number.
Am I wrong? Is there a good reason to say that the wavefunction is a function of particle positions (or momenta)?
