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I'm studying Hartree-Fock theory. It is written that, in Hartree approach, for the two electrons in positions r1 and r2 (these are vectors, of course), the joint wavefunction of this 2-particle system is given as

Psi(r1, r2)=Psi1(r1) * Psi2(r2)

I'm confused a lot at this point. I have a basic knowledge of quantum mechanics and as much as I know, the wavefunction of a particle is a function of spatial coordinates (r), as in infinite square well, the wavefunction is given as

Psi(x)=sqrt(2/L) * sin(n*pi*x/L)

here, wavefunction is a function of x. It can be plotted as a function of position x and the squared modulus of wavefunction is also a function of x which gives the probability that we find it there.

Let us turn back to the 2-particle case. The wavefunction of the 2-particle system is Psi(r1, r2) that seems to be meaningless to me since there is only one coordinate system and the wavefunction had to be given something like

Psi(r)=somefunction( Psi1(r), Psi2(r)) (All Psi's are a function of independent variable r)

but it is given as

Psi(r1, r2)=Psi1(r1) * Psi2(r2)

r1 and r2 being the coordinates of particle-1 and particle-2.

Could anybody please give an explanation of this situation? We do not know r1 and r2 exactly in quantum mechanics. But how can we write the wavefunction of 2-particles as a function of r1 and r2?