Understanding the Schrodinger Wave Equation for Multiple Electrons

[nobahar]

Hello!
No maths involved. I am just trying to qualitatively understand the Shrodinger wave equation.
So, the square:
$$(\psi(r,\theta,\phi))^2$$
is the probability of finding an electron at some distance r, and some angle$$\theta,\phi$$ from the nucleus. THis is for one elctron. For two electrons, this becomes:
$$(\psi(r_{1},\theta_{1},\phi_{1},r_{2},\theta_{2},\phi_{2})^2$$
What does this mean? Is it saying that, if I choose an electron and place it at r₁, theta₁... and choose another and place it at r₂..., then the probability that the two electrons will be in these places is the equation given above? I could keep one in the same place, and move the other?
Is this correct?
Any help appreciated.

 

[fzero]

$$|\psi(r_{1},\theta_{1},\phi_{1},r_{2},\theta_{2},\phi_{2})|^2$$ is the probability that one of the electrons will be at $$\vec{r}_1$$ while the other is at $$\vec{r}_2$$.

Your example is slightly complicated by the fact that the electrons are indistinguishable from each other. If the electrons have the same spin, we would not be able to tell one from the other after we've prepared the state. However, by the spin-statistics theorem, if the electrons have the same spin, the spatial wavefunction must be antisymmetric:

$$\psi(\vec{r}_1,\vec{r}_2) = - \psi(\vec{r}_2,\vec{r}_1).$$

If the electrons have opposite spins, we can use that to distinguish them and spin-statistics tells us that the spatial wavefunction must be symmetric.

 

[nobahar]

Thanks fzero.