# What does $\psi_j(r_i)$ mean?

I have a mean-field Hamiltonian for N electrons. The mean-field potential felt by electron $i$ at position ${\bf r}_i$ is given by

$V^{(i)}_{int}({\bf r}_i)=\sum_{j\ne i}|\psi_j({\bf r}_i)|^2$

I can understand why this is the case. However, I need to clear up my understanding of the term $\psi_j({\bf r}_i)$. Is it simply the wavefunction of the $j^{th}$ electron at position ${\bf r}_i$?

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The key idea in the mean field approach is taking into account the contribution, to the potential(the cause; see below) at each point ${\bf r}$, of the complete electronic configuration. By electronic configuration, we mean the probability density distribution of each of the one electron states $|\psi_j({\bf r}^')|^2$. So the cause of the $j^{th}$ electron(at ${\bf r}^'$) "on" the' electron at ${\bf r}$ is given by
$u({\bf r}) = -e \frac{|\psi_j({\bf r}^')|^2}{|{\bf r} - {\bf r}^'|}$.
Hence the total cause would be integrating over all the possible positions ${\bf r}^'$ of the $j^{th}$ electron and summing over all the states $j$ including 'the' electron which feels the effect. This total cause is used then to deduce the wavefunction of the' electron using Schr\:odinger equation (the effect).
Some caveats : While integrating over all possible ${\bf r}^'$ we encounter a singularity when ${\bf r}^' = {\bf r}$. This is dealt with what is called "self-interaction" term and correction. I dont have good understanding of this.