the general solution for the wave function of Helium is a linear combination of the two possible states: [![][1]][1] [1]: https://i.sstatic.net/1Jdyo.png
$$ \text{state 1:} \quad \psi_a(\vec{r}_{1}) \, \psi_b(\vec{r}_{2}) \\ \text{state 2:} \quad \psi_a(\vec{r}_{2}) \, \psi_b(\vec{r}_{1}) $$
(where r1/r2$\vec{r}_{1}/\vec{r}_{2}$ denotes electron 1/2.)
But aren't the products of these wavefunction the same? Why then make a linear combination; it would be just like same function +/- same function?
$$
\psi_a(\vec{r}_{1}) \, \psi_b(\vec{r}_{2})
=
\psi_a(\vec{r}_{2}) \, \psi_b(\vec{r}_{1}) \, ?
$$