Why is the helium wavefunction a linear combination?

the general solution for the wave function of Helium is a linear combination of the two possible states:


$$
\text{state 1:} \quad \psi\_a(\vec{r}\_{1}) \, \psi\_b(\vec{r}\_{2}) \\
\text{state 2:} \quad \psi\_a(\vec{r}\_{2}) \, \psi\_b(\vec{r}\_{1})
$$


(where $\vec{r}\_{1}/\vec{r}\_{2}$ 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?


$$
\psi\_a(\vec{r}\_{1}) \, \psi\_b(\vec{r}\_{2})
=
\psi\_a(\vec{r}\_{2}) \, \psi\_b(\vec{r}\_{1}) \, ?
$$




First of all, note that 
$$
\psi\_a(\vec{r}\_{1}) \, \psi\_b(\vec{r}\_{2})
\neq
\psi\_a(\vec{r}\_{2}) \, \psi\_b(\vec{r}\_{1}) \, ,
$$
since same functions in these products ($\psi\_a$ and $\psi\_b$) have *different arguments* ($\vec{r}\_{1}$ and $\vec{r}\_{2}$) to the left and to the right of the inequality sign.




---


Now, to the question, why do we have a linear combination. For a system of two indistinguishable particles (say, electrons) the obvious requirement is that there should be no observable difference between the system in state $\psi(1, 2)$ and the system in state $\psi(2, 1)$,
\begin{equation}
 |\psi(1, 2)|^{2} = |\psi(2, 1)|^{2} \, ,
\end{equation}
where $1$ and $2$ stand for coordinates of particles.
This implies that
\begin{equation\*}
 \psi(1, 2) = \pm \psi(2, 1) \, .
\end{equation\*}




---


And in order to satisfy this equality in what is often called the *orbital approximation*, where the state of a many electron system $\psi$ is represented as a product of a single-particle states $\psi\_i$, the wave function $\psi(1, 2)$ should be actually written as not just a product of $\psi\_1$ and $\psi\_2$ (none of which, neither $\psi\_1(1) \psi\_2(2)$ nor $\psi\_2(1) \psi\_1(2)$, is symmetric or antisymmetric), but rather as the following linear combination of these products,
\begin{equation\*}
 \psi(1, 2) = \frac{1}{\sqrt{2}} \Big( \psi\_1(1) \psi\_2(2) \pm \psi\_1(2) \psi\_2(1) \Big) \, ,
\end{equation\*}
where $1 / \sqrt{2}$ is the normalization factor.




---


The choice of the sign in the expressions above is not arbitrary, but rather determined by the particles spin $s$, namely,
\begin{equation\*}
 \psi(1, 2) = (-1)^{2s} \psi(2, 1).
\end{equation\*}
Particles with *integer spins*, called *bosons* (photons, alpha particles), are described by a *symmetric* wave function
\begin{equation\*}
 \psi\_{\mathrm{s}}(1, 2)
 = 
 \frac{1}{\sqrt{2}}
 \Big(
 \psi\_1(1) \psi\_2(2)
 +
 \psi\_1(2) \psi\_2(1)
 \Big) \, ,
\end{equation\*}
while the particles with *half-integer spins*, called *fermions* (electrons, protons, neutrons), are described by the *anti-symmetric* wave function
\begin{equation\*}
 \psi\_{\mathrm{a}}(1, 2)
 = 
 \frac{1}{\sqrt{2}}
 \Big( 
 \psi\_1(1) \psi\_2(2)
 -
 \psi\_1(2) \psi\_2(1)
 \Big) \, .
\end{equation\*}


 12