I came across this basic exercise but I don't fully get the gist of it.
Consider two neutral particles in a 1D Box with the interval $0\leq x \leq L$. The interaction between the two particles is negligible.
We first compute the hamiltonian operator for the two non-interacting particles.
$$ \hat{H}=\frac{-\hbar}{2m_e}\big(\frac{d^2}{dx_1^2}+\frac{d^2}{dx_2^2}\big) + V(x_1, x_2) \tag{1} $$
with
$$V(x_1,x_2) = \begin{cases} 0, & x_1,x_2\in[0,L] \\ \infty, & \text{else}\end{cases} \tag{2}$$
We further calculate the eigenfunctions and the eigenvalues
$$E_{n_1,n_2}=\frac{h^2}{8mL^2}(n_1^2 + n_2^2) \tag{3.1}$$
$$\Psi_{n_1,n_2}(x_1,x_2) = \sqrt{\frac{2}{L}}\sin(\frac{n_1\pi x_1}{L})\sqrt{\frac{2}{L}}\sin(\frac{n_2\pi x_2}{L}) \tag{3.2}$$
We now consider two neutrons. Neutrons are Fermions with a spin quantum number of $s=1/2$. We know that the projection of the spin vector onto the z-Axis is given by:
$$\hat{S}_z | S, M_S\rangle = \hbar M_S|S,M_S\rangle \tag{4}$$
So for $S=1/2$ we get $M_S=\pm 1/2$ and thus we find the two possible projections
$$\pm\frac{\hbar}{2} \tag{5}$$
From the Pauli-Principle we know that the wave equation for Fermions must be anti-symmetric. Now the exercise wants me to modify the previously found wave equation such that
i) it is compatible with the Pauli-Principle
ii) it is a product of two functions whereas the first functions should be a function of the x-coordinate of both Neutrons and the second should be a function of the spin states of the two-neutron-system.
So they say that the wave equations can be written as a product of the "space-part" and the "spin-part" of the wave equation. For the "space-part" we can find an symmetric and an anti-symmetric function:
$$\Psi_{s/a}^{\text{Space}}(x_1, x_2) = \frac{1}{\sqrt{2}}\big( \Psi_{n_1, n_2}(x_1, x_2) \pm \Psi_{n_1,n_2}(x_2, x_1)\big) \tag{6}$$
The "spin-part" of the wave equation can be written as follows. The symmetric functions:
$$\Psi^{\text{Spin}}_{1,1} = |\alpha(1)\alpha(2)\rangle \tag{7.1}$$ $$\Psi^{\text{Spin}}_{1,0} = \frac{1}{\sqrt{2}}(|\alpha(1)\alpha(2)\rangle + |\beta(1)\alpha(2)\rangle \tag{7.2}$$ $$\Psi^{\text{Spin}}_{1,-1} = |\beta(1)\beta(2)\rangle \tag{7.3}$$ $$\Psi^{\text{Spin}}_{0,0} = \frac{1}{\sqrt{2}}(|\alpha(1)\alpha(2)\rangle - |\beta(1)\alpha(2)\rangle \tag{7.4}$$
For the wave equation to be anti-symmetric, we need
$$\Psi(x_1,x_2)=\Psi_{s}^{\text{Space}}(x_1,x_2)\cdot\Psi_{0,0}^{ \text{Spin}} \tag{8}$$ resp. $$\Psi(x_1,x_2)=\Psi_{a}^{\text{Space}}(x_1,x_2)\cdot\Psi_{1,M}^{ \text{Spin}}, \quad M=0,\pm 1 \tag{8}$$
Questions: Now I'm a bit confused about what's going on here. Why do we first calculate the wave equation for neutral particles and then somehow change it? What I don't get is: What is there to change? The wave equation is explicitly given in 3.2. There isn't anything we can change? Do we maybe use this wave equation in 3.2 as a basis for the one we are looking for the Fermions. Furthermore I don't fully get 7.* - what kind of basis do we use here and why is it two dimensional? We have two particles, each particle is a 1/2-spin and so we have a 2-dimensional space for each particle resulting in a 4-dimensional space, so what exactly is $\alpha, \beta$?
In general, the whole process is way too fast and it just lacks details for me to really get it. What's the main idea behind what we do here?