Mixed states and total wave function for three-Fermion-systems

[Like Tony Stark]

Homework Statement

Find the total wave function (including the spatial part) of a system of three spin $\frac{1}{2}$ particles.

Relevant Equations

$\Psi = \psi_s(x_1, x_2, x_3) \xi_a (S_1, S_2, S_3) + \psi_a(x_1, x_2, x_3) \xi_s (S_1, S_2, S_3)$

I've already calculated the total spin of the system in the addition basis:

$\ket{1 \frac{3}{2} \frac{3}{2}}; \ket{1 \frac{3}{2} \frac{-3}{2}}; \ket{1 \frac{3}{2} \frac{1}{2}}; \ket{1 \frac{3}{2} \frac{-3}{2}}; \ket{0 \frac{1}{2} \frac{1}{2}}; \ket{0 \frac{1}{2} \frac{-1}{2}}; \ket{1 \frac{1}{2} \frac{1}{2}}; \ket{1 \frac{1}{2} \frac{-1}{2}}$

The states corresponding to the $j=\frac{3}{2}$-subspace are symmetric and I'll call it $\xi_s (S_1, S_2, S_3)$, while the other states are neither symmetric nor antisymmetric.

The total wave function must be antisymmetric since the system is fermionic. If there were antisymmetric states, the wave function would be:

$\Psi = \psi_s(x_1, x_2, x_3) \xi_a (S_1, S_2, S_3) + \psi_a(x_1, x_2, x_3) \xi_s (S_1, S_2, S_3)$

with

$\psi_s(x_1, x_2, x_3)=\frac{1}{\sqrt{3!}} [\psi_1 (x_1) \psi_2 (x_2) \psi_3 (x_3)+\psi_1 (x_1) \psi_2 (x_3) \psi_3 (x_2)+\psi_1 (x_2) \psi_2 (x_1) \psi_3 (x_3)+\psi_1 (x_2) \psi_2 (x_3) \psi_3 (x_1)+\psi_1 (x_3) \psi_2 (x_1) \psi_3 (x_2)+\psi_1 (x_3) \psi_2 (x_2) \psi_3 (x_1)]$

$\psi_a(x_1, x_2, x_3)=\frac{1}{\sqrt{3!}} [\psi_1 (x_1) \psi_2 (x_2) \psi_3 (x_3)-\psi_1 (x_1) \psi_2 (x_3) \psi_3 (x_2)-\psi_1 (x_2) \psi_2 (x_1) \psi_3 (x_3)+\psi_1 (x_2) \psi_2 (x_3) \psi_3 (x_1)+\psi_1 (x_3) \psi_2 (x_1) \psi_3 (x_2)-\psi_1 (x_3) \psi_2 (x_2) \psi_3 (x_1)]$

But we don't have $\xi_a (S_1, S_2, S_3)$ states.

What should I do?

 

[kuruman]

Are you familiar with Slater determinants?