'Deriving' superposition representations for 3 particles I'm trying to write up all the possible superposition states for 3 spin-1/2 particles (one spin-up, 2 spin-down). Lets denote $|\uparrow \rangle = |0\rangle, |\downarrow \rangle = |1\rangle$. Intuitively, the states in superposition should look something like:
$$
|011\rangle\ \pm|101\rangle\ \pm|110\rangle\ 
$$ (normalization constants are omitted).
However, I'm wondering if this notation could be related to the symmetry and anti-symmetry superpositions of 2 electrons (still one spin-up), in other words:
$$
|01\rangle\ \pm|10\rangle
$$
So my questions are how can I correctly write all the superposition of 3 electrons (one spin-up)? Can I 'derive' the 3-particle superposition states from the 2-particle superposition states? Do I need to worry about permutation? How can I tell which state is symmetry or anti-symmetry for 3 particles? Thanks:)
 A: The widely recommended approach for deriving completely asymmetric states of many particles is by using the Slater determinant. Thus, the asymmetric state of two particles can be expressed as
$$
\Psi(x_1, x_2) = 
\frac{1}{\sqrt{2!}}
\left|\begin{matrix}
\phi_1(x_1) & \phi_2(x_1)\\
\phi_1(x_2) & \phi_2(x_2)
\end{matrix}\right|
= 
\frac{1}{\sqrt{2}}\left[\phi_1(x_1)\phi_2(x_2) - \phi_2(x_1)\phi_1(x_2)\right],
$$
for three particles we have
$$
\Psi(x_1, x_2, x_3) = 
\frac{1}{\sqrt{3!}}
\left|\begin{matrix}
\phi_1(x_1) & \phi_2(x_1) & \phi_3(x_1)\\
\phi_1(x_2) & \phi_2(x_2) & \phi_3(x_2)\\
\phi_1(x_3) & \phi_2(x_3) & \phi_3(x_3)\\
\end{matrix}\right| = ...
$$
and so on.
The important thing here is to keep the track of the number of particles and the one-particle state index: $\phi_\alpha(x_k)$ is $k$-th particle in state $\alpha$. This may get particularly confusing in the bra-ket notation, since the index of the particle is not explicitly stated, but marked by the order of indices, and the notation might get easily confused with second quantization (where the order of indices refers to different states). Thus, in your case the notation really stands for
$$
|100\rangle = |1\rangle_1 \otimes|0\rangle_2 \otimes|0\rangle_3,
$$
where the subscripts refer to particles $1,2,3$. In other words, this notation is translated to Slater determinants by associating
$\phi_\alpha(x_n)$ to $|\alpha\rangle_n$.
