Possible states for two electrons in the helium atom

Question

Consider the helium atom with two electrons, but ignore coupling of angular momenta, relativistic effects, etc.

The spin state of the system is a combination of the triplet states and the singlet state. I will denote a linear combination of the three triplet states as $\lvert\chi_+\rangle$ (because it's symmetrical under exchange of electrons) and $\lvert\chi_-\rangle$ the singlet state (because it's anti-symmetric).

Then, the orbital state of the electrons. Suppose one electron is in the state $\lvert\phi_a\rangle$; the other in the state $\lvert\phi_b\rangle$. The orbital state of the system is:

$$ \lvert\phi_{\pm}\rangle = \frac{1}{\sqrt 2} \left (\lvert\phi_a\rangle\lvert\phi_b\rangle \pm \lvert\phi_b\rangle\lvert\phi_a\rangle \right )$$

Because the overal state $\lvert\psi\rangle$ of the electrons must be anti-symmetric, is it correct to construct it as following:

$$\lvert\psi\rangle = \lvert\phi_{\pm}\rangle \lvert\chi_{\mp}\rangle \text{ ?}$$

Answer 1

For two electrons your are right! The absolute must is for the total state to be anti-symmetric. Technically, this means that the total wave-function must belong to a one-dimensional representation of the permutation group such that each permutation is represented by either +1 or -1 depending on it's parity.

You can ask about permutational symmetry separately for the spin part and the orbital part. If there number of electrons $n>2$, then the orbital part and the spin part may belong to these may belong to more complicated (e.g., more than one-dimensional) representations of the permutations group, those get classified by Young tableux..There are rules how to combine two "conjugate"(not sure if this is mathematically correct term) representations to construct the total wave-function anti-symmetric under any odd permutation. For $n=2$ these rules reduce to a simple product as you quote. For more on this you can consult Landau and Lifshitz, Quantum Mechanics (The Course of Theoretical Physics vol.III).

There is also a systematic connection between the representations of the permutation group for the spin part and the $SU(2)$ of the spin. Unfortuntally, I can't find the refernce which I have learn this from (hopefully others can help).