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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:

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

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    $\begingroup$ Tex tip of the day: while | and \lvert may produce the same symbol in your documents, the latter is favored because it explicitly is the left delimiter. Besides, MathJax renders the latter better, making for nicer-looking kets on this site :) $\endgroup$
    – user10851
    May 10, 2013 at 0:08
  • $\begingroup$ look at marty-green having-trouble-counting-spin-states $\endgroup$
    – Helder Velez
    May 21, 2013 at 11:22

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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).

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  • $\begingroup$ Thanks! I was just asked to do this in an exercise and while what I did seemed intuitively obvious, I just wanted to make sure. The case for $n>2$ seems really complicated, though. $\endgroup$
    – Samantha
    May 9, 2013 at 22:15

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