# parallel/anti-parallel vs. triplet/singlet description of two spins

If we consider two spins, we can think of the spins as being either parallel (up|up or down|down)or anti-parallel (up|down or down|up).

Or we can think of them as being in the triplet or singlet configuration.

Is one description more correct than the other? Or is it just a matter of choice between two basis sets? It would seem to me that using T/S is correct because it accurately reflects the symmetry needed in the wavefunction.

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## 2 Answers

It's just a choice of basis. Whether you use $$\bigl\{|\uparrow\downarrow\rangle,|\downarrow\uparrow\rangle\bigr\}$$ (individual spins) or $$\biggl\{\frac{1}{2}\bigl(|\uparrow\downarrow\rangle + |\downarrow\uparrow\rangle\bigr),\frac{1}{2}\bigl(|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle\bigr)\biggr\}$$ (triplet/singlet) they span the same space. But usually the T/S basis is more useful because those states are also eigenstates of the total spin operator $S^2$. As a side benefit, they reflect the (anti-)symmetrization requirements of identical particles; for example, if you have two identical fermions with no other quantum numbers (neglecting the fact that such particles don't exist :-P) in a bound state, you know that they have to take the singlet configuration in order for the wavefunction to be antisymmetric.

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Here is my problem then. Consider a single spin $\uparrow$ at a site. Now bring in another spin. In the first basis you write, there is a 50 % that the added spin can coexist at the site since there is a 50 % chance it will be anti-parallel (so by Pauli exclusion). In the second basis, it would be a 75 % chance they could exist on  the same site since there is a 75 % chance they will form triplet state. Where am I wrong here? – BeauGeste Aug 14 '11 at 20:13
– David Zaslavsky♦ Aug 14 '11 at 20:43

The parallel/antiparallel picture is not quite correct. Mostly because it hides the fact that the electrons are indistinguishable from our intuition. In a (very rare) cases when you may consider electrons distinguishable, this picture is correct. In more natural case when electrons are completely equal triplet/singlet picture is much better.

Maybe few more explanations needed. If you say that electron spins are antiparallel what you actually say is "spin of one electron is antiparallel to spin of another". But electrons are completely equal! You can not freely point to electron and say that it is "the one". Singlet/triplet picture however stays on a more solid ground: what is the total angular momentum of system which consists of two electrons. Here you do not choose electrons. You explicitly use the fact that electrons are equal.

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