# 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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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.
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