Adding 3 electron spins

Question

I've learned how to add two 1/2-spins, which you can do with C-G-coefficients. There are 4 states (one singlet, three triplet states). States are symmetric or antisymmetric and the quantum numbers needed are total spin and total z-component.

But how do you add three 1/2-spins? It should yield 8 different eigenstates. Which quantum numbers do you need to characterise the 8 states?

It is not as easy as using C-G-coefficients and the usual quantum numbers as for the total momentum the doubly degenerate 1/2 state and quadruple degenerate 3/2 state can describe only 6 or the 8 states. You will need an additional quantum number for the degeneracy.

So how do you get the result?

(I actually tried out myself with a large 8x8 matrix. The total spin 1/2 is each doubly degenerate. For the additional quantum number I chose the cyclic permutation. Spin 1/2 states are neither symmetric nor antisymmetric. But what is the usual way to derive this?)

Answer 1

$\newcommand{\Ket}[1]{\left|#1\right>}$You can build them just from the highest one, $\Ket{\frac{1}{2}\frac{1}{2}\frac{1}{2}}$, using the lowering operator $S_{-}=S_{1,-}+S_{2,-}+S_{3,-}$. Now remember that each operator in this sum acts only on it's respective space of states. Also, it gets messy with the numerical coefficients, but remember, after each step you can check to see if the norm is 1. Lets do one together. $$S_{-}\Ket{\frac{3}{2}\frac{3}{2}}=\sqrt{\left(\frac{3}{2}+\frac{3}{2}\right)\left(\frac{3}{2}-\frac{3}{2}+1\right)}\hbar\Ket{\frac{3}{2}\frac{1}{2}}=(S_{1,-}+S_{2,-}+S_{3,-})\Ket{\frac{1}{2}\frac{1}{2}\frac{1}{2}}$$ $$=\sqrt{\left(\frac{1}{2}+\frac{1}{2}\right)\left(\frac{1}{2}-\frac{1}{2}+1\right)}\hbar\left(\Ket{-\frac{1}{2}\frac{1}{2}\frac{1}{2}}+\Ket{\frac{1}{2}-\frac{1}{2}\frac{1}{2}}+\Ket{\frac{1}{2}\frac{1}{2}-\frac{1}{2}}\right)$$ You can do this a couple more times to to get the other $3/2$ ones. But after this first one, you can build a $\Ket{\frac{1}{2}\frac{1}{2}}$ one thats perpendicular to the other ones. I think you can just use Gram-Schmidt, or eye-ball it. Then with the $\Ket{\frac{1}{2}\frac{1}{2}}$ you just use the lowering operator some more.

I did this for learning a little while back, I hope this helps/is correct.

Answer 2

I looked in Edmonds, which is usually the standard reference, and he doesn't mention any standard approach at breaking the degeneracy.

You need two linearly independent $s=1/2,\,m=1/2$ solutions, and you can get three different solutions by first coupling one of the three different pairs to the singlet $s=0$ state and then adding an up state. This yields the three vectors $\newcommand{\ket}[1]{|#1\rangle}$ $$\ket{\psi_1}={1\over\sqrt{2}}\left(\ket{\uparrow\uparrow\downarrow}-\ket{\uparrow\downarrow\uparrow}\right),$$ $$\ket{\psi_2}={1\over\sqrt{2}}\left(\ket{\downarrow\uparrow\uparrow}-\ket{\uparrow\uparrow\downarrow}\right),$$ $$\ket{\psi_3}={1\over\sqrt{2}}\left(\ket{\uparrow\downarrow\uparrow}-\ket{\downarrow\uparrow\uparrow}\right),$$ which add to zero so only two are linearly independent.

Edmonds shows, in particular, that there is a unitary transformation linking any of the three representations linked to the three vectors above (which is of course no surprise) and that this unitary transformation is independent of spatial orientation (which is not automatic but by the Wigner-Eckart theorem ought to happen). He then goes on to define appropriate invariant transformation coefficients (the Wigner $6j$ symbols) and spends a good deal of time exploring them, but he doesn't say how to (canonically) break the degeneracy.

If it's a basis you want, then take any two of the three above. If you need (like you should!) an orthonormal basis, then you can take linear combinations like $$\ket{\psi_{23}}={1\over\sqrt{6}}\left(\ket{\uparrow\uparrow\downarrow}-2\ket{\downarrow\uparrow\uparrow}+\ket{\uparrow\downarrow\uparrow}\right)$$ which obeys $\langle\psi_1|\psi_{23}\rangle=0$.

However, I don't think there is any way to treat the problem symmetrically in the three electrons. I had a quick go and I think one can prove there are no linear combinations of the three states that are symmetric or antisymmetric w.r.t. all three electron exchanges.

One way to see this is noting that you have three linearly dependent, unit-norm vectors that span a two-dimensional vector space and sum to zero. This is like having three unit vectors on a plane, symmetrically arranged at $120^\circ$ to each other. (The analogy is precise: the Gram matrices, $G_{ij}=\langle\psi_i|\psi_j\rangle$, which encode all the geometrical information about any set of vectors (see cf. problem 8.5), coincide.) There is then no way to choose a basis for the plane that is symmetric in the three "electrons", i.e. one whose symmetry group is the same as the three original vectors.

There is one thing, though, that you can do symmetrically: you can form an electron-exchange invariant resolution of the identity, of the form $$\frac{2}{3}\sum_{j=1}^3\ket{\psi_j}\langle\psi_j|=1|_{S={1\over2},m={1\over2}}$$ This also holds for the three vectors in the plane and expresses the fact that they form a tight vector space frame for $\mathbb{R}^2$.

Answer 3

I did not sort it out myself, but I guess 6j coefficients are used instead of C-G for three angular momenta.