Symmetry of Combined States

Exchange and the swap operator

When two particles carry the same kind of angular momentum — two electron spins, say — we can ask how a state behaves under exchange: swapping particle 1 with particle 2. Define the swap operator $P_{12}$ by

P_{12}\,|a\rangle_1|b\rangle_2 = |b\rangle_1|a\rangle_2 .

Since swapping twice returns the original state, $P_{12}^2 = \mathbb{1}$ , so its eigenvalues are $\pm 1$ . A state with eigenvalue $+1$ is symmetric; one with eigenvalue $-1$ is antisymmetric.

Total spin determines the symmetry

For two spin- $\tfrac12$ particles the coupled states sort cleanly by exchange symmetry. Look back at the triplet and singlet:

\begin{aligned} \text{triplet } (s=1):\quad &|\!\uparrow\uparrow\rangle,\;\; \tfrac{1}{\sqrt2}(|\!\uparrow\downarrow\rangle + |\!\downarrow\uparrow\rangle),\;\; |\!\downarrow\downarrow\rangle &&\xrightarrow{P_{12}}\; +1 ,\\ \text{singlet } (s=0):\quad &\tfrac{1}{\sqrt2}(|\!\uparrow\downarrow\rangle - |\!\downarrow\uparrow\rangle) &&\xrightarrow{P_{12}}\; -1 . \end{aligned}

So the three triplet states are symmetric and the singlet is antisymmetric. This is not a coincidence: in general, combining two equal angular momenta $j_1 = j_2 = j$ , the resulting multiplet of total $J$ is symmetric when $2j - J$ is even and antisymmetric when it is odd. For $j = \tfrac12$ , $J = 1$ gives symmetric and $J = 0$ gives antisymmetric, matching the above.

The link to the Pauli principle

For identical fermions (electrons, spin- $\tfrac12$ ), the total wavefunction — spatial part times spin part — must be antisymmetric under exchange. The spin part alone is therefore tied to the spatial part:

Singlet spin state (antisymmetric) $\Rightarrow$ symmetric spatial wavefunction.
Triplet spin state (symmetric) $\Rightarrow$ antisymmetric spatial wavefunction.

This single rule has enormous consequences. In helium it explains why the spin-triplet (ortho) states sit lower in energy than the spin-singlet (para) states of the same configuration: the antisymmetric spatial wavefunction keeps the two electrons apart, reducing their Coulomb repulsion. The energy difference — the exchange splitting — is the origin of Hund's first rule and ultimately of ferromagnetism.

Summary

The total-spin quantum number does double duty: it labels the magnitude of the combined angular momentum and it fixes the exchange symmetry of the spin state. For identical fermions, that symmetry then dictates the symmetry of the spatial wavefunction through the Pauli principle — linking the abstract addition of angular momenta directly to the structure of atoms and the periodic table.