Spin and Spatial Symmetry

For two electrons the full wavefunction factorizes into a spatial part and a spin part. Because the total state must be antisymmetric, the symmetries of those two parts are locked together: choosing one fixes the other. This coupling is the key to atomic spectra and magnetism.

Two spin-1/2 particles combine into four states

Two spin- $\tfrac12$ particles have $2 \times 2 = 4$ spin states. They organize into one antisymmetric singlet (total spin $S=0$ ) and three symmetric triplet states (total spin $S=1$ ):

\text{singlet } (S=0): \quad \tfrac{1}{\sqrt2}\big(|\!\uparrow\downarrow\rangle - |\!\downarrow\uparrow\rangle\big),

\text{triplet } (S=1): \quad |\!\uparrow\uparrow\rangle, \;\; \tfrac{1}{\sqrt2}\big(|\!\uparrow\downarrow\rangle + |\!\downarrow\uparrow\rangle\big), \;\; |\!\downarrow\downarrow\rangle.

Under exchange, the singlet picks up a minus sign (antisymmetric) while all three triplet states are unchanged (symmetric). You can check the middle triplet state directly: swapping the two spins maps $|\!\uparrow\downarrow\rangle \leftrightarrow |\!\downarrow\uparrow\rangle$ , leaving their sum invariant and flipping the sign of their difference.

The locking rule

Write the total state as (spatial) $\times$ (spin). For electrons the product must be antisymmetric. There are exactly two consistent pairings:

\underbrace{\text{symmetric spatial}}_{\psi_S} \times \underbrace{\text{antisymmetric spin}}_{\text{singlet}}, \qquad \underbrace{\text{antisymmetric spatial}}_{\psi_A} \times \underbrace{\text{symmetric spin}}_{\text{triplet}}.

So a spin singlet always rides a symmetric spatial wavefunction, and a spin triplet always rides an antisymmetric spatial wavefunction. You never need to antisymmetrize space and spin separately — fixing the total symmetry to $-1$ does it.

Why this is observable

The spatial symmetry controls how close the electrons get (the exchange force from the previous lesson). The antisymmetric spatial state ( $\psi_A$ , paired with the triplet) keeps electrons apart, reducing their Coulomb repulsion and lowering the energy. So the spin-triplet, parallel-spin configuration is often the lower-energy state even though no magnetic interaction is involved — this is the microscopic content of Hund's first rule and the seed of ferromagnetism. The spin you can measure is a proxy for a spatial correlation you usually cannot.

Try it

Build the symmetric $m_s = 0$ triplet state $\tfrac{1}{\sqrt2}(|01\rangle + |10\rangle)$ on two qubits, reading $|0\rangle$ as spin up and $|1\rangle$ as spin down. This is the spin partner of an antisymmetric spatial wavefunction.

Run your code to see the quantum state.