Exchange Forces

Symmetry under exchange is not a passive bookkeeping rule. It changes where particles are likely to be found relative to one another, and that change looks like an attractive or repulsive force even when no force is written into the Hamiltonian. This apparent interaction is the exchange force.

The setup: two particles, two states

Put two identical particles in distinct single-particle states $\psi_a$ and $\psi_b$ . Compare three ways of writing the spatial state:

Distinguishable (no symmetrization): $\psi_a(x_1)\psi_b(x_2)$ .
Symmetric: $\psi_S = \tfrac{1}{\sqrt{2}}[\psi_a(x_1)\psi_b(x_2) + \psi_b(x_1)\psi_a(x_2)]$ .
Antisymmetric: $\psi_A = \tfrac{1}{\sqrt{2}}[\psi_a(x_1)\psi_b(x_2) - \psi_b(x_1)\psi_a(x_2)]$ .

We ask: what is the expected squared separation $\langle (x_1 - x_2)^2\rangle$ in each case?

The exchange term

Expanding $\langle (x_1 - x_2)^2\rangle = \langle x_1^2\rangle + \langle x_2^2\rangle - 2\langle x_1 x_2\rangle$ for the symmetric and antisymmetric states gives the same result as the distinguishable case plus an extra cross term that comes from the interference between the two orderings:

\langle (x_1 - x_2)^2 \rangle_\pm = \langle (x_1 - x_2)^2 \rangle_\text{dist} \mp 2\,\big|\langle x \rangle_{ab}\big|^2,

where the exchange overlap is

\langle x \rangle_{ab} = \int \psi_a^*(x)\, x\, \psi_b(x)\, dx,

the upper sign ( $-$ ) applying to the symmetric state and the lower sign ( $+$ ) to the antisymmetric state. Everything hinges on $\langle x\rangle_{ab}$ : if the two orbitals do not overlap, this integral vanishes and there is no exchange effect at all.

Reading the result

When $\langle x\rangle_{ab} \neq 0$ :

The symmetric spatial state has a smaller mean-square separation than the distinguishable case — the particles are drawn together. This effective attraction is sometimes called an exchange attraction.
The antisymmetric spatial state has a larger mean-square separation — the particles are pushed apart. The antisymmetric wavefunction must vanish when $x_1 = x_2$ , carving out an exchange hole around each particle.

No term in the Hamiltonian produced this. It is purely a consequence of the (anti)symmetry the identical-particle postulate imposes on the wavefunction.

Why it has real, measurable consequences

For electrons the spin and spatial parts are tied together (the next lesson makes this precise), so the exchange force feeds into energy differences between spin configurations. This is the origin of Hund's first rule in atoms (parallel spins lower the energy by keeping electrons apart and reducing their Coulomb repulsion) and of ferromagnetism in solids (the exchange interaction makes aligned spins energetically favourable). The "force" that aligns the spins in a bar magnet is, at bottom, the exchange force — antisymmetry plus the ordinary Coulomb repulsion, with no magnetic force doing the work.

The takeaway

Exchange symmetry correlates the positions of identical particles even with no interaction term in the Hamiltonian: symmetric states bunch, antisymmetric states avoid. The size of the effect is set by the overlap of the single-particle orbitals. When combined with Coulomb repulsion and spin, this "statistical force" explains atomic ground-state spins and the very existence of magnetism.