Identical Particles

For two distinguishable particles, the wave function is $\psi(\mathbf{r}_1, \mathbf{r}_2) = \psi_a(\mathbf{r}_1) \psi_b(\mathbf{r}_2)$. For identical particles, this is not allowed because we cannot tell which particle is which.

The exchange operator $\hat{P}_{12}$ swaps the coordinates: $\hat{P}_{12} \psi(\mathbf{r}_1, \mathbf{r}_2) = \psi(\mathbf{r}_2, \mathbf{r}_1)$. For identical particles, the Hamiltonian must be symmetric under exchange: $\hat{P}_{12} \hat{H} \hat{P}_{12}^{-1} = \hat{H}$. This implies $[\hat{P}_{12}, \hat{H}] = 0$, so $\hat{P}_{12}$ is conserved.

Since $\hat{P}_{12}^2 = \hat{I}$, the eigenvalues of $\hat{P}_{12}$ are $\pm 1$. Thus the wave function must be either symmetric ($\hat{P}_{12} \psi = +\psi$) or antisymmetric ($\hat{P}_{12} \psi = -\psi$). These are the only possibilities for identical particles.

The spin-statistics theorem (a result of relativistic quantum field theory) states that particles with integer spin ($s = 0, 1, 2, \ldots$) are bosons and have symmetric wave functions. Particles with half-integer spin ($s = 1/2, 3/2, \ldots$) are fermions and have antisymmetric wave functions.

For bosons, any number of particles can occupy the same single-particle state. This leads to Bose-Einstein condensation at low temperatures, where a macroscopic fraction of particles occupies the ground state.

For fermions, the Pauli exclusion principle follows from antisymmetry: if two fermions occupy the same state, $\psi_A = 0$. No two fermions can be in the same quantum state. This explains the structure of atoms, the periodic table, and the stability of matter.

Electrons, protons, and neutrons are all spin-$1/2$ fermions. Photons are spin-$1$ bosons. The difference between bosons and fermions is one of the most profound consequences of quantum mechanics.

The helium atom provides a textbook example of the effects of identical particles and exchange symmetry. The Hamiltonian for helium (ignoring spin-orbit coupling) is:

where the last term is the electron-electron repulsion. This repulsion prevents an exact analytic solution.

In the independent-particle approximation, we ignore the electron-electron repulsion. Each electron moves in the nuclear Coulomb field, giving hydrogen-like orbitals. The ground state has both electrons in the $1s$ orbital, with the antisymmetric spin singlet.

The spatial wave function is symmetric: $\psi_S = \psi_{100}(\mathbf{r}_1) \psi_{100}(\mathbf{r}_2)$. The spin state must be antisymmetric (singlet) because the total wave function must be antisymmetric for fermions.

When the spatial wave function is symmetric (triplet spin), the electrons tend to be closer together on average than when it is antisymmetric (singlet spin). Wait — actually, for fermions, the antisymmetric spatial wave function gives $\psi = 0$ when $\mathbf{r}_1 = \mathbf{r}_2$, so the electrons avoid each other.

The energy difference between the symmetric spatial state (triplet spin) and antisymmetric spatial state (singlet spin) is called the exchange splitting. This is not a real force but a consequence of the symmetry requirement.

In helium, the triplet states (parahelium) lie lower in energy than the singlet states (orthohelium) when electron-electron repulsion is included, because the antisymmetric spatial state keeps the electrons apart, reducing the Coulomb repulsion.

The exchange interaction is the origin of ferromagnetism in solids, the hyperfine structure of atomic spectra, and many other phenomena in condensed matter and atomic physics.

Two identical spin-$1/2$ particles are in a 1D infinite square well of width $a$. Particle 1 is in the $n=1$ state and particle 2 is in the $n=2$ state. Write the total wave function for both bosons and fermions.

Step 1 — Single-particle states. $\phi_1(x) = \sqrt{2/a} \sin(\pi x/a)$, $\phi_2(x) = \sqrt{2/a} \sin(2\pi x/a)$.

Step 2 — Bosons (spin-$0$ or spin-$1$, symmetric spatial). $\psi_S(x_1, x_2) = (1/\sqrt{2})[\phi_1(x_1)\phi_2(x_2) + \phi_2(x_1)\phi_1(x_2)]$. The spin state can be anything (symmetric for integer spin).

Step 3 — Fermions (spin-$1/2$, antisymmetric total). The spatial part can be symmetric with antisymmetric spin (singlet) or antisymmetric spatial with symmetric spin (triplet).

Case A: Symmetric spatial + singlet spin. $\psi = \psi_S(x_1,x_2) \times (|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle)/\sqrt{2}$.

Case B: Antisymmetric spatial + triplet spin. $\psi_A(x_1, x_2) = (1/\sqrt{2})[\phi_1(x_1)\phi_2(x_2) - \phi_2(x_1)\phi_1(x_2)]$. The spin state can be any of the three triplet states.

Step 4 — Ground state for two fermions. Both in $n=1$, symmetric spatial. Must have antisymmetric spin (singlet). $\psi = \phi_1(x_1) \phi_1(x_2) \times \chi_{\text{singlet}}$. Energy = $2E_1 = \pi^2 \hbar^2 / (ma^2)$.