Identical Particles and Exchange Symmetry

9.1 Symmetrisation Postulate

For a system of $N$ identical particles, the wavefunction must satisfy:

$\psi(\ldots, \mathbf{r}_i, \ldots, \mathbf{r}_j, \ldots) = \pm\psi(\ldots, \mathbf{r}_j, \ldots, \mathbf{r}_i, \ldots)$

• Bosons (integer spin): symmetric ($+$ sign). Any number can occupy the same state.
• Fermions (half-integer spin): antisymmetric ($-$ sign). Pauli exclusion: no two fermions can occupy the same state.

For two particles, the properly symmetrised states are:

$\psi_S = \frac{1}{\sqrt{2}}\left[\psi_a(1)\psi_b(2) + \psi_b(1)\psi_a(2)\right] \quad \text{(bosons)}$

$\psi_A = \frac{1}{\sqrt{2}}\left[\psi_a(1)\psi_b(2) - \psi_b(1)\psi_a(2)\right] \quad \text{(fermions)}$

9.2 Exchange Interaction

Even without an explicit interaction potential, the requirement of (anti)symmetry leads to an effective exchange interaction. For two electrons in a box, the probability of finding them close together differs between the triplet (spatially antisymmetric, spin symmetric) and singlet (spatially symmetric, spin antisymmetric) states:

$|\psi_{\text{triplet}|^2 = 0 \quad \text{when} \mathbf{r}_1 = \mathbf{r}_2}$

$|\psi_{\text{singlet}|^2 > 0 \quad \text{when} \mathbf{r}_1 = \mathbf{r}_2}$

The triplet state keeps electrons apart (effective repulsion), while the singlet allows them to be close. This is the origin of the Hund”s first rule: parallel spins are energetically favourable for atoms because the exchange interaction lowers the Coulomb repulsion.

9.3 The Helium Atom

The helium Hamiltonian (ignoring nuclear motion):

$\hat{H} = -\frac{\hbar^2}{2m_e}\left(\nabla_1^2 + \nabla_2^2\right) - \frac{2e^2}{4\pi\varepsilon_0 r_1} - \frac{2e^2}{4\pi\varepsilon_0 r_2} + \frac{e^2}{4\pi\varepsilon_0|\mathbf{r}_1 - \mathbf{r}_2|}$

Ground state (parahelium): Both electrons in the $1s$ orbital with opposite spins (singlet). The spatial part is symmetric: $\psi_{100}(\mathbf{r}_1)\psi_{100}(\mathbf{r}_2)$.

First-order perturbation theory for the electron-electron repulsion:

$E^{(1)} = \frac{5}{4}\frac{e^2}{4\pi\varepsilon_0 a_0} = \frac{5}{2}\times 13.6\ \text{eV} = 34.0\ \text{eV}$

The unperturbed ground state energy is $E^{(0)} = 2 \times (-54.4\ \text{eV}) = -108.8$ eV (two electrons in $Z = 2$ Coulomb potential). Including perturbation: $E \approx -108.8 + 34.0 = -74.8$ eV. The experimental value is $-79.0$ eV.

Excited states: When one electron is excited to $1s\,nl$The spin configuration matters:

• Parahelium (singlet, $S = 0$): symmetric spatial, antisymmetric spin. Lower energy for given configuration.
• Orthohelium (triplet, $S = 1$): antisymmetric spatial, symmetric spin. Higher energy.

The exchange integral $K$ and direct integral $J$:

$J = \iint |\psi_a(1)|^2\frac{e^2}{4\pi\varepsilon_0 r_{12}}|\psi_b(2)|^2\, d^3r_1 d^3r_2$

$K = \iint \psi_a^*(1)\psi_b^*(2)\frac{e^2}{4\pi\varepsilon_0 r_{12}}\psi_b(1)\psi_a(2)\, d^3r_1 d^3r_2$

The energy splitting between singlet and triplet is $2K$With the triplet lower by $2K$.

9.4 Slater Determinants

For $N$ fermions, the antisymmetric wavefunction is efficiently written as a Slater determinant:

$\Psi(1, 2, \ldots, N) = \frac{1}{\sqrt{N!}}\begin{vmatrix} \phi_1(1) & \phi_2(1) & \cdots & \phi_N(1) \\ \phi_1(2) & \phi_2(2) & \cdots & \phi_N(2) \\ \vdots & \vdots & \ddots & \vdots \\ \phi_1(N) & \phi_2(N) & \cdots & \phi_N(N) \end{vmatrix}$

Properties:

• Swapping any two rows (particles) changes the sign
• If any two columns (orbitals) are identical, the determinant vanishes (Pauli exclusion)
• The normalisation is correct if the spin-orbitals $\phi_i$ are orthonormal