Identical Particles and Exchange Symmetry

Attempt these from memory before reading.

• What happens to the wave function when two identical particles are exchanged?
• What distinguishes a boson from a fermion?
• State the Pauli exclusion principle precisely.
• What is the exchange interaction, and where does it come from?
• Write the ground-state configuration of helium.

Check your answers against the text. Prompt 4 is the one with the most misleading name in the subject: the exchange interaction is not an interaction.

The symmetrisation postulate is usually stated as a rule about wave functions. It is better understood as a statement about what “identical” means.

If two particles are genuinely identical, then exchanging them cannot change anything measurable — not the energy, not the probability density, not any expectation value. Let $\hat{P}$ be the exchange operator. Then $\hat{P}$ and every observable commute, which means $\hat{P}$ maps a state to a physically indistinguishable state. Doing it twice returns the original state, so

$$\hat{P}^2 = I \quad\Longrightarrow\quad \hat{P} = \pm 1 .$$

That is the entire derivation. There is no third option, because $\hat{P}$ is Hermitian and its eigenvalues must square to one. Bosons are the $+1$ eigenstates, fermions the $-1$ eigenstates, and which one a particle is turns out to be fixed by its spin.

The exchange interaction is bookkeeping that behaves like a force. Take two electrons in a box. If their spins form a triplet, the spatial part is antisymmetric and vanishes at $\mathbf{r}_1 = \mathbf{r}_2$; if singlet, it does not. So the two states have different Coulomb energies, purely because the electrons avoid each other in one case and not the other. No interaction term was added. The “exchange energy” in ferromagnetism is exactly this, and it is why magnets exist.

What the Pauli principle actually forbids. The usual phrasing — “no two fermions in the same state” — is ambiguous. Two electrons can share the same spatial wave function, as in helium’s ground state, provided their spins differ. What is forbidden is two fermions in the same complete state, including spin. The precise statement is about the antisymmetry of the total wave function, not about a rule for counting boxes.

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:

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

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:

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

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 $2K$.

Worked Example 9.1: Helium $1s2s$ States

For the $1s\,2s$ configuration of helium:

$$J_{1s,2s} = \frac{e^2}{4\pi\varepsilon_0}\int |\psi_{1s}(1)|^2\frac{1}{r_{12}}|\psi_{2s}(2)|^2\, d^3r_1 d^3r_2$$ $$K_{1s,2s} = \frac{e^2}{4\pi\varepsilon_0}\int \psi_{1s}^*(1)\psi_{2s}^*(2)\frac{1}{r_{12}}\psi_{2s}(1)\psi_{1s}(2)\, d^3r_1 d^3r_2$$

Evaluating these (using the multipole expansion $1/r_{12} = \sum_l r_<^l/r_>^{l+1}\,P_l(\cos\theta)$):

$$J_{1s,2s} \approx 0.42\ \text{Ry} = 5.7\ \text{eV}$$ $$K_{1s,2s} \approx 0.032\ \text{Ry} = 0.43\ \text{eV}$$

The singlet (parahelium) has energy $E = E_0 + J + K$ And the triplet (orthohelium) has $E = E_0 + J - K$.

The splitting: $E_{\text{singlet} - E_{\text{triplet} = 2K \approx 0.86}}$ eV. This is the exchange splitting.

The orthohelium $2^3S$ state is metastable: it cannot decay to the ground state by electric dipole transition (because $\Delta S = 0$ for E1 transitions, and the ground state is a singlet). Its lifetime is $\sim 10^4$ s.

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

• Spin-statistics connection: Particles with integer spin are bosons; half-integer spin are fermions. No exceptions in 3+1 dimensions.
• Exchange energy: $E_{\text{exchange}} = J \mp K$ where the sign depends on the spin configuration. The triplet (parallel spins) has energy $J - K$ and the singlet (antiparallel) has $J + K$.
• Slater determinant size: For $N$ particles each with $M$ available states, the Hilbert space dimension is $\binom{M}{N}$ for fermions versus $M^N$ for bosons.
• Pfaffian for pairs: For an even number of fermions, the antisymmetric state can also be written as a Pfaffian, which is computationally efficient for specific pairing structures.

• Forgetting normalisation: The symmetrisation prefactor $1/\sqrt{2}$ in two-particle states is essential. Without it, the states are not normalised and probability conservation fails.
• Confusing exchange with interaction: The exchange splitting $2K$ arises from symmetry requirements, not from an explicit interaction potential between particles.
• Assuming all particles are fermions or bosons: Composite particles can be either. For example, $^4$ He atoms (2 protons, 2 neutrons, 2 electrons) are bosons, while $^3$ He atoms are fermions.
• Neglecting spin in antisymmetrisation: The full two-particle wavefunction (spatial $\times$ spin) must be antisymmetric for fermions. Using only the spatial part leads to incorrect results.

• Electron gas in metals: The Pauli exclusion principle forces electrons into progressively higher energy states, creating the Fermi sea. This accounts for the electronic specific heat and the stability of matter.
• White dwarf and neutron star stability: Electron degeneracy pressure (from the Pauli principle) supports white dwarfs against gravitational collapse. Neutron degeneracy pressure supports neutron stars.
• Bose-Einstein condensation: Below a critical temperature, a macroscopic fraction of bosons occupies the lowest energy state, producing superfluidity and coherent emission (atom lasers).
• Magnetic ordering: Hund’s rules and exchange interactions determine whether a material is ferromagnetic or antiferromagnetic. The exchange integral $K > 0$ favours parallel alignment (ferromagnetism).

Consider three electrons confined to a one-dimensional box of length $L$. The single-particle energies are $E_n = n^2\pi^2\hbar^2/(2mL^2)$. The lowest configuration has two electrons in $n = 1$ (opposite spins) and one in $n = 2$.

The spatial part of the wavefunction must be antisymmetric under exchange of any two electrons. Using the Slater determinant with orbitals $\phi_1$, $\phi_1$, $\phi_2$ (where $\phi_1$ and $\phi_2$ are the spatial wavefunctions of the box), the antisymmetric state is:

$$\Psi = \frac{1}{\sqrt{3!}}\begin{vmatrix} \phi_1(r_1)\alpha(1) & \phi_1(r_1)\beta(1) & \phi_2(r_1)\alpha(1) \\ \phi_1(r_2)\alpha(2) & \phi_1(r_2)\beta(2) & \phi_2(r_2)\alpha(2) \\ \phi_1(r_3)\alpha(3) & \phi_1(r_3)\beta(3) & \phi_2(r_3)\alpha(3) \end{vmatrix}$$

The total energy is $E_1 + E_1 + E_2 = 6\pi^2\hbar^2/(2mL^2)$. The exchange splitting between the two possible spin configurations (total $S = 1/2$) depends on the exchange integral $K_{12}$ between the $n = 1$ and $n = 2$ states.

Identical particles are truly indistinguishable, not just similar. Swapping two electrons must give the same physics, which forces their wavefunction to be antisymmetric. This creates the exchange interaction, an effective force arising purely from symmetry, not from any physical interaction. The Pauli exclusion principle is a consequence: two fermions cannot occupy the same state because the antisymmetric wavefunction would vanish. Bosons, with symmetric wavefunctions, crowd together instead. This explains why electrons fill up energy levels in atoms, why white dwarfs resist collapse, and why laser light is coherent.

The first two rows are the same fact from different angles, and both are frequently stated wrongly. The exchange interaction is the best example in undergraduate physics of an effect that is real, measurable and large, and yet is not a force.

The last row is the frontier item: in two dimensions, exchange is a braid rather than a permutation, and $\hat{P}$ can have any phase. Anyons are real, they have been observed, and they are the reason the topological quantum computing programme exists.

• Anyons and braiding. In two dimensions the exchange group is the braid group, whose representations allow arbitrary phases. Abelian anyons have been observed in fractional quantum Hall systems; non-Abelian anyons are the proposed resource for topological quantum computing.
• Bose-Einstein condensation. A macroscopic number of bosons in one state. Predicted in 1925, first observed in 1995, now a working tool — and its finite-size and interacting versions are still active.
• Superconductivity. Electrons are fermions and cannot condense; bound pairs are bosons and can. Superconductivity is exchange statistics with a mechanism for making the pairs.
• Entanglement between identical particles. Whether two indistinguishable particles can be said to be “entangled” at all is genuinely contested, since the particles have no individual identities to assign.
• Fractional statistics in optical lattices. Cold-atom experiments now realise statistical behaviour that has no condensed-matter analogue, and the theory of what can be measured is still being written.

Mix these with spin, approximation methods, Linear Algebra, Thermodynamics and Statistical Mechanics and Stochastic Calculus.

1. (Linear algebra.) Show that the exchange operator is Hermitian and that $\hat{P}^2 = I$, and hence that its eigenvalues are $\pm 1$. Then show that any two-particle state can be written as a sum of a symmetric and an antisymmetric part.
2. (Spin.) Show that the triplet states are symmetric and the singlet state is antisymmetric, and hence that the spatial wave function of a two-electron system must be antisymmetric in the triplet. Which combination gives the helium ground state, and why?
3. (Linear algebra.) Write the $N$-electron wave function as a Slater determinant, and show that setting two rows equal makes the determinant vanish. That is the Pauli principle, derived rather than asserted.
4. (Thermal physics.) Show that the Bose-Einstein distribution follows from maximising entropy subject to fixed particle number, and that the Fermi-Dirac distribution follows from the same argument with the occupation of each state capped at one. The difference is exactly the Pauli principle.
5. (Approximation methods.) Use first-order perturbation theory to estimate the ground-state energy of helium, treating the electron-electron repulsion as the perturbation. Compare with experiment and explain what the difference is called.
6. (Research.) In two dimensions, exchange is described by the braid group rather than the symmetric group. Explain why the braid group allows phases other than $\pm 1$, and what that implies about the existence of particles that are neither bosons nor fermions.

• Spin: Spin is an intrinsic angular momentum that determines particle statistics and exchange symmetry.

• Angular Momentum and the Hydrogen Atom: Angular momentum theory provides the foundation for understanding spin statistics.

• Approximation Methods: Perturbation theory and variational methods are essential for solving many-body quantum problems.

• Calculus

• Linear Algebra

• Vector Calculus

• Quantum Computing

• Angular Momentum and the Hydrogen Atom
• Approximation Methods
• Density Functional Theory: Conceptual Overview
• Historical Motivation