Two billiard balls painted alike can still be told apart: follow each along its path, and the one that ends in the corner pocket is the one that started on the left. Quantum particles have no paths to follow. Once the wavefunctions of two electrons overlap, there is no fact of the matter about which is which, and electrons are identical in every property — mass, charge, spin. So the state with the first particle at $x_{1}$ and the second at $x_{2}$ must describe the same physics as the state with them swapped. The exchange operator $\hat{P}_{12}$ swaps the labels. It squares to one, so its eigenvalues are $+1$ and $-1$, and because the Hamiltonian of identical particles treats them alike, it commutes with the Hamiltonian: a state that is symmetric or antisymmetric stays so for ever.

Two identical coins in a bag: after a shake, it makes no sense to ask whether the coin on the left is the one you put in first. For classical coins the question is merely unanswerable in practice; for electrons it has no answer at all.

The plane of the two positions, $x_{1}$ across and $x_{2}$ up. Exchanging the particles reflects a point in the diagonal $x_{1}=x_{2}$. For identical particles the configurations and are one and the same, so takes the same value at both.

For two identical particles, with $x$ standing for a position and a spin component together, the exchange operator swaps their labels:

• $\hat{P}_{12}$ is Hermitian and unitary, with the eigenvalues $+1$, the symmetric states, and $-1$, the antisymmetric states.
• The Hamiltonian of identical particles is unchanged by swapping them; if it were not, the particles could be told apart by how they move.

Swapping twice changes nothing, so the operator squares to one. Renaming the variables of integration shows that it is Hermitian. An eigenvalue must then square to one, and since the Hamiltonian is symmetric the operator commutes with it and the symmetry is conserved.

Swap, and swap back.

Rename $x_{1}\leftrightarrow x_{2}$ in the integral.

A Hermitian operator has real eigenvalues, and these square to one.

Then $\hat{P}_{12}$ is conserved, by the equation of motion of the dynamics chapter.

Classical particles can be told apart by following their paths. Quantum particles have none, so identical ones cannot be tracked once their wavefunctions overlap.

For identical particles, a state and the same state with the labels swapped describe the same physics.

The exchange operator swaps the labels of two particles. Swapping twice changes nothing.

The exchange operator squares to the identity, so its eigenvalues are $+1$ and $-1$.

Great work! You now know how to:

• explain why identical quantum particles cannot be told apart
• define the exchange operator and find its eigenvalues
• show that the symmetry under exchange is conserved