Identical particles and the exchange operator

Write ψ(r₁, r₂) and you have already done something nature did not: you named one electron 1 and the other 2. No apparatus returns that name. An electron carries no serial number, and a detector reports a click at a place, not an identity. The formal version of that complaint is an operator: P̂₁₂|α⟩₁|β⟩₂ = |β⟩₁|α⟩₂, or in the position representation (P̂₁₂ψ)(r₁, r₂) = ψ(r₂, r₁). Notice what it swaps. It exchanges the two slots of the tensor product H ⊗ H, not the coordinates inside one slot, and it is defined by its action on product states and then extended linearly to every superposition. Notice also what it does not require. The swap map can be written down for any two particles whose one-particle spaces are the same space — an electron and a proton included, both living in L²(R³) ⊗ C². What makes a pair identical is the extra fact that Ĥ is unchanged by the swap, and only then does the swap commute with the dynamics. So the opening move of this topic is to convert a philosophical worry about names into a linear operator you can diagonalise beside the Hamiltonian.

Two properties do all the work. Hermiticity follows from relabelling the dummy integration variables in ⟨Φ|P̂₁₂Ψ⟩ = ∫∫ Φ*(r₁, r₂) ψ(r₂, r₁) d³r₁ d³r₂, so P̂₁₂† = P̂₁₂ and its eigenvalues are real. Involution is even cheaper: swapping the slots twice restores them, so P̂₁₂² = Î exactly, which forces λ² = 1 and hence λ = ±1. The two facts together make P̂₁₂ unitary as well, since P̂₁₂†P̂₁₂ = Î. Now take orthonormal orbitals α ≠ β and restrict to the two-dimensional span of |αβ⟩ and |βα⟩. In that ordered basis P̂₁₂ carries zeros on the diagonal and ones off it — it is σₓ, the same matrix you diagonalised for spin along the x axis. Its eigenvectors are therefore the familiar pair (|αβ⟩ ± |βα⟩)/√2, and the projectors onto the two sectors are Ŝ₊ = ½(Î + P̂₁₂) and Ŝ₋ = ½(Î − P̂₁₂), built exactly like ½(Î ± σₓ), with Ŝ₊ + Ŝ₋ = Î and Ŝ₊Ŝ₋ = 0.

For identical particles Ĥ is symmetric under the swap: the kinetic term is a sum over particles, an external potential is the same function for each, and a pair interaction written in |r₁ − r₂| is untouched by relabelling. So [Ĥ, P̂₁₂] = 0, and Ehrenfest's theorem gives d⟨P̂₁₂⟩/dt = 0. That is genuinely useful — exchange parity is a good quantum number, and a state prepared antisymmetric stays antisymmetric under any symmetric interaction, however strong. But read carefully what it does not say. Take |Ψ⟩ = (3|αβ⟩ − |βα⟩)/√10, a perfectly legitimate vector in the two-particle space. Its symmetric weight is ‖Ŝ₊|Ψ⟩‖² = 0.20, its antisymmetric weight 0.80, and ⟨P̂₁₂⟩ = 0.20 − 0.80 = −0.60. Every one of those numbers is frozen for all time by the very commutator that was supposed to help. Conservation preserves what it is handed; it never manufactures an eigenstate. If nature contains no such state, something outside the Schrödinger equation has to say so.

With N = 2 there is one transposition and the story is a two-level system. With N ≥ 3 the permutation group S_N is non-abelian, and the postulate changes shape. Write |αβγ⟩ for particle 1 in α, 2 in β, 3 in γ. Then P̂₁₂P̂₂₃|αβγ⟩ = P̂₁₂|αγβ⟩ = |γαβ⟩, while P̂₂₃P̂₁₂|αβγ⟩ = P̂₂₃|βαγ⟩ = |βγα⟩. Those are different states, so [P̂₁₂, P̂₂₃] ≠ 0 and no basis diagonalises every transposition at once. Demanding one simultaneous ±1 eigenvalue for all of them is therefore a real restriction, not bookkeeping: it can be met only inside a one-dimensional irreducible representation of S_N, and S_N has exactly two, the trivial one and the sign one. For three distinct orbitals the six orderings span six dimensions, decomposing as 6 = 1 + 1 + 2 × 2 — one totally symmetric state, one totally antisymmetric state, and a two-dimensional mixed-symmetry representation appearing twice. The postulate keeps two of those six dimensions and throws away four.

Non-relativistic quantum mechanics cannot derive the choice; relativistic quantum field theory can. Pauli's spin-statistics theorem shows that Lorentz invariance, a Hamiltonian bounded below, and microcausality — local observables commuting at spacelike separation — force integer-spin fields to be quantised with commutators and half-integer-spin fields with anticommutators, which is precisely the symmetric and the antisymmetric sector. A composite counts by its constituents, as long as nothing probes its internal structure: helium-4 carries 2 protons, 2 neutrons and 2 electrons, six fermions in all, so it is a boson and goes superfluid at 2.17 K, while helium-3 carries 2 protons, 1 neutron and 2 electrons, five fermions, so it is a fermion and must pair before it can flow — which it manages only in the millikelvin range, at 0.93 mK on the saturated vapour curve and 2.5 mK at the melting pressure. The postulate also has a boundary. In two dimensions the configuration space of two identical particles is not simply connected, exchange is a braid carrying a winding number rather than a permutation, and the exchange phase may be any e^(iθ). Those are anyons — not a loophole in the algebra, a different topology.

Symmetrising costs nothing when the one-particle states do not overlap, because every cross term carries an overlap integral that has already vanished — two electrons in two different laboratories may be labelled without error. The scale that decides is the thermal de Broglie wavelength Λ = h/√(2πmk_BT) set against the mean spacing n^(−1/3). Argon at 300 K and one atmosphere has Λ = 0.016 nm against a spacing of 3.4 nm, so nΛ³ ≈ 1 × 10⁻⁷ and Maxwell–Boltzmann counting is safe. Liquid helium-4 has a spacing of 0.36 nm and Λ = 0.59 nm at 2.17 K, giving nΛ³ = 4.5; an ideal Bose gas at that density would condense at 3.13 K, against the measured λ-point of 2.17 K. Electrons at 300 K have Λ = 4.3 nm, some seventeen times the 0.25 nm spacing of atoms in a metal — which is why conduction electrons are degenerate at room temperature, and why two electrons on one atom can never be treated as distinguishable.