Identical particles

In classical physics, it is possible to distinguish individual particles in a system, even if they have the same mechanical properties. One might paint each particle a unique color to distinguish it from the rest, or track the trajectory of each particle.

This does not work for elementary particles, which are truly identical. Every particle of the same species has the same physical property, and track each particle necessarily disturbs the system by the laws of quantum mechanics.

Consider a system with two particles. If the state vector of particle 1 is ψ and the state vector of particle 2 is ψ′, the state of the combined system is denoted

|ψ ψ′>

If the particles have the same species, they occupy identical Hilbert spaces. The probability amplitude for |ψψ′> and |ψ′ ψ> to collapse to an arbitrary state |φ> must be equal:

|<φ|ψψ′>|² = |<φ|ψ′ψ>|²

This is possible provided the permutation operation simply introduces a phase:

|ψψ′> = e^iπ |ψ′ψ>

However, two permutations are the identity, so we require e^2iπ = 1. Then either

|ψψ′> = + |ψ′ψ>

which is called a totally symmetric state, or

|ψψ′> = - |ψ′ψ>

which is called a totally antisymmetric state.

It is an empirical fact that states in Nature are either totally symmetric or totally antisymmetric, with one caveat: in two space dimensions, a technical loophole in the above argument allows states with mixed symmetry, giving rise to [fractional statistics]?.

Particles that produce totally antisymmetric states are called fermions. The Pauli exclusion principle forbids identical fermions from occupying a single quantum state. Fermions obey [[Pauli-Dirac statistics]].

Particles that produce totally symmetric states are called bosons. Bosons can and do occupy the same quantum state, giving rise to [Bose-Einstein statistics]?. This "clumping" gives rise to such varied phenomena as the laser, [Bose-Einstein condensation]?, and superfluidity?.

According to the [spin-statistics theorem]?, particles with integer spin are bosons, and particles with half-integer spin are fermions. Particles obeying [fractional statistics]? are known as anyons, and possess fractional spin.