In quantum mechanics, indistinguishable particles (also called identical or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, elementary particles (such as electrons), composite subatomic particles (such as atomic nuclei), as well as atoms and molecules. Although all known indistinguishable particles only exist at the quantum scale, there is no exhaustive list of all possible sorts of particles nor a clear-cut limit of applicability, as explored in quantum statistics. They were first discussed by Werner Heisenberg and Paul Dirac in 1926. There are two main categories of identical particles: bosons, which can share quantum states, and fermions, which cannot (as described by the Pauli exclusion principle). Examples of bosons are photons, gluons, phonons, helium-4 nuclei and all mesons. Examples of fermions are electrons, neutrinos, quarks, protons, neutrons, and helium-3 nuclei. The fact that particles can be identical has important consequences in statistical mechanics, where calculations rely on probabilistic arguments, which are sensitive to whether or not the objects being studied are identical. As a result, identical particles exhibit markedly different statistical behaviour from distinguishable particles. For example, the indistinguishability of particles has been proposed as a solution to Gibbs' mixing paradox.
Distinguishing between particles There are two methods for distinguishing between particles. The first method relies on differences in the intrinsic physical properties of the particles, such as mass, electric charge, and spin. If differences exist, it is possible to distinguish between the particles by measuring the relevant properties. If microscopic particles with equivalent physical properties were described by classical physics, a second method for distinguishing between particles is to track the trajectory of each particle. As long as the position of each particle can be measured, even when the particles collide, then there would be no ambiguity about which particle is which. However, microscopic identical particles are described by principles of quantum mechanics. According to quantum theory, the particles do not possess definite positions. The particles are then said to be indistinguishable.
Quantum mechanical description
Symmetrical and antisymmetrical states
What follows is an example to make the above discussion concrete, using the formalism developed in the article on the mathematical formulation of quantum mechanics. Let n denote a complete set of (discrete) quantum numbers for specifying single-particle states (for example, for the particle in a box problem, take n to be the quantized wave vector of the wavefunction.) For simplicity, consider a system composed of two particles that are not interacting with each other. Suppose that one particle is in the state n1, and the other is in the state n2. The quantum state of the system is denoted by the expression
| n 1 ⟩ | n 2 ⟩ {\displaystyle |n_{1}\rangle |n_{2}\rangle }
where the order of the tensor product matters ( if | n 2 ⟩ | n 1 ⟩ {\displaystyle |n_{2}\rangle |n_{1}\rangle } , then the particle 1 occupies the state n2 while the particle 2 occupies the state n1). This is the canonical way of constructing a basis for a tensor product space H ⊗ H {\displaystyle H\otimes H} of the combined system from the individual spaces. This expression is valid for distinguishable particles, however, it is not appropriate for indistinguishable particles since | n 1 ⟩ | n 2 ⟩ {\displaystyle |n_{1}\rangle |n_{2}\rangle } and | n 2 ⟩ | n 1 ⟩ {\displaystyle |n_{2}\rangle |n_{1}\rangle } as a result of exchanging the particles are generally different states.
"the particle 1 occupies the n1 state and the particle 2 occupies the n2 state" ≠ "the particle 1 occupies the n2 state and the particle 2 occupies the n1 state". Two states are physically equivalent only if they differ at most by a complex phase factor. For two indistinguishable particles, a state before the particle exchange must be physically equivalent to the state after the exchange, so these two states differ at most by a complex phase factor. This fact suggests that a state for two indistinguishable (and non-interacting) particles is given by following two possibilities:
| n 1 ⟩ | n 2 ⟩ ± | n 2 ⟩ | n 1 ⟩ {\displaystyle |n_{1}\rangle |n_{2}\rangle \pm |n_{2}\rangle |n_{1}\rangle }
States where it is a sum are known as symmetric, while states involving the difference are called antisymmetric. More completely, symmetric states have the form
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