Editing Indistinguishable particles (section)

=== Exchange symmetry ===
{{anchor|Exchange symmetry}}
The importance of symmetric and antisymmetric states is ultimately based on empirical evidence. It appears to be a fact of nature that identical particles do not occupy states of a mixed symmetry, such as
: <math> |n_1, n_2; ?\rang = \mbox{constant} \times \bigg( |n_1\rang |n_2\rang + i |n_2\rang |n_1\rang \bigg) </math>

There is actually an exception to this rule, which will be discussed later. On the other hand, it can be shown that the symmetric and antisymmetric states are in a sense special, by examining a particular symmetry of the multiple-particle states known as '''exchange symmetry'''.

Define a linear operator ''P'', called the exchange operator. When it acts on a tensor product of two state vectors, it exchanges the values of the state vectors:
: <math>P \bigg(|\psi\rang |\phi\rang \bigg) \equiv |\phi\rang |\psi\rang </math>

''P'' is both [[Hermitian operator|Hermitian]] and [[Unitary operator|unitary]]. Because it is unitary, it can be regarded as a [[symmetry (physics)|symmetry operator]]. This symmetry may be described as the symmetry under the exchange of labels attached to the particles (i.e., to the single-particle Hilbert spaces).

Clearly, <math>P^2 = 1</math> (the identity operator), so the [[eigenvalue]]s of ''P'' are +1 and &minus;1. The corresponding [[eigenvector]]s are the symmetric and antisymmetric states:
: <math>P|n_1, n_2; S\rang = + |n_1, n_2; S\rang</math>
: <math>P|n_1, n_2; A\rang = - |n_1, n_2; A\rang</math>

In other words, symmetric and antisymmetric states are essentially unchanged under the exchange of particle labels: they are only multiplied by a factor of +1 or &minus;1, rather than being "rotated" somewhere else in the Hilbert space. This indicates that the particle labels have no physical meaning, in agreement with the earlier discussion on indistinguishability.

It will be recalled that ''P'' is Hermitian. As a result, it can be regarded as an observable of the system, which means that, in principle, a measurement can be performed to find out if a state is symmetric or antisymmetric. Furthermore, the equivalence of the particles indicates that the [[Hamiltonian (quantum mechanics)|Hamiltonian]] can be written in a symmetrical form, such as
: <math>H = \frac{p_1^2}{2m} + \frac{p_2^2}{2m} + U(|x_1 - x_2|) + V(x_1) + V(x_2) </math>

It is possible to show that such Hamiltonians satisfy the [[Commutator|commutation relation]]
: <math>\left[P, H\right] = 0</math>

According to the [[Heisenberg picture|Heisenberg equation]], this means that the value of ''P'' is a constant of motion. If the quantum state is initially symmetric (antisymmetric), it will remain symmetric (antisymmetric) as the system evolves. Mathematically, this says that the state vector is confined to one of the two eigenspaces of ''P'', and is not allowed to range over the entire Hilbert space. Thus, that eigenspace might as well be treated as the actual Hilbert space of the system. This is the idea behind the definition of [[Fock space]].