Editing Indistinguishable particles (section)

== Quantum mechanical description ==

=== Symmetrical and antisymmetrical states ===
[[Image:Asymmetricwave2.png|right|thumb|Antisymmetric wavefunction for a (fermionic) 2-particle state in an infinite square well potential]]
[[Image:Symmetricwave2.png|right|thumb|Symmetric wavefunction for a (bosonic) 2-particle state in an infinite square well potential]]

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 ''n''<sub>1</sub>, and the other is in the state ''n''<sub>2</sub>. The quantum state of the system is denoted by the expression
: <math> | n_1 \rang | n_2 \rang </math>
where the order of the tensor product matters ( if <math> | n_2 \rang | n_1 \rang </math>, then the particle 1 occupies the state ''n''<sub>2</sub> while the particle 2 occupies the state ''n''<sub>1</sub>). This is the canonical way of constructing a basis for a [[tensor product]] space <math>H \otimes H</math> of the combined system from the individual spaces. This expression is valid for distinguishable particles, however, it is not appropriate for indistinguishable particles since <math> |n_1\rang |n_2\rang</math> and <math>|n_2\rang |n_1\rang </math> as a result of exchanging the particles are generally different states.
* "the particle 1 occupies the ''n''<sub>1</sub> state and the particle 2 occupies the ''n''<sub>2</sub> state" ≠ "the particle 1 occupies the ''n''<sub>2</sub> state and the particle 2 occupies the ''n''<sub>1</sub> 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:<ref>Haynes, P. [http://www.tcm.phy.cam.ac.uk/~pdh1001/thesis/node14.html Linear-scaling methods in ab initio quantum-mechanical calculations]. Diss. University of Cambridge, 1998. Section 2.3 Identical particles</ref><ref>{{harvtxt|Tuckerman|2010|p=385}}</ref><ref>{{Cite book|title=Introductory Quantum Mechanics|last=Liboff|first=Richard|publisher=Addison-Wesley|year=2003|isbn=978-0805387148|pages=597}}</ref>
: <math> |n_1\rang |n_2\rang \pm |n_2\rang |n_1\rang </math>

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
: <math> |n_1, n_2; S\rang \equiv \mbox{constant} \times \bigg( |n_1\rang |n_2\rang + |n_2\rang |n_1\rang \bigg) </math>
while antisymmetric states have the form
: <math> |n_1, n_2; A\rang \equiv \mbox{constant} \times \bigg( |n_1\rang |n_2\rang - |n_2\rang |n_1\rang \bigg) </math>

Note that if ''n''<sub>1</sub> and ''n''<sub>2</sub> are the same, the antisymmetric expression gives zero, which cannot be a state vector since it cannot be normalized. In other words, more than one identical particle cannot occupy an antisymmetric state (one antisymmetric state can be occupied only by one particle). This is known as the [[Pauli exclusion principle]], and it is the fundamental reason behind the [[chemistry|chemical]] properties of atoms and the stability of [[matter]].

=== 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]].

=== Fermions and bosons ===

The choice of symmetry or antisymmetry is determined by the species of particle. For example, symmetric states must always be used when describing [[photon]]s or [[helium-4]] atoms, and antisymmetric states when describing [[electron]]s or [[proton]]s.

Particles which exhibit symmetric states are called [[boson]]s. The nature of symmetric states has important consequences for the statistical properties of systems composed of many identical bosons. These statistical properties are described as [[Bose–Einstein statistics]].

Particles which exhibit antisymmetric states are called [[fermion]]s. Antisymmetry gives rise to the [[Pauli exclusion principle]], which forbids identical fermions from sharing the same quantum state. Systems of many identical fermions are described by [[Fermi–Dirac statistics]].

[[Parastatistics]] are mathematically possible, but no examples exist in nature.<ref>{{Cite journal |last1=Baker |first1=David John |last2=Halvorson |first2=Hans |last3=Swanson |first3=Noel |date=2015-12-01 |title=The Conventionality of Parastatistics |url=https://www.journals.uchicago.edu/doi/10.1093/bjps/axu018 |journal=The British Journal for the Philosophy of Science |language=en |volume=66 |issue=4 |pages=929–976 |doi=10.1093/bjps/axu018 |issn=0007-0882}}</ref>

In certain two-dimensional systems, mixed symmetry can occur. These exotic particles are known as [[anyon]]s, and they obey [[fractional statistics]]. Experimental evidence for the existence of anyons exists in the [[quantum Hall effect|fractional quantum Hall effect]], a phenomenon observed in the two-dimensional electron gases that form the inversion layer of [[MOSFET]]s. There is another type of statistic, known as [[braid statistics]], which are associated with particles known as [[plekton]]s.

The [[spin-statistics theorem]] relates the exchange symmetry of identical particles to their [[Spin (physics)|spin]]. It states that bosons have integer spin, and fermions have half-integer spin. Anyons possess fractional spin.

=== ''N'' particles ===

The above discussion generalizes readily to the case of ''N'' particles. Suppose there are ''N'' particles with quantum numbers ''n''<sub>1</sub>, ''n''<sub>2</sub>, ..., ''n''<sub>''N''</sub>. If the particles are bosons, they occupy a '''totally symmetric state''', which is symmetric under the exchange of ''any two'' particle labels:
: <math>|n_1 n_2 \cdots n_N; S\rang = \sqrt{\frac{\prod_n m_n!}{N!}} \sum_p \left|n_{p(1)}\right\rang \left|n_{p(2)}\right\rang \cdots \left|n_{p(N)}\right\rang </math>

Here, the sum is taken over all different states under [[permutation]]s ''p'' acting on ''N'' elements. The square root left to the sum is a [[normalizing constant]]. The quantity ''m<sub>n</sub>'' stands for the number of times each of the single-particle states ''n'' appears in the ''N''-particle state. Note that {{nowrap|1=Σ<sub>''n''</sub> ''m''<sub>''n''</sub> = ''N''}}.

In the same vein, fermions occupy '''totally antisymmetric states''':
: <math>|n_1 n_2 \cdots n_N; A\rang = \frac{1}{\sqrt{N!}} \sum_p \operatorname{sgn}(p) \left|n_{p(1)}\right\rang \left|n_{p(2)}\right\rang \cdots \left|n_{p(N)}\right\rang\ </math>

Here, {{math|sgn(''p'')}} is the [[parity of a permutation|sign]] of each permutation (i.e. <math>+1</math> if <math>p</math> is composed of an even number of transpositions, and <math>-1</math> if odd). Note that there is no <math>\Pi_n m_n</math> term, because each single-particle state can appear only once in a fermionic state. Otherwise the sum would again be zero due to the antisymmetry, thus representing a physically impossible state. This is the [[Pauli exclusion principle]] for many particles.

These states have been normalized so that
: <math> \lang n_1 n_2 \cdots n_N; S | n_1 n_2 \cdots n_N; S\rang = 1, \qquad \lang n_1 n_2 \cdots n_N; A | n_1 n_2 \cdots n_N; A\rang = 1. </math>

=== Measurement ===

Suppose there is a system of ''N'' bosons (fermions) in the symmetric (antisymmetric) state
: <math>|n_1 n_2 \cdots n_N; S/A \rang</math>
and a measurement is performed on some other set of discrete observables, ''m''. In general, this yields some result ''m''<sub>1</sub> for one particle, ''m''<sub>2</sub> for another particle, and so forth. If the particles are bosons (fermions), the state after the measurement must remain symmetric (antisymmetric), i.e.
: <math>|m_1 m_2 \cdots m_N; S/A \rang</math>

The probability of obtaining a particular result for the ''m'' measurement is
: <math>P_{S/A}\left(n_1, \ldots, n_N \rightarrow m_1, \ldots, m_N\right) \equiv \big|\left\lang m_1 \cdots m_N; S/A \,|\, n_1 \cdots n_N; S/A \right\rang \big|^2 </math>

It can be shown that
: <math>\sum_{m_1 \le m_2 \le \dots \le m_N} P_{S/A}(n_1, \ldots, n_N \rightarrow m_1, \ldots, m_N) = 1</math>
which verifies that the total probability is 1. The sum has to be restricted to ''ordered'' values of ''m''<sub>1</sub>, ..., ''m<sub>N</sub>'' to ensure that each multi-particle state is not counted more than once.

=== Wavefunction representation ===

So far, the discussion has included only discrete observables. It can be extended to continuous observables, such as the [[position (vector)|position]]&nbsp;''x''.

Recall that an eigenstate of a continuous observable represents an infinitesimal ''range'' of values of the observable, not a single value as with discrete observables. For instance, if a particle is in a state |''ψ''⟩, the probability of finding it in a region of volume ''d''<sup>3</sup>''x'' surrounding some position ''x'' is
: <math> |\lang x | \psi \rang|^2 \; d^3 x </math>

As a result, the continuous eigenstates |''x''⟩ are normalized to the [[dirac delta function|delta function]] instead of unity:
: <math> \lang x | x' \rang = \delta^3 (x - x') </math>

Symmetric and antisymmetric multi-particle states can be constructed from continuous eigenstates in the same way as before. However, it is customary to use a different normalizing constant:
: <math>\begin{align}
  |x_1 x_2 \cdots x_N; S\rang &= \sqrt{\frac{\prod_j n_j!}{N!}} \sum_p \left|x_{p(1)}\right\rang \left|x_{p(2)}\right\rang \cdots \left|x_{p(N)}\right\rang \\
  |x_1 x_2 \cdots x_N; A\rang &= \frac{1}{\sqrt{N!}} \sum_p \mathrm{sgn}(p) \left|x_{p(1)}\right\rang \left|x_{p(2)}\right\rang \cdots \left|x_{p(N)}\right\rang
\end{align}</math>

A many-body [[wavefunction]] can be written,
: <math>\begin{align}
  \Psi^{(S)}_{n_1 n_2 \cdots n_N} (x_1, x_2, \ldots, x_N)
    & \equiv \lang x_1 x_2 \cdots x_N; S | n_1 n_2 \cdots n_N; S \rang \\[4pt]
    & = \sqrt{\frac{\prod_j n_j!}{N!}} \sum_p \psi_{p(1)}(x_1) \psi_{p(2)}(x_2) \cdots \psi_{p(N)}(x_N) \\[10pt]
  \Psi^{(A)}_{n_1 n_2 \cdots n_N} (x_1, x_2, \ldots, x_N)
    & \equiv \lang x_1 x_2 \cdots x_N; A | n_1 n_2 \cdots n_N; A \rang \\[4pt]
    & = \frac{1}{\sqrt{N!}} \sum_p \mathrm{sgn}(p) \psi_{p(1)}(x_1) \psi_{p(2)}(x_2) \cdots \psi_{p(N)}(x_N)
\end{align}</math>
where the single-particle wavefunctions are defined, as usual, by
: <math>\psi_n(x) \equiv \lang x | n \rang </math>

The most important property of these wavefunctions is that exchanging any two of the coordinate variables changes the wavefunction by only a plus or minus sign. This is the manifestation of symmetry and antisymmetry in the wavefunction representation:
: <math>\begin{align}
  \Psi^{(S)}_{n_1 \cdots n_N} (\cdots x_i \cdots x_j\cdots) =
     \Psi^{(S)}_{n_1 \cdots n_N} (\cdots x_j \cdots x_i \cdots) \\[3pt]
  \Psi^{(A)}_{n_1 \cdots n_N} (\cdots x_i \cdots x_j\cdots) =
    -\Psi^{(A)}_{n_1 \cdots n_N} (\cdots x_j \cdots x_i \cdots)
\end{align}</math>

The many-body wavefunction has the following significance: if the system is initially in a state with quantum numbers ''n''<sub>1</sub>, ..., n<sub>N</sub>, and a position measurement is performed, the probability of finding particles in infinitesimal volumes near ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''N''</sub> is
: <math> N! \; \left|\Psi^{(S/A)}_{n_1 n_2 \cdots n_N} (x_1, x_2, \ldots, x_N) \right|^2 \; d^{3N}\!x </math>

The factor of ''N''! comes from our normalizing constant, which has been chosen so that, by analogy with single-particle wavefunctions,
: <math> \int\!\int\!\cdots\!\int\; \left|\Psi^{(S/A)}_{n_1 n_2 \cdots n_N} (x_1, x_2, \ldots, x_N)\right|^2 d^3\!x_1 d^3\!x_2 \cdots d^3\!x_N = 1 </math>

Because each integral runs over all possible values of ''x'', each multi-particle state appears ''N''! times in the integral. In other words, the probability associated with each event is evenly distributed across ''N''! equivalent points in the integral space. Because it is usually more convenient to work with unrestricted integrals than restricted ones, the normalizing constant has been chosen to reflect this.

Finally, antisymmetric wavefunction can be written as the [[determinant]] of a [[Matrix (mathematics)|matrix]], known as a [[Slater determinant]]:
: <math>\Psi^{(A)}_{n_1 \cdots n_N} (x_1, \ldots, x_N) =
  \frac{1}{\sqrt{N!}} \left|
  \begin{matrix}
    \psi_{n_1}(x_1) & \psi_{n_1}(x_2) & \cdots & \psi_{n_1}(x_N) \\
    \psi_{n_2}(x_1) & \psi_{n_2}(x_2) & \cdots & \psi_{n_2}(x_N) \\
             \vdots &          \vdots & \ddots &          \vdots \\
    \psi_{n_N}(x_1) & \psi_{n_N}(x_2) & \cdots & \psi_{n_N}(x_N) \\
  \end{matrix}
  \right|
</math>

=== Operator approach and parastatistics ===

The Hilbert space for <math>n</math> particles is given by the tensor product <math display="inline"> \bigotimes_n H </math>. The permutation group of <math> S_n </math> acts on this space by permuting the entries. By definition the expectation values for an observable <math>a</math> of <math>n</math> indistinguishable particles should be invariant under these permutation. This means that for all <math> \psi \in H </math> and <math> \sigma \in S_n </math>
: <math> (\sigma \Psi )^t a  (\sigma \Psi)   = \Psi^t a \Psi,</math>
or equivalently for each <math> \sigma \in S_n </math>
: <math> \sigma^t a \sigma = a </math>.
Two states are equivalent whenever their expectation values coincide for all observables. If we restrict to observables of <math>n </math> identical particles, and hence observables satisfying the equation above, we find that the following states (after normalization) are equivalent
: <math> \Psi \sim  \sum_{\sigma \in S_n} \lambda_{\sigma} \sigma \Psi</math>.
The equivalence classes are in [[bijective relation]] with irreducible subspaces of <math display="inline"> \bigotimes_n H </math> under <math> S_n </math>.

Two obvious irreducible subspaces are the one dimensional symmetric/bosonic subspace and anti-symmetric/fermionic subspace. There are however more types of irreducible subspaces. States associated with these other irreducible subspaces are called [[Parastatistics|parastatistic states]].<ref>{{Cite journal|last=Bach|first=Alexaner|date=1993|title=Classification of Indistinguishable Particles|journal=[[Europhysics Letters]]|volume=21|issue=5|pages=515–520|doi=10.1209/0295-5075/21/5/002|bibcode=1993EL.....21..515B|s2cid=250835341 }}</ref> [[Young tableau#Applications in representation theory|Young tableaux]] provide a way to classify all of these irreducible subspaces.