Editing Fock state (section)

==Definition==
One specifies a multiparticle state of ''N'' non-interacting identical particles by writing the state as a sum of [[Tensor product of Hilbert spaces|tensor products]] of ''N'' one-particle states.  Additionally, depending on the integrality of the particles' [[Spin (physics)|spin]], the tensor products must be [[Antisymmetric tensor|alternating]] (anti-symmetric) or [[Symmetric tensor|symmetric products]] of the underlying one-particle [[Hilbert space|Hilbert spaces]].  Specifically:

* [[Fermion]]s, having half-integer spin and obeying the [[Pauli exclusion principle]], correspond to antisymmetric tensor products. 
* [[Boson]]s, possessing integer spin (and not governed by the exclusion principle) correspond to symmetric tensor products.

If the number of particles is variable, one constructs the [[Fock space]] as the [[Direct sum of modules|direct sum]] of the tensor product Hilbert spaces for each [[particle number]]. In the Fock space, it is possible to specify the same state in a new notation, the occupancy number notation, by specifying the number of particles in each possible one-particle state.

Let <math display="inline">\left\{\mathbf{k}_{i}\right\}_{i \in I}</math> be an [[orthonormal basis]] of states in the underlying one-particle Hilbert space. This induces a corresponding basis of the Fock space called the "occupancy number basis". A quantum state in the Fock space is called a '''Fock state''' if it is an element of the occupancy number basis.

A Fock state satisfies an important criterion: for each ''i'', the state is an eigenstate of the [[particle number operator]] <math>\widehat{N_{{\mathbf{k}}_i}}</math> corresponding to the ''i''-th elementary state '''k'''<sub>i</sub>. The corresponding eigenvalue gives the number of particles in the state. This criterion nearly defines the Fock states (one must in addition select a phase factor).

A given Fock state is denoted by <math>|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},..n_{{\mathbf{k}}_i}...\rangle</math>. In this expression, <math>n_{{\mathbf{k}}_i}</math> denotes the number of particles in the i-th state '''k'''<sub>i</sub>, and the particle number operator for the i-th state, <math>\widehat{N_{{\mathbf{k}}_i}}</math>, acts on the Fock state in the following way:

: <math>\widehat{N_{{\mathbf{k}}_i}}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},..n_{{\mathbf{k}}_i}...\rangle = n_{{\mathbf{k}}_i}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},..n_{{\mathbf{k}}_i}...\rangle</math>

Hence the Fock state is an eigenstate of the number operator with eigenvalue <math>n_{{\mathbf{k}}_i}</math>.<ref name="Mandel">{{Cite book
| last = Mandel
| first = Wolf
| title = Optical coherence and quantum optics
| publisher = Cambridge University Press
| date = 1995
| isbn = 0521417112
}}</ref>{{rp|478}}

Fock states often form the most convenient [[basis (linear algebra)|basis]] of a Fock space. Elements of a Fock space that are [[Quantum superposition|superpositions]] of states of differing [[particle number]] (and thus not eigenstates of the number operator) are not Fock states. For this reason, not all elements of a Fock space are referred to as "Fock states".

If we define the aggregate particle number operator <math display="inline">\widehat{N}</math> as
: <math>\widehat{N} = \sum_i \widehat{N_{{\mathbf{k}}_i}},</math>

the definition of Fock state ensures that the [[variance]] of measurement <math>\operatorname{Var}\left(\widehat{N}\right) = 0</math>, i.e., measuring the number of particles in a Fock state always returns a definite value with no fluctuation.