In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a welldefined number of particles (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play important role in second quantization formulation of quantum mechanics.
The particle representation was first treated in detail by Paul Dirac for bosons and by Jordan and Wigner for fermions. ^{[1]}^{:35}
Definition
Instead of specifying a multiparticle state of N noninteracting particles by writing the state as a tensor product of N oneparticle states, it is possible to specify the same state in a new notation, the Fock space representation, by specifying the number of particles in each possible oneparticle state.
However, this notation loses the ordering of tensor products, which is an important part of the specification of quantum states. To retain the same information in the multiparticle state, one constructs Fock space as the direct sum of Hilbert spaces for different particle numbers.
A quantum state is called a Fock state if it satisfies two criteria:
(i) the state belongs to a Fock space.
(ii) the state is an eigenstate of the particle number operator.
The particle number operator operating on a Fock state gives the number of particles in that particular state.
A given Fock state is denoted by $n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},..n_{{\mathbf {k} }_{i}}...\rangle$. In this expression, $n_{{\mathbf {k} }_{i}}$ denotes number of particles in the ith state, and the
particle number operator for the ith state, ${\widehat {N_{{\mathbf {k} }_{i}}}}$ acts on the Fock state in the following way:
${\widehat {N_{{\mathbf {k} }_{i}}}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},..n_{{\mathbf {k} }_{i}}...\rangle =n_{{\mathbf {k} }_{i}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},..n_{{\mathbf {k} }_{i}}...\rangle$
Hence the Fock state is an eigenstate of the number operator with eigenvalue $n_{{\mathbf {k} }_{i}}$.^{[2]}^{:478}
Fock states form the most convenient basis of a Fock space. Elements of a Fock space which are superpositions of states of differing particle number (and thus not eigenstates of the number operator) are not Fock states. Thus, not all elements of a Fock space are referred to as "Fock states".
The definition of Fock state ensures that ${\rm {Var}}({\widehat {N}})=0$, i.e., measuring the number of particles in a Fock state always returns a definite value with no fluctuation.
Example using two particles
For any final state $f\rangle$, any Fock state of two identical particles given by $1_{\mathbf {k} _{1}},1_{\mathbf {k} _{2}}\rangle$, and any operator ${\widehat {\mathbb {O} }}$, we have the following condition for indistinguishability:^{[3]}^{:191}
$\langle f{\widehat {\mathbb {O} }}1_{\mathbf {k} _{1}},1_{\mathbf {k} _{2}}\rangle ^{2}=\langle f{\widehat {\mathbb {O} }}1_{\mathbf {k} _{2}},1_{\mathbf {k} _{1}}\rangle ^{2}$.
So, we must have, $\langle f{\widehat {\mathbb {O} }}1_{\mathbf {k} _{1}},1_{\mathbf {k} _{2}}\rangle =e^{i\delta }\langle f{\widehat {\mathbb {O} }}1_{\mathbf {k} _{2}},1_{\mathbf {k} _{1}}\rangle$
where $e^{i\delta }=+1$ for bosons and $1$ for fermions.
As $\langle f$ and ${\widehat {\mathbb {O} }}$ are arbitrary, we can say,
$1_{\mathbf {k} _{1}},1_{\mathbf {k} _{2}}\rangle =+1_{\mathbf {k} _{2}},1_{\mathbf {k} _{1}}\rangle$ for bosons
and $1_{\mathbf {k} _{1}},1_{\mathbf {k} _{2}}\rangle =1_{\mathbf {k} _{2}},1_{\mathbf {k} _{1}}\rangle$ for fermions.
^{[3]}^{:191}
Operation by a Number operator
Suppose, for a number operator given by ${\widehat {N_{{\mathbf {k} }_{1}}}}$ , we have for bosons, ${\widehat {N_{{\mathbf {k} }_{1}}}}{1_{\mathbf {k} _{1}},1_{\mathbf {k} _{2}}\rangle }_{bosonic}=1_{\mathbf {k} _{1}}1_{\mathbf {k} _{1}},1_{\mathbf {k} _{2}}\rangle$
and for fermions ${\widehat {N_{{\mathbf {k} }_{1}}}}{1_{\mathbf {k} _{1}},1_{\mathbf {k} _{2}}\rangle }_{fermionic}=1_{\mathbf {k} _{1}}1_{\mathbf {k} _{1}},1_{\mathbf {k} _{2}}\rangle$
Hence, using number operator on a state, we can never identify the state whether it is a bosonic one or a fermionic one. For that we need different algebra, which follows in the next sections.
Bosonic Fock state
Bosons, which are particles with integer spin, follow a simple rule: their composite eigenstate is symmetric^{[4]} under operation by an exchange operator. For example, in a two particle system in the tensor product representation we have ${\hat {P}}\leftx_{1},x_{2}\right\rangle =\leftx_{2},x_{1}\right\rangle$ .
Boson Creation and Annihilation operators
We should be able to express the same symmetric property in this new Fock space representation. For this we introduce nonHermitian bosonic creation and annihilation operators,^{[4]} denoted by $b^{\dagger }$ and $b$ respectively. The action of these operators on a Fock state are given by the following two equations:
 $b_{{\mathbf {k} }_{l}}^{\dagger }n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}},...\rangle ={\sqrt {n_{{\mathbf {k} }_{l}}+1}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}+1,...\rangle$ ^{[4]}
 $b_{{\mathbf {k} }_{l}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}},...\rangle ={\sqrt {n_{{\mathbf {k} }_{l}}}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}1,...\rangle$ ^{[4]}
Hermiticity of Creation and Annihilation operator
Creation and Annihilation operators are not Hermitian operators.^{[4]}
Proof that Creation and Annihilation operators are not Hermitian.

For a Fock state, $n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}},...\rangle$,
 $\langle n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}1,...b_{{\mathbf {k} }_{l}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}},...\rangle ={\sqrt {n_{{\mathbf {k} }_{l}}}}\langle n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}1,...n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}1,...\rangle$
 $(\langle n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}},...b_{{\mathbf {k} }_{l}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}1,...\rangle )^{*}$
 $=\langle n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}1...b_{{\mathbf {k} }_{l}}^{\dagger }n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}},...\rangle ={\sqrt {n_{{\mathbf {k} }_{l}}+1}}\langle n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}1...n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}+1...\rangle$
Therefore, it is clear that adjoint of Creation (Annihilation) operator doesn't go into itself. Hence, they are not Hermitian operators.
But adjoint of Creation (Annihilation) operator is Annihilation (Creation) operator.^{[5]}^{:45}

Operator Identities
The commutation relations of creation and annihilation operators in a bosonic system are
 $[b_{i}^{\,},b_{j}^{\dagger }]\equiv b_{i}^{\,}b_{j}^{\dagger }b_{j}^{\dagger }b_{i}^{\,}=\delta _{ij},$ ^{[4]}
 $[b_{i}^{\dagger },b_{j}^{\dagger }]=[b_{i}^{\,},b_{j}^{\,}]=0,$ ^{[4]}
where $[\ \ ,\ \ ]$ is the commutator and $\delta _{ij}$ is the Kronecker delta.
Number of Particles (N) 
Bosonic basis states^{[6]}^{:11}

0 
$0,0,0...\rangle$

1 
$1,0,0...\rangle$,$0,1,0...\rangle$,$0,0,1...\rangle$,...

2 
$2,0,0...\rangle$,$1,1,0...\rangle$,$0,2,0...\rangle$,...

... 
...

Action on some specific Fock states
 $b_{{\mathbf {k} }_{l}}^{\dagger }0_{{\mathbf {k} }_{1}},0_{{\mathbf {k} }_{2}},0_{{\mathbf {k} }_{3}}...0_{{\mathbf {k} }_{l}},...\rangle =0_{{\mathbf {k} }_{1}},0_{{\mathbf {k} }_{2}},0_{{\mathbf {k} }_{3}}...1_{{\mathbf {k} }_{l}},...\rangle$
and, $b_{{\mathbf {k} }_{l}}0_{{\mathbf {k} }_{1}},0_{{\mathbf {k} }_{2}},0_{{\mathbf {k} }_{3}}...0_{{\mathbf {k} }_{l}},...\rangle =0$ ^{[4]}
 We can generate any Fock state by operating on the vacuum state with an appropriate number of creation operators:
$n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}}...\rangle ={\frac {(b_{{\mathbf {k} }_{1}}^{\dagger })^{{\mathbf {k} }_{1}}}{\sqrt {{{\mathbf {k} }_{1}}!}}}{\frac {(b_{{\mathbf {k} }_{2}}^{\dagger })^{{\mathbf {k} }_{2}}}{\sqrt {{{\mathbf {k} }_{2}}!}}}...0_{{\mathbf {k} }_{1}},0_{{\mathbf {k} }_{2}},...\rangle$
 $b_{\mathbf {k} }^{\dagger }n_{\mathbf {k} }\rangle ={\sqrt {n_{\mathbf {k} }+1}}n_{\mathbf {k} }+1\rangle$
and, $b_{\mathbf {k} }n_{\mathbf {k} }\rangle ={\sqrt {n_{\mathbf {k} }}}n_{\mathbf {k} }1\rangle$
Action of Number operator
The number operators for a bosonic system are given by ${\widehat {N_{{\mathbf {k} }_{l}}}}=b_{{\mathbf {k} }_{l}}^{\dagger }b_{{\mathbf {k} }_{l}}$, where ${\widehat {N_{{\mathbf {k} }_{l}}}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}...\rangle =n_{{\mathbf {k} }_{l}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}...\rangle$ ^{[4]}
Number operators are Hermitian operators.
Symmetric behaviour of Bosonic Fock states
The commutation relations of the creation and annihilation operators ensure that the bosonic Fock states have the appropriate symmetric behaviour under particle exchange. Here, exchange of particles between two states is done by annihilating one particle in one state and creating one in another.
If we start with a Fock state, $n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},....n_{{\mathbf {k} }_{m}}...n_{{\mathbf {k} }_{l}}...\rangle$,
and want to shift a particle from state $k_{l}$ to state $k_{m}$, then we operate the Fock state by $b_{{\mathbf {k} }_{m}}^{\dagger }b_{{\mathbf {k} }_{l}}$ in the following way:
Using the commutation relation we have,
$b_{{\mathbf {k} }_{m}}^{\dagger }.b_{{\mathbf {k} }_{l}}=b_{{\mathbf {k} }_{l}}.b_{{\mathbf {k} }_{m}}^{\dagger }$
$b_{{\mathbf {k} }_{m}}^{\dagger }.b_{{\mathbf {k} }_{l}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},....n_{{\mathbf {k} }_{m}}...n_{{\mathbf {k} }_{l}}...\rangle =b_{{\mathbf {k} }_{l}}.b_{{\mathbf {k} }_{m}}^{\dagger }n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},....n_{{\mathbf {k} }_{m}}...n_{{\mathbf {k} }_{l}}...\rangle ={\sqrt {n_{{\mathbf {k} }_{m}}+1}}{\sqrt {n_{{\mathbf {k} }_{l}}}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},....n_{{\mathbf {k} }_{m}}+1...n_{{\mathbf {k} }_{l}}1...\rangle$
So, the Bosonic Fock state behaves to be symmetric under operation by Exchange operator.
Fermionic Fock state
Fermion Creation and Annihilation operators
To be able to retain the antisymmetric behaviour of fermions, for Fermionic fock states we introduce nonHermitian Fermion Creation and annihilation operators,^{[4]} defined as, for a Fermionic fock state, $n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}},...\rangle$,
Creation operator acts as:
 $c_{{\mathbf {k} }_{l}}^{\dagger }n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}},...\rangle ={\sqrt {n_{{\mathbf {k} }_{l}}+1}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}+1,...\rangle$ ^{[4]}
and Annihilation operator acts as:
 $c_{{\mathbf {k} }_{l}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}},...\rangle ={\sqrt {n_{{\mathbf {k} }_{l}}}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}1,...\rangle$ ^{[4]}
These two actions are done antisymmetrically, which we shall discuss later.
Operator Identities
The anticommutation relations of creation and annihilation operators in a fermionic system are,
 $\{c_{i}^{\,},c_{j}^{\dagger }\}\equiv c_{i}^{\,}c_{j}^{\dagger }+c_{j}^{\dagger }c_{i}^{\,}=\delta _{ij},$ ^{[4]}
 $\{c_{i}^{\dagger },c_{j}^{\dagger }\}=\{c_{i}^{\,},c_{j}^{\,}\}=0,$ ^{[4]}
where ${\ \ ,\ \ }$ is the anticommutator and $\delta _{ij}$ is the Kronecker delta.
These anticommutation relation will be used to show antisymmetric behaviour of Fermionic Fock states.
Action of Number operator
Number Operator for Fermions is given by ${\widehat {N_{{\mathbf {k} }_{l}}}}=c_{{\mathbf {k} }_{l}}^{\dagger }.c_{{\mathbf {k} }_{l}}$
and, ${\widehat {N_{{\mathbf {k} }_{l}}}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}...\rangle =n_{{\mathbf {k} }_{l}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}...\rangle$ ^{[4]}
Maximum Occupation number
Action of Number operator, as well as, creation and annihilation operators might seem same as the Bosonic ones, but the real twist comes from the maximum occupation number of each state in the Fermionic Fock state.
Suppose, a Fermionic Fock state $n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}...\rangle$ be obtained by using some operator from the tensor product of eigenkets as follows:
$n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}...\rangle =S_{}i_{1},i_{2},i_{3}...i_{l}...\rangle ={\frac {1}{\sqrt {N!}}}{\begin{vmatrix}i_{1}\rangle _{1}&\cdots &i_{1}\rangle _{N}\\\vdots &\ddots &\vdots \\i_{N}\rangle _{1}&\cdots &i_{N}\rangle _{N}\end{vmatrix}}$ ^{[7]}^{:16}
This determinant is called Slater determinant. If any of the single particle states are same, two rows of the Slater determinant would be same and hence the determinant would be zero. Hence, two identical fermions must not occupy the same state. Therefore, occupation number of any single state is either 0 or 1.
Eigenvalue of fermionic Fock state ${\widehat {N_{{\mathbf {k} }_{l}}}}$ will be either 0 or 1.
Number of Particles (N) 
Fermionic basis states^{[6]}^{:11}

0 
$0,0,0...\rangle$

1 
$1,0,0...\rangle$,$0,1,0...\rangle$,$0,0,1...\rangle$,...

2 
$1,1,0...\rangle$,$0,1,1...\rangle$,$0,1,0,1...\rangle$,$1,0,1,0...\rangle$...

... 
...

Action on some specific Fock states
 $c_{\mathbf {k} }^{\dagger }0_{\mathbf {k} }\rangle =1_{\mathbf {k} }\rangle$
and, $c_{\mathbf {k} }^{\dagger }1_{\mathbf {k} }\rangle =0$, as maximum occupation number of any state is 1, more than 1 fermions cannot occupy the same state.
 $c_{\mathbf {k} }1_{\mathbf {k} }\rangle =0_{\mathbf {k} }\rangle$
and, $c_{\mathbf {k} }0_{\mathbf {k} }\rangle =0$, as particle number cannot be less than zero.
$c_{{\mathbf {k} }_{\alpha }}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},...n_{{\mathbf {k} }_{\beta }},n_{{\mathbf {k} }_{\alpha }},...\rangle =(1)^{\sum _{\beta <\alpha }n\beta }n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},...n_{{\mathbf {k} }_{\beta }},1n_{{\mathbf {k} }_{\alpha }},...\rangle$,
where, $(1)^{\sum _{\beta <\alpha }n\beta }$ is called JordanWigner String, which depends on the ordering of the involved singleparticle states and counting of fermion occupation number of all preceding states.^{[5]}^{:88}
Antisymmetric behaviour of Fermionic Fock state
Antisymmetric behaviour of Fermionic states under Exchange operator is taken care of the anticommutation relations. Here, exchange of particles between two states is done by annihilating one particle in one state and creating one in other.
If we start with a Fock state, $n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},....n_{{\mathbf {k} }_{m}}...n_{{\mathbf {k} }_{l}}...\rangle$,
and want to shift a particle from state $k_{l}$ to state $k_{m}$, then we operate the Fock state by $c_{{\mathbf {k} }_{m}}^{\dagger }.c_{{\mathbf {k} }_{l}}$ in the following way:
Using the anticommutation relation we have,
$c_{{\mathbf {k} }_{m}}^{\dagger }.c_{{\mathbf {k} }_{l}}=c_{{\mathbf {k} }_{l}}.c_{{\mathbf {k} }_{m}}^{\dagger }$
$c_{{\mathbf {k} }_{m}}^{\dagger }.c_{{\mathbf {k} }_{l}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},....n_{{\mathbf {k} }_{m}}...n_{{\mathbf {k} }_{l}}...\rangle ={\sqrt {n_{{\mathbf {k} }_{m}}+1}}{\sqrt {n_{{\mathbf {k} }_{l}}}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},....n_{{\mathbf {k} }_{m}}+1...n_{{\mathbf {k} }_{l}}1...\rangle$
but, $c_{{\mathbf {k} }_{l}}.c_{{\mathbf {k} }_{m}}^{\dagger }n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},....n_{{\mathbf {k} }_{m}}...n_{{\mathbf {k} }_{l}}...\rangle =c_{{\mathbf {k} }_{m}}^{\dagger }.c_{{\mathbf {k} }_{l}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},....n_{{\mathbf {k} }_{m}}...n_{{\mathbf {k} }_{l}}...\rangle ={\sqrt {n_{{\mathbf {k} }_{m}}+1}}{\sqrt {n_{{\mathbf {k} }_{l}}}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},....n_{{\mathbf {k} }_{m}}+1...n_{{\mathbf {k} }_{l}}1...\rangle$
So, the Fermionic Fock state behaves to be antisymmetric under operation by Exchange operator.
Fock states are not Energy eigenstates in general
In Second quantization theory, Hamiltonian density function is given by
 ${\mathfrak {H}}={\frac {1}{2m}}\nabla _{i}\psi ^{*}(x)\nabla _{i}\psi (x)$ ^{[3]}^{:189}
Total Hamiltonian is given by
 ${\mathcal {H}}=\int d^{3}x\,{\mathfrak {H}}=\int d^{3}x\psi ^{*}(x)\left({\tfrac {\nabla ^{2}}{2m}}\right)\psi (x)$
$\therefore {\mathfrak {H}}={\tfrac {\nabla ^{2}}{2m}}$
For free Schrodinger Theory,^{[3]}^{:189}
 ${\mathfrak {H}}\psi _{n}^{(+)}(x)={\tfrac {\nabla ^{2}}{2m}}\psi _{n}^{(+)}(x)=E_{n}^{0}\psi _{n}^{(+)}(x)$
and
 $\int d^{3}x\,\psi _{n}^{(+)^{*}}(x)\psi _{n'}^{(+)}(x)=\delta _{nn'}$
and
 $\psi (x)=\sum _{n}a_{n}\psi _{n}^{(+)}(x)$,
where $a_{n}$ is the annihilation operator.
$\therefore {\mathcal {H}}=\sum _{n,n'}\int d^{3}x\,a_{n'}^{\dagger }\psi _{n'}^{(+)^{*}}(x){\mathfrak {H}}a_{n}\psi _{n}^{(+)}(x)$
Only for noninteracting particles ${\mathfrak {H}}$ and $a_{n}$ commute; but in general they do not commute.
For noninteracting particles, ${\mathcal {H}}=\sum _{n,n'}\int d^{3}x\,a_{n'}^{\dagger }\psi _{n'}^{(+)^{*}}(x)E_{n}^{0}\psi _{n}^{(+)}(x)a_{n}=\sum _{n,n'}E_{n}^{0}a_{n'}^{\dagger }a_{n}\delta _{nn'}=\sum _{n}E_{n}^{0}a_{n}^{\dagger }a_{n}=\sum _{n}E_{n}^{0}{\widehat {N}}$
If they do not commute, Hamiltonian will not have the above expression. Therefore, in general, fock states are not energy eigenstates of a system.
Vacuum fluctuations
The vacuum state or $0\rangle$ is the state of lowest energy and the expectation values of $a$ and $a^{\dagger }$ vanish in this state:
 $a0\rangle =0=\langle 0a^{\dagger }$
The electrical and magnetic fields and the vector potential have the mode expansion of the same general form:
 $F({\vec {r}},t)=\varepsilon ae^{i{\vec {k}}x\omega t}+h.c$
Thus it is easy to see that the expectation values of these field operators vanishes in the vacuum state:
 $\langle 0F0\rangle =0$
However, it can be shown that the expectation values of the square of these field operators is nonzero. Thus there are fluctuations in the field about the zero ensemble average. These vacuum fluctuations are responsible for many interesting phenomenon including the Lamb shift in quantum optics.
Multimode Fock states
In a multimode field each creation and annihilation operator operates on its own mode. So $a_{{\mathbf {k} }_{l}}$ and $a_{{\mathbf {k} }_{l}}^{\dagger }$ will operate only on $n_{{\mathbf {k} }_{l}}\rangle$. Since operators corresponding to different modes operate in different subspaces of the Hilbert space, the entire field is a direct product of $n_{{\mathbf {k} }_{l}}\rangle$ over all the modes:
 $n_{{\mathbf {k} }_{1}}\rangle n_{{\mathbf {k} }_{2}}\rangle n_{{\mathbf {k} }_{3}}\rangle ...\equiv n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}...\rangle \equiv \{n_{\mathbf {k} }\}\rangle$
The creation and annihilation operators operate on the multimode state by only raising or lowering the number state of their own mode:
 $a_{{\mathbf {k} }_{l}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}},...\rangle ={\sqrt {n_{{\mathbf {k} }_{l}}}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}1,...\rangle$
 $a_{{\mathbf {k} }_{l}}^{\dagger }n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}},...\rangle ={\sqrt {n_{{\mathbf {k} }_{l}}+1}}n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}+1,...\rangle$
We also define the total number operator for the field which is a sum of number operators of each mode:
 ${\hat {n}}_{\mathbf {k} }=\sum {\hat {n}}_{\mathbf {k} _{l}}$
The multimode Fock state is an eigenvector of the total number operator whose eigenvalue is the total occupation number of all the modes
 ${\hat {n}}_{\mathbf {k} }\{n_{\mathbf {k} }\}\rangle ={\bigg (}\sum n_{\mathbf {k} _{l}}{\bigg )}\{n_{\mathbf {k} }\}\rangle$
In case of noninteracting particles, number operator and Hamiltonian commute with each other and hence multimode Fock states become eigenstates of the multimode Hamiltonian
 ${\hat {H}}\{n_{\mathbf {k} }\}\rangle ={\bigg (}\sum \hbar \omega {\big (}n_{\mathbf {k} _{l}}+{\frac {1}{2}}{\big )}{\bigg )}\{n_{\mathbf {k} }\}\rangle$
Source of single photon state
Single photons are routinely generated using single emitters (atoms, Nitrogenvacancy center
,^{[8]} Quantum dot ^{[9]}). However, these sources are not always very efficient (low probability of actually getting a single photon on demand) and often complex and unsuitable out of a laboratory environment.
Other sources are commonly used that overcome these issues at the expense of a nondeterministic behavior. Heralded single photon sources are probabilistic twophoton sources from whom the pair is split and the detection of one photon heralds the presence of the remaining one. These sources usually rely on the optical nonlinearity of some materials like periodically poled Lithium niobate (Spontaneous parametric downconversion), or silicon (spontaneous Fourwave mixing) for example.
Nonclassical behaviour
The GlauberSudarshan Prepresentation of Fock states shows that these states are purely quantum mechanical and have no classical counterpart. The $\scriptstyle \varphi (\alpha )\,$ of these states in the representation is a $2n$'th derivative of the Dirac delta function and therefore not a classical probability distribution.
See also
References
 ↑ Template:Cite Book
 ↑ {{#invoke:citation/CS1citation
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 ↑ ^{3.0} ^{3.1} ^{3.2} ^{3.3} {{#invoke:citation/CS1citation
CitationClass=book
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 ↑ ^{4.00} ^{4.01} ^{4.02} ^{4.03} ^{4.04} ^{4.05} ^{4.06} ^{4.07} ^{4.08} ^{4.09} ^{4.10} ^{4.11} ^{4.12} ^{4.13} ^{4.14} Template:Cite web
 ↑ ^{5.0} ^{5.1} {{#invoke:citation/CS1citation
CitationClass=book
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 ↑ ^{6.0} ^{6.1} {{#invoke:citation/CS1citation
CitationClass=book
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 ↑ {{#invoke:citation/CS1citation
CitationClass=book
}}
 ↑ C. Kurtsiefer, S. Mayer, P. Zarda, Patrick and H. Weinfurter, (2000), "Stable SolidState Source of Single Photons",
Phys. Rev. Lett. 85 (2) 290293, doi 10.1103/PhysRevLett.85.290
 ↑ C. Santori, M. Pelton, G. Solomon, Y. Dale and Y. Yamamoto (2001), "Triggered Single Photons from a Quantum Dot", Phys. Rev. Lett. 86 (8):15021505 DOI 10.1103/PhysRevLett.86.1502
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