In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play an important role in the second quantization formulation of quantum mechanics.
The particle representation was first treated in detail by Paul Dirac for bosons and by Pascual Jordan and Eugene Wigner for fermions. The Fock states of bosons and fermions obey useful relations with respect to the Fock space creation and annihilation operators.
Definition
One specifies a multiparticle state of N non-interacting identical particles by writing the state as a sum of tensor products of N one-particle states. Additionally, depending on the integrality of the particles' spin, the tensor products must be alternating (anti-symmetric) or symmetric products of the underlying one-particle Hilbert spaces. Specifically:
- Fermions, having half-integer spin and obeying the Pauli exclusion principle, correspond to antisymmetric tensor products.
- Bosons, 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 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__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__1},n__2},..n__i}...\rangle</math>. In this expression, <math>n__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__i</math>, acts on the Fock state in the following way:
: <math>\widehat{N__i|n__1},n__2},..n__i}...\rangle = n__i}|n__1},n__2},..n__i}...\rangle</math>
Hence the Fock state is an eigenstate of the number operator with eigenvalue <math>n__i}</math>.
Fock states often form the most convenient basis of a Fock space. Elements of a Fock space that are 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__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.
Example using two particles
For any final state <math>|f\rangle</math>, any Fock state of two identical particles given by <math>|1_, 1_\rangle</math>, and any operator <math> \widehat{\mathbb{O </math>, we have the following condition for indistinguishability:
: <math>
\left|\left\langle f\left|\widehat{\mathbb{O\right|1_{\mathbf{k}_1}, 1_{\mathbf{k}_2}\right\rangle\right|^2 =
\left|\left\langle f\left|\widehat{\mathbb{O\right|1_{\mathbf{k}_2}, 1_{\mathbf{k}_1}\right\rangle\right|^2
</math>.
So, we must have <math>\left\langle f\left|\widehat{\mathbb{O\right|1_, 1_\right\rangle = e^{i\delta}\left\langle f\left|\widehat{\mathbb{O\right|1_, 1_\right\rangle</math>
where <math>e^{i\delta} = +1</math> for bosons and <math>-1</math> for fermions. Since <math>\langle f| </math> and <math>\widehat{\mathbb{O</math> are arbitrary, we can say,
: <math>\left|1_{\mathbf{k}_1}, 1_{\mathbf{k}_2}\right\rangle = +\left|1_{\mathbf{k}_2}, 1_{\mathbf{k}_1}\right\rangle</math> for bosons and
: <math>\left|1_{\mathbf{k}_1}, 1_{\mathbf{k}_2}\right\rangle = -\left|1_{\mathbf{k}_2}, 1_{\mathbf{k}_1}\right\rangle</math> for fermions. under operation by an exchange operator. For example, in a two particle system in the tensor product representation we have <math>\hat{P}\left|x_1, x_2\right\rangle = \left|x_2, x_1\right\rangle</math> .
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 non-Hermitian bosonic creation and annihilation operators,
Operator identities
The commutation relations of creation and annihilation operators in a bosonic system are
: <math>\left[b^{\,}_i, b^\dagger_j\right] \equiv b^{\,}_i b^\dagger_j - b^\dagger_jb^{\,}_i = \delta_{i j},</math>
|-
| 0 || <math>|0,0,0...\rangle</math>
|-
| 1 || <math>|1,0,0...\rangle</math>, <math>|0,1,0...\rangle</math>, <math>|0,0,1...\rangle</math>,...
|-
| 2 || <math>|2,0,0...\rangle</math>, <math>|1,1,0...\rangle</math>, <math>|0,2,0...\rangle</math>,...
|-
| <math>n</math>|| <math>|n__{1, n__{2 ,n__{3...n__{l,...\rangle</math>
|}
Action on some specific Fock states
, 0__{2, 0__{3...0__{l, ...\rangle</math>, we have:
: <math>b^{\dagger}__l}|0__{1, 0__{2, 0__{3...0__{l, ...\rangle = |0__{1, 0__{2, 0__{3...1__{l, ...\rangle </math>
and, <math>b_{\mathbf{k}_l}|0_{\mathbf{k}_1}, 0_{\mathbf{k}_2}, 0_{\mathbf{k}_3}...0_{\mathbf{k}_l}, ...\rangle = 0</math>.
This determinant is called the Slater determinant. If any of the single particle states are the same, two rows of the Slater determinant would be the same and hence the determinant would be zero. Hence, two identical fermions must not occupy the same state (a statement of the Pauli exclusion principle). Therefore, the occupation number of any single state is either 0 or 1. The eigenvalue associated to the fermionic Fock state <math>\widehat{N__l</math> must be either 0 or 1.
Action on some specific Fock states
center|The operation of creation and annihilation operators on Fermionic Fock states.