Fock state

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A Fock state, in quantum mechanics, is any state of the Fock space with a well-defined number of particles in each state. The name is for V. A. Fock.

If we limit to a single mode for simplicity (doing so we formally describe a mere harmonic oscillator), a Fock state is of the type $|n\rangle$ with n an integer value. This means that there are n quanta of excitation in the mode. $|0\rangle$ corresponds to the ground state (no excitation). It is different from 0, which is the null vector.

Fock states form the most convenient basis of the Fock space. They are defined to obey the following relations in the bosonic algebra:

$a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle$

$a|n\rangle=\sqrt{n}|n-1\rangle$

$|n\rangle={1\over\sqrt{n!}}(a^{\dagger})^n|0\rangle$

with a (resp. $a^{\dagger}$ ) the annihilation (resp. creation) bose operator. Similar relations hold for fermionic algebra.

This allows to check that $\langle a^{\dagger} a \rangle = n$ and $Var(a^{\dagger}a)=0$ , i.e., that measuring the number of particles $a^{\dagger}a$ in a Fock state returns always a definite value with no fluctuation.