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^†) the annihilation (resp. creation) bose operator. Similar relations hold for fermionic algebra.

This allows to check that <a^†a>=n and Var(a^†a)=0, i.e., that measuring the number of particles a^†a in a Fock state returns always a definite value with no fluctuation.