particle number operator

In many cases, the particle number operator does not commute with the Hamiltonian for the system.

In quantum optics specifically, often the operators of interest, especially the particle number operator, is naturally expressed in normal order.

Analogous to the Quantum harmonic oscillator case, we can define particle number operator as .

The concept is therefore generalized to the particle number operator, that is, the observable that counts the number of constituent particles.

In quantum field theory, the particle number operator (see Fock state) is conjugate to the phase of the classical wave (see coherent state).

A more mathematical definition is that Fock states are those elements of a Fock space which are eigenstates of the particle number operator.

In the modern view, energy is always conserved, but the eigenstates of the Hamiltonian (energy observable) are not the same as (i.e., the Hamiltonian doesn't commute with) the particle number operators.

In formal terms, a particle is considered to be an eigenstate of the particle number operator aa, where a is the particle annihilation operator and a the particle creation operator (sometimes collectively called ladder operators).

In quantum optics, this representation, formally equivalent to several other representations, is sometimes championed over alternative representations to describe light in optical phase space, because typical optical observables, such as the particle number operator, are naturally expressed in normal order.