Reference articles on history, science, culture and more
Surprise meSurprise meSubscribe
Encyclopedia

Particle number operator

Operator in quantum mechanics

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated April 16, 2025

In quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles.

The following is in bra-ket notation: The number operator acts on Fock space. Let

|\Psi \rangle _{\nu }=|\phi _{1},\phi _{2},\cdots ,\phi _{n}\rangle _{\nu }

be a Fock state, composed of single-particle states |\phi _{i}\rangle drawn from a basis of the underlying Hilbert space of the Fock space. Given the corresponding creation and annihilation operators a^{\dagger }(\phi _{i}) and a(\phi _{i})\, we define the number operator by

{\hat {N_{i}}}\ {\stackrel {\mathrm {def} }{=}}\ a^{\dagger }(\phi _{i})a(\phi _{i})

and we have

{\hat {N_{i}}}|\Psi \rangle _{\nu }=N_{i}|\Psi \rangle _{\nu }

where N_{i} is the number of particles in state |\phi _{i}\rangle. The above equality can be proven by noting that {\begin{matrix}a(\phi _{i})|\phi _{1},\phi _{2},\cdots ,\phi _{i-1},\phi _{i},\phi _{i+1},\cdots ,\phi _{n}\rangle _{\nu }&=&{\sqrt {N_{i}}}|\phi _{1},\phi _{2},\cdots ,\phi _{i-1},\phi _{i+1},\cdots ,\phi _{n}\rangle _{\nu }\\a^{\dagger }(\phi _{i})|\phi _{1},\phi _{2},\cdots ,\phi _{i-1},\phi _{i+1},\cdots ,\phi _{n}\rangle _{\nu }&=&{\sqrt {N_{i}}}|\phi _{1},\phi _{2},\cdots ,\phi _{i-1},\phi _{i},\phi _{i+1},\cdots ,\phi _{n}\rangle _{\nu }\end{matrix}} then {\begin{array}{rcl}{\hat {N_{i}}}|\Psi \rangle _{\nu }&=&a^{\dagger }(\phi _{i})a(\phi _{i})\left|\phi _{1},\phi _{2},\cdots ,\phi _{i-1},\phi _{i},\phi _{i+1},\cdots ,\phi _{n}\right\rangle _{\nu }\\[1ex]&=&{\sqrt {N_{i}}}a^{\dagger }(\phi _{i})\left|\phi _{1},\phi _{2},\cdots ,\phi _{i-1},\phi _{i+1},\cdots ,\phi _{n}\right\rangle _{\nu }\\[1ex]&=&{\sqrt {N_{i}}}{\sqrt {N_{i}}}\left|\phi _{1},\phi _{2},\cdots ,\phi _{i-1},\phi _{i},\phi _{i+1},\cdots ,\phi _{n}\right\rangle _{\nu }\\[1ex]&=&N_{i}|\Psi \rangle _{\nu }\\[1ex]\end{array}}

Watch videos about Particle number operatorExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Particle number operator”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Continue exploring

Related topics

Harmonic oscillator

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x: F → = − k x → , {\displaystyle {\vec {F}}=-k{\vec {x}},} where k is a positive constant. The harmonic oscillator model is important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations.

Quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics.

Second quantization

Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields are thought of as field operators, in a manner similar to how the physical quantities (position, momentum, etc.) are thought of as operators in first quantization.