Particle number operator

 ( Quantum Operators ) 

In quantum mechanics, for systems where the total Particle number 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

$$|\backslash Psi\backslash rangle\_\backslash nu=|\backslash phi\_1,\backslash phi\_2,\backslash cdots,\backslash phi\_n\backslash rangle\_\backslash nu$$

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

$$\backslash hat\{N\_i\}\; \backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; a^\{\backslash dagger\}(\backslash phi\_i)a(\backslash phi\_i)$$

and we have

$$\backslash hat\{N\_i\}|\backslash Psi\backslash rangle\_\backslash nu=N\_i|\backslash Psi\backslash rangle\_\backslash nu$$

where $N\_i$ is the number of particles in state $|\backslash phi\_i\backslash rangle$. The above equality can be proven by noting that then

• Harmonic oscillator
• Quantum harmonic oscillator
• Second quantization
• Quantum field theory
• Thermodynamics
• (-1)^F

• Bruus, Henrik (2025). 9780198566335, Oxford University Press. ISBN 9780198566335
• Second quantization notes by Fradkin