Particle number operator

In quantum mechanics, for systems where the total number of particles

$|\Psi \rangle _{\nu }$composed of single-particle basis states $|\phi _{i}\rangle$:

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

with creation and annihilation operators $a^{\dagger }(\phi _{i})$ and $a(\phi _{i})\,$ we define the number operator ${\hat {N_{i}}}\ {\stackrel {\mathrm {def} }{=}}\ a^{\dagger }(\phi _{i})a(\phi _{i})$ 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

${\begin{matrix}{\hat {N_{i}}}|\Psi \rangle _{\nu }=a^{\dagger }(\phi _{i})a(\phi _{i})|\phi _{1},\phi _{2},\cdots ,\phi _{i-1},\phi _{i},\phi _{i+1},\cdots ,\phi _{n}\rangle _{\nu }&=&{\sqrt {N_{i}}}a^{\dagger }(\phi _{i})|\phi _{1},\phi _{2},\cdots ,\phi _{i-1},\phi _{i+1},\cdots ,\phi _{n}\rangle _{\nu }\\&=&{\sqrt {N_{i}}}{\sqrt {N_{i}}}|\phi _{1},\phi _{2},\cdots ,\phi _{i-1},\phi _{i},\phi _{i+1},\cdots ,\phi _{n}\rangle _{\nu }\\&=&N_{i}|\Psi \rangle _{\nu }\\\end{matrix}}$

See also

• Harmonic oscillator
• Quantum harmonic oscillator
• Second quantization
• Quantum field theory
• Thermodynamics
• Fermion number operator

References

• Bruus, Henrik, Flensberg, Karsten. (2004). Many-body Quantum Theory in Condensed Matter Physics: An Introduction. Oxford University Press. ISBN 0-19-856633-6.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Second quantization notes by Fradkin