Linear Harmonic Oscillator Hamiltonian
of a particle
is defined as

where operators
Anihilation Operator
is defined as
Creation Operator
is defined as
Lets assume Linear Harmonic Oscillator Hamiltonian. Then
![{\displaystyle {\hat {a}}[{\hat {a}}^{+}(|n>)]={\frac {{\hat {H}}(|n>)}{\hbar \omega }}+{\frac {{\hat {1}}(|n>)}{2}},\qquad {\hat {a}}^{+}[{\hat {a}}(|n>)]={\frac {{\hat {H}}(|n>)}{\hbar \omega }}-{\frac {{\hat {1}}(|n>)}{2}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05fdd5398fc96b43cb68c23b474fc0ac8cdc331f)
Proof:
Directly form a, a+ definitions with respect to LHO Hamiltonian definition and [x,p] commutator
![{\displaystyle a[a^{+}(|n>)]{\overset {def\,a,a^{+}}{=}}\left({\sqrt {\frac {m\omega }{2\hbar }}}{\hat {x}}+i{\sqrt {\frac {1}{2\hbar m\omega }}}{\hat {p}}\right)\left({\sqrt {\frac {m\omega }{2\hbar }}}{\hat {x}}-i{\sqrt {\frac {1}{2\hbar m\omega }}}{\hat {p}}\right)=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c67121e73d11c6514f7d94749d69db068b2028e)
![{\displaystyle ={\frac {m\omega }{2\hbar }}{\hat {x}}^{2}(|n>)-i^{2}{\frac {1}{2\hbar m\omega }}{\hat {p}}^{2}(|n>)-i{\sqrt {\frac {m\omega }{2\hbar }}}{\sqrt {\frac {1}{2\hbar m\omega }}}{\hat {x}}[{\hat {p}}(|n>)]+i{\sqrt {\frac {m\omega }{2\hbar }}}{\sqrt {\frac {1}{2\hbar m\omega }}}{\hat {p}}[{\hat {x}}(|n>)]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5238c0a70c8af35f0b61e8d69974271f498c8e20)

Lets assume Linear Harmonic Oscillator Hamiltonian. Then
Proof:
The first expression may be evaluated from commutator definition directly from a, a+ compound expression theorem
![{\displaystyle [{\hat {a}},{\hat {a}}^{+}]={\hat {a}}{\hat {a}}^{+}-{\hat {a}}^{+}{\hat {a}}{\overset {theorem}{=}}{\frac {{\hat {H}}|n>}{2\hbar \omega }}+{\frac {{\hat {1}}|n>}{2}}-{\frac {{\hat {H}}|n>}{2\hbar \omega }}-{\frac {{\hat {1}}|n>}{2}}={\hat {1}}|n>}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06626ff8f25b23542e22891977b011f0435e0857)
The second and the third expressions needs
first, which can be obtained from a,a+ compound expression theorem again: