harmonic oscillator creation operator
and has energy 





1
2


ℏ
ω


{\displaystyle {\frac {1}{2}}\hbar \omega }


One can easily check that 



[



H
^



,




a
^




†


]
=
ℏ
ω
,


{\displaystyle [{\hat {H}},{\hat {a}}^{\dagger }]=\hbar \omega ,}

 which implies that 







a
^




†




{\displaystyle {\hat {a}}^{\dagger }}

 increases the energy of the simple harmonic oscillator by 



ℏ
ω


{\displaystyle \hbar \omega }

. For example, the state 







a
^




†



|

0
⟩


{\displaystyle {\hat {a}}^{\dagger }|0\rangle }

 is an eigenstate of energy 



3
ℏ
ω

/

2


{\displaystyle 3\hbar \omega /2}

.
Any energy eigenstate state of a single harmonic oscillator can be obtained from 




|

0
⟩


{\displaystyle |0\rangle }

 by successively applying the creation operator 







a
^




†




{\displaystyle {\hat {a}}^{\dagger }}

: and any state of the system can be expressed as a linear combination of the states
