and has energy 1 2 ℏ ℏ ω ω . One can easily check that [ H ^ ^ , a ^ ^ † † ] = ℏ ℏ ω ω a ^ ^ † † , which implies that a ^ ^ † † increases the energy of the simple harmonic oscillator by ℏ ℏ ω ω . For example, the state a ^ ^ † † | 0 ⟩ ⟩ is an eigenstate of energy 3 ℏ ℏ ω ω / 2 . Any energy eigenstate state of a single harmonic oscillator can be obtained from | 0 ⟩ ⟩ by successively applying the creation operator a ^ ^ † † : and any state of the system can be expressed as a linear combination of the states