Ladder of States

Assume there is an energy eigenstate ∣E⟩\ket{E}

H^∣E⟩=E∣E⟩,⟨E∣E⟩=1\hat{H}\ket{E} = E\ket{E}, \qquad \braket{E|E} = 1

via the number operator

N^∣E⟩=NE∣E⟩,E=ℏω(NE+12)\hat{N}\ket{E} = N_E\ket{E}, \qquad E = \hbar\omega\left(N_E + \tfrac{1}{2}\right)

Let these be called raised/lowered states respectively

∣E+⟩≡a^†∣E⟩,∣E−⟩≡a^∣E⟩\ket{E_+} \equiv \hat{a}^\dagger\ket{E}, \qquad \ket{E_-} \equiv \hat{a}\ket{E}

Note that

H^∣E+⟩=(ℏω a^†+a^†H^)∣E⟩\hat{H}\ket{E_+} = (\hbar\omega\,\hat{a}^\dagger + \hat{a}^\dagger\hat{H})\ket{E} =ℏω a^†∣E⟩+a^†E∣E⟩=(E+ℏω) a^†∣E⟩=(E+ℏω)= \hbar\omega\,\hat{a}^\dagger\ket{E} + \hat{a}^\dagger E\ket{E} = (E + \hbar\omega)\,\hat{a}^\dagger\ket{E} = (E + \hbar\omega) =(E+ℏω)∣E+⟩= (E + \hbar\omega)\ket{E_+}

so ∣E+⟩\ket{E_+} has eigenstate of H^\hat{H} with energy eigenvalue E+ℏωE+\hbar\omega similarly, ∣E−⟩\ket{E_-} has eigenstate of H^\hat{H} with energy eigenvalue E−ℏωE-\hbar \omega so

⟨E+∣E+⟩=⟨E∣a^a^†∣E⟩=⟨E∣(1+N^)∣E⟩=(1+NE)⟨E∣E⟩\braket{E_+ | E_+} = \bra{E}\hat{a}\hat{a}^\dagger\ket{E} = \bra{E}(1 + \hat{N})\ket{E} = (1 + N_E)\braket{E|E}

and

⟨E−∣E−⟩=⟨E∣N^∣E⟩=NE⟨E∣E⟩\braket{E_-|E_-} = \bra{E}\hat{N}\ket{E} = N_E\braket{E|E}