I just finished deriving the commutators:

\begin{align} [\hat{H}, \hat{a}] &= -\hbar \omega \hat{a}\\ [\hat{H}, \hat{a}^\dagger] &= \hbar \omega \hat{a}^\dagger\\ \end{align}

On the Wikipedia it is said that these commutators can be used to find energy eigenstates of Quant. harm. oscilator, but explaination is a bit too fast there. Anyway i strive to be able to derive the equation $W_n = \hbar \omega \left(n + \tfrac{1}{2}\right)$ in full, but first i need to clarify why theese two relations hold:

\begin{align} \hat{H}\hat{a} \psi_n &= (W_n - \hbar \omega) \hat{a} \psi_n\\ \hat{H}\hat{a}^\dagger \psi_n &= (W_n + \hbar \omega) \hat{a}^\dagger \psi_n \end{align}

I can't see any commutators in above relations, so how do the commutators i just calculated help us to get and solve these two relations?

I am sorry for asking such a basic questions. I am a selftaught and a real freshman to commutators algebra.

I just finished deriving the commutators:

\begin{align} [\hat{H}, \hat{a}] &= -\hbar \omega \hat{a}\\ [\hat{H}, \hat{a}^\dagger] &= \hbar \omega \hat{a}^\dagger\\ \end{align}

On the Wikipedia it is said that these commutators can be used to find energy eigenstates of Quant. harm. oscillator, but explanation is a bit too fast there. Anyway i strive to be able to derive the equation $W_n = \hbar \omega \left(n + \tfrac{1}{2}\right)$ in full, but first i need to clarify why theese two relations hold:

\begin{align} \hat{H}\hat{a} \psi_n &= (W_n - \hbar \omega) \hat{a} \psi_n\\ \hat{H}\hat{a}^\dagger \psi_n &= (W_n + \hbar \omega) \hat{a}^\dagger \psi_n \end{align}

I can't see any commutators in above relations, so how do the commutators i just calculated help us to get and solve these two relations?

I am sorry for asking such a basic questions. I am a self-taught and a real freshman to commutators algebra.