For context, I am currently studying perturbation theory, as well as variational methods in quantum physics. My professor uses natural units when solving problems, and he states in every problem that $\hbar / 2m = 1$ will be assumed. This makes sense to me, as I can see that Schrödinger's time independent equation looks simpler this way, but I have trouble understanding how to derive certain things in this unit system.
For example, there's this perturbation theory problem where we're studying the potential $V(x) = x^2$, with the perturbation being $V_1(x) = ax$. Therefore:
$$H = H_0 + V_1(x) = -\frac{d^2}{dx^2} + x^2 + ax$$
($H_0$ being the original harmonic oscillator, and $ax$ the perturbation we're working with).
He states that $\hbar / 2m = 1$, and then he says that the energy associated to $H_0$ is:
$$E_n^{(0)} = \hbar \omega \left(n + \frac{1}{2}\right) = 2 \left(n+\frac{1}{2}\right)$$
I do not understand how you get $\hbar\omega = 2$ from $\hbar / 2m = 1$.
I guess this is usually covered in quantum physics textbooks, but I have checked the one I normally use (Messiah's) and I can't find an answer. If someone could help me understand this I'd be very thankful.