quantum mechanics - What are the units of the creation and annihilation operators?

The creation and annihilation operators - also known as ladder operators are; $ \hat{a}^\dagger$ and $\hat{a}$ respectively.

Using the equation $\hat{H} = \hbarω\left(\hat{a}^\dagger \hat{a} + \frac{1}{2}\right)$

and knowing that the units of $\hat{H}$ are J,

the units of $ω$ are Rad/s

and the units of $\hbar$ are J.s

I think that the ladder operators should have units of $\frac{1}{\sqrt{Rad}}$

But I have never seen a square root of an angle in units before. Is this correct?

Answer

The units of $\hbar$ are in fact J.s/rad. (thanks AV23) this is because $\hbar = \frac{h}{2\pi}$ the units of h are J.s and the units of $\pi$ are rad. Thus we have J.s/rad. (thanks Noiralef)

Thus the ladder operators are in fact unitless.

On reflection this is the only logical possibility as they move between different eigenstates - which must all be in the same units.