# 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?

• The units of $\hbar$ are in fact J.s/rad.
– AV23
Commented May 3, 2015 at 11:52
• @AV23 Well that would make things much neater but wikipedia (that exceedingly trustworthy source) does not mention that. Nor does hyperphysics. Is this because Rad are not considered 'real' units? Commented May 3, 2015 at 12:24
• @Jekowl Yes. But $\hbar = \frac{h}{2\pi}$, $h$ is an angular momentum and $\pi$ has the unit rad. Commented May 3, 2015 at 12:26
• @Noiralef that make sense! thanks. Not really sure what the correct procedure here is but I guess I will leave false assumptions in the question and they can be corrected in the answer. Commented May 3, 2015 at 12:33
• That is the right way to do it, by the way: leave your question as is and post an answer that explains it. Commented May 3, 2015 at 12:44

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)