What happens to the $2\pi$ factor in natural units?

Question

In natural units when we define $c=\hbar=1$ and we have that energy and mass have the same units because of $E=mc^2$. The same happens for time and space due to $x=ct$. Now, when we want to relate energy-mass to space-time we have $$E=\hbar\omega$$ so if $\hbar=1$ then energy and time must be related by $$\text{energy}=\frac{2\pi}{\text{time}}$$ However, I cannot find this $2\pi$ factor anywhere, for example here. Where did it go?

Answer 1

One physical interpretation of natural units is that we will use photons as measuring devices. We can measure all quantities in terms of the energy of a photon:

• When we say the mass of a proton is about $1\ {\rm GeV}$, we mean if we took a photon with energy $1\ {\rm GeV}$ and somehow converted this energy completely to the rest energy of some chunk of matter, that chunk of matter would have a mass about the mass of the proton.
• When we say the radius of a proton is about $(1\ {\rm GeV})^{-1}$? It means a photon with energy $1\ {\rm GeV}$ (and therefore momentum of $1\ {\rm GeV}$) has a reduced wavelength of about $1$ proton radius.
• When we say the neutron lifetime is about $(5\times 10^{-18}\ {\rm eV})^{-1}$, we mean a photon with energy $5 \times 10^{-18}\ {\rm eV}$ has a reduced period of about 15 minutes. (Actually to be totally honest I've never heard anyone quote the neutron lifetime in eV rather than minutes, but this example is just to prove a point)

Above you will note that the photon's wavelength $\lambda$ and period $T$ do not appear, but rather its reduced wavelength $\lambda_{\rm red}=\lambda/2\pi$ and reduced period $T_{\rm red}=T/2\pi$. If we had set $h$ instead of $\hbar$ equal to one, we would use the actual wavelength and period, with no $2\pi$. This is exactly the factor of $2\pi$ you are asking about.

Having said that, people usually don't think this much about units. Since $2\pi$ is dimensionless, the key point is that any time has units of 1 / energy, and factors of $2$ and $\pi$ come out in the wash. The specific numerical value can be worked out once you fix all your conventions in a given calculation.

Answer 2

Et has units of ℏ , hence no units in the natural system, as you might know from QM evolving phases. The same holds for any multiple of it.

Thus, the equation connecting the energy of a photon with its period, $$ E=\hbar \omega=\hbar ~2\pi/T~~~\leadsto ~~~ET=2\pi \hbar ~, $$ tells you that ET is equal to 2π in units of ℏ. In natural units, it is equal to 2π. To get to engineering units, you multiply it by ℏ.

Not every time times any energy has to be equal to 1, as you incorrectly surmised, and this is not the equation "defining" ℏ. Dirac used the exponential phase equation, effectively, to define ℏ.