Undergrad Confused about unit conversion involving natural units

[kelly0303]

Hello! I have an expression whose natural units are Joules, but all the terms are expressed in terms of cm$^{-1}$ (it is for an atomic transition). I have a term in the expression whose units are 1/A (angstrom) and I am not sure how to convert it to what I need. On one hand, if I were to go from 1/A to 1/cm, I would need to multiply that term by $10^8$. On the other hand, if I want to convert to J, I need to multiply by $\hbar$ c, then convert from J to cm$^{-1}$, which gives $15927759.569$. The difference between the 2 approaches is $2\pi$, but I am not sure why. Should I actually use hc instead of $\hbar$c? The issue is that the formula is defined using $\hbar$ and I am not sure what I am doing wrong. For reference I am talking about equation 1 in this paper ($\alpha_5$, $A_1$ and $A_2$ are unitless). Thank you!

 

[Vanadium 50]

You have implicit conversion factors somewhere. You need to make them explicit. (Well, you don't need to, but the chances of screwing up are smaller if you do)

 

[WernerQH]

On one hand, if I were to go from 1/A to 1/cm, I would need to multiply that term by $10^8$.

Correct.

Should I actually use hc instead of $\hbar$c?

Yes. What spectroscopists refer to as wave number is $1/\lambda$, not $k = 2 \pi / \lambda$, as theorists often do. In terms of energy ($E=hc/\lambda$), $\rm 1 ~eV = 8065.5~cm^{-1} = 1.6022 \times 10^{-19}~J$.