Converting energy from units of joules into units of wavenumber If we have an expression for energy $E$ in Joules why is it that if we wish to convert the energy into wavenumber we divide the expression by $hc$?
I know that $$E = \frac{hc}{\lambda}$$
So surely this implies that $$\frac{E}{hc} = \frac{1}{\lambda}$$
and since $$ k = \frac{2\pi}{\lambda}$$
Would it not make sense that in order to convert energy in joules to wavenumber we multiply $E$ by $\frac{2\pi}{hc} = \frac{1}{\hbar c}$.
So my question is why in converting energy from the units of joules to the units of wavenumber do we divide $E$ by $ hc $ rather than by $\hbar c$
 A: You're using the 'angular' wavenumber $k = \frac{2\pi}{\lambda}$ (i.e. the number of radians accumulated per unit length), but the wavenumber used by spectroscopists is the "spectroscopic" wavenumber,
$$
\tilde\nu=\frac1\lambda,
$$
which measures the number of wavelengths accumulated per unit length. If you want to get $k$ then indeed you should divide by $\hbar c$, and for $\tilde\nu$ you should divide by $hc$.
Unfortunately, the notation for these choices is very muddy and there are very few universal and objective standards, so the same word ends up getting used for distinct concepts. This is highly suboptimal, but it's mostly an unavoidable consequence of the fact that there isn't that much linguistic room around this notation to begin with. (Or maybe it is just that we are too lazy to say "angular wavenumber" like we do to distinguish angular frequencies from frequencies.) It isn't too terrible (you just need to be careful to check what factors of $2\pi$ need to go where), but it does require some attention (you need to be careful with the factors of $2\pi$).
As a general rule, though, if it's reported in $\rm cm^{-1}$, then it is almost certainly $\tilde\nu$ instead of $k$.
A: From the Planck relation
$$E=h\nu = \frac{hc}{\lambda} = hc\tilde{\nu} $$
where in spectroscopy $\tilde{\nu}$ is the wavenumber and it has the following relation with angular wavenumber $k$
$$\tilde{\nu}=\frac{k}{2\pi}$$
