# Difference between equations for spectroscopy wavenumbers

What is the difference between these two equations and when is appropriate to use which one?

We were told that $$F(J)$$ in this case is

$$F(J)= E/h$$ for units in frequency

And

$$F(J) = BJ(J+1)$$

$$\nu = 2B(J+1)$$

(where $$\nu$$ (nu) is meant to signify wave number)

I have seen difference example problems use difference ones but it appears to me as though these two equations give the same result: wavenumber. So how is it possible that it has two completely separate values?

• The $\nu=$ equation is obtained for a rotational transition $J\to J+1$ in a diatomic molecule so is not the same as the second equation $F(J)=BJ(J+1)$. This equation is that for the energy level itself so $\nu =F(J+1)-F(J)$. The first equation has no meaning unless you define $E$ in some way. $B$ is often given in wavenumbers. – porphyrin Dec 7 '19 at 12:01
• @porphyrin What do you mean by the "first equation has no meaning"? Which equation are you referring to? Please clarify – Harley McFarlen Dec 7 '19 at 14:05
• $F(J)=E/h$ unless you specify what $E$ is and in what units. – porphyrin Dec 8 '19 at 9:35

Ultimately, they are just different expressions for the same fundamental physical quantity: the energies of quantum states and the differences between them. There are different units you can use to express that quantity - energy, frequency, wavelength, wavenumber - but at the end of the day they are all equivalent, and can be converted among each other, as $$E=h\nu = h\frac{c}{\lambda}=hc\tilde{\nu}.$$
NB: The $$E/h$$ you have written yields a frequency, not a wavenumber. Things may sometimes differ depending on how the rotational constant is defined, but again a look at the units will tell you what you need to know.
Also, $$\nu$$ without the tilde is only ever a symbol for a frequency, not a wavenumber. To get the wavenumber $$\tilde{\nu}$$, write $\tilde{\nu}$.