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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?

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  • $\begingroup$ 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. $\endgroup$ – porphyrin Dec 7 '19 at 12:01
  • $\begingroup$ @porphyrin What do you mean by the "first equation has no meaning"? Which equation are you referring to? Please clarify $\endgroup$ – Harley McFarlen Dec 7 '19 at 14:05
  • $\begingroup$ $F(J)=E/h$ unless you specify what $E$ is and in what units. $\endgroup$ – porphyrin Dec 8 '19 at 9:35
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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 $$E=h\nu = h\frac{c}{\lambda}=hc\tilde{\nu}.$$

There's no hard and fast rule what to use when, just a lot of conventions that can unfortunately differ between closely related fields. For example, microwave spectroscopists will typically talk about their spectra in units of frequencies, and infrared spectroscopists about theirs in units of wavenumbers. Throw electronic states into the mix and you'll also get electronvolts (a unit of energy). - If you want people from a certain field to understand what you're talking about, it makes sense to stick with the units they are familiar with. Otherwise, pick your favorite, or what most suits your problem and available data at hand.

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 $\tilde{\nu}$.

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