Derive the relation between the P & R branches

[Bananen]

Hi,

I have an assignment to derive the two following expressions:

R(J)-P(J)=2B'(2J+1)
R(J-1)-P(J+1)=2B''(2J+1)
where Bis the rotational constant and prime ' stands for upper level and bis '' for lower level.
B_v=B_e-α(v+1/2)

using the selection rules (I guess in this case ΔJ=±1) and ΔT=G(v')-G(v'')+F(J')-F(J'') where ΔT is the transition/energy difference
between two energy levels expressed in wavenumbers and F(J)=BJ(J+1)
G(v)=ω_e(v+1/2)-ω_ex_e(v+1/2)².

I don't understand how I'm going to put all of this together and I don't understand what they mean with R(J) and P(J) etc.
Thankful for any help!

 

[DrClaude]

You'll have to give more background. Can you provide with the exact problem statement?

 

[TeethWhitener]

It's difficult to determine if there is a precise question in OP's post.

R(J) refers to the energy of an R-branch transition originating from the J'th rotational state. Ditto for P(J) (for a P-branch transition). You might spend some time poring over this diagram:
https://en.wikipedia.org/wiki/Rotat...scopy#/media/File:Vibrationrotationenergy.svg

Also, presumably this is comparing P and R branches of the same vibrational transition, so you don't have to worry about most of the expressions you've listed involving vibrational quantum numbers. Maybe, as @DrClaude suggested, you could give us a more focused question?