Show that starting with: S (v,J)= (v+1/2) We - ( v+1/2)^2 WeXe + BvJ(J+1) you ge

Question

Show that starting with: S (v,J)= (v+1/2) We - ( v+1/2)^2 WeXe + BvJ(J+1) you get the expression Vr(J")= We-2WeXe+B1(J"+1)(J"+2)-BoJ"(J"+1) For the R branch of the vibrational-rotational spectrum. Be very careful about your quantum numbers, symbols you use and showing each step. No credit will be given for this problem if you do not carefully show each step and use the symbols in the equations above. (Can you please follow the instructions and explain in details?)( there is an answer was posted in chegg for this question, but the instructor didnt accepte it)

Explanation / Answer

SV,J = (v+1/2) w            e - (v+1/2)2 we ce + BvJ(J+1)

Selection rule for R-branch is ?J = +1

Consider a transition from J” to J”+1 from the v=0 to v=1 (ground vibrational level to first excited level)

S0,J” = (0+1/2) w           e - (0+1/2)2 we ce + B0J”(J”+1) = (1/2) we - (1/4) we ce + B0J”(J”+1)

S1,J”+1 = (1+1/2) we - (1+1/2)2 we ce + B1(J”+1)(J”+2) = (3/2) we - (9/4) we ce + B1(J”+1)(J”+2)

For the transition from J” to J”+1 from the v=0 to v=1

VR = S1,J”+1 - S0,J” = (3/2) we - (9/4) we ce + B1(J”+1)(J”+2) – [(1/2) we - (1/4) we ce + B0J”(J”+1)]

= we - 2 we ce + B1(J”+1)(J”+2) – B0J”(J”+1)

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