Question #277917

In Chem 465, we learn that the Hamiltonian of a rigid rotor is H=BJ2 where B is the rotational constant and J is the rotational angular momentum. The rotational energies are: E=BJ(J+1) 5 where J=0,1,2… is the rotational angular momentum quantum number. Consider a perturbation: š»š»ā€™ = 1 2 šœ†šœ†ļæ½š½š½Ģ‚+ + š½š½Ģ‚āˆ’ļæ½. where Ī» is a constant. The operators š½š½Ģ‚+ and š½š½Ģ‚āˆ’ have the following effects on the spherical harmonics š‘Œš‘Œš½š½ š‘šš‘šš½š½ : š½š½Ģ‚āˆ’š‘Œš‘Œš½š½ š‘šš‘šš½š½ = š½š½š‘Œš‘Œš½š½āˆ’1 š‘šš‘šš½š½āˆ’1 , š½š½Ģ‚āˆ’š‘Œš‘Œ0 0 = 0 š½š½Ģ‚+š‘Œš‘Œš½š½ š‘šš‘šš½š½ = (š½š½ + 1)š‘Œš‘Œš½š½+1 š‘šš‘šš½š½+1 (1) [5 pts] Prove that the first-order perturbation vanishes based on symmetry arguments. (2) [10 pts] For the ground rotational state (J=0), compute the second-order contribution to its energy due to the perturbation by the J=1 state. (3) [3 pts] For the ground rotational state (J=0), prove that the second-order contribution to its energy due to the perturbation by the J=2 state vanishes


1
Expert's answer
2021-12-10T03:26:32-0500

Selection rules only permit transitions between consecutive rotational levels: Ī”J=J±1, and require the molecule to contain a permanent dipole moment. Due to the dipole requirement, molecules such as HF and HCl have pure rotational spectra and molecules such as H2 and N2 are rotationally inactive.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS