Rotation Spectrum of a diatomic molecule (QM)

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SUMMARY

The discussion focuses on the rotation spectrum of the diatomic molecule 12C16O, specifically analyzing the transitions l = 4 -> 3 at 461.04077 GHz and l = 36 -> 35 at 4115.6055 GHz. It is established that the intra-nuclear distance is approximately 0.113 nm and that the electrons create a spring-like force between the nuclei with a force constant of approximately 1904 N/m. The vibrational frequency of CO is calculated to be near 6.47 × 1013 Hz, closely matching the measured value of 6.43 × 1013 Hz. The correct formula for frequency is confirmed to be f = jh/(2πI).

PREREQUISITES
  • Understanding of quantum mechanics, specifically rotational spectra
  • Familiarity with the concepts of moment of inertia and eigenvalues
  • Knowledge of classical mechanics, particularly the relationship between angular momentum and moment of inertia
  • Proficiency in using the Planck constant in calculations
NEXT STEPS
  • Explore the derivation of rotational spectra in diatomic molecules using quantum mechanics
  • Study the calculation of moment of inertia for various molecular geometries
  • Investigate the classical mechanics principles that relate force constants to molecular vibrations
  • Learn about the application of the Planck constant in spectral analysis
USEFUL FOR

This discussion is beneficial for students and researchers in quantum mechanics, particularly those studying molecular spectroscopy, as well as chemists and physicists interested in the properties of diatomic molecules like CO.

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Homework Statement


In the rotation spectrum of 12C16O the line arising from the transition l = 4 -> 3 is at 461.04077GHz, while that arising from l = 36 -> 35 is at 4115.6055GHz. Show from these data that in a non-rotating CO molecule the intra-nuclear distance is s ~ 0.113 nm, and that the electrons provide a spring between the nuclei that has force constant ~1904Nm−1. Hence show that the vibrational frequency of CO should lie near 6.47×10^13Hz (measured value 6.43 × 1013 Hz). Hint: show from classical mechanics that the distance of O from the centre of mass is (3/7) s and that the molecule’s moment of inertia is (48/7) ms^2. Recall also the classical relation L = Iw.


Homework Equations


f = j h/((2pi)^2*I)

Where f is the frequency of the emitted photon and j(j+1) is the eigenvalue of J^2 (I think j is the same as l in this question?)

The Attempt at a Solution


I can derive the two classical bits in the hint and the above formula. I then tried taking one of the frequencies given, working out I using the above formula, and plugging it into the second classical expression to get s. It didn't work!

Thanks
 
Physics news on Phys.org
It's not (2pi)^2 on the bottom, it's just 2pi. Plugging in the numbers with the modified formula gives the correct value.
 

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