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Classical Model of Rigid Rotor

Classical Model of Rigid Rotor. A particle rotating around a fixed point, as shown below, has angular momentum and rotational kinetic energy (“rigid rotor”). The classical kinetic energy is given by:

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Classical Model of Rigid Rotor

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  1. Classical Model of Rigid Rotor A particle rotating around a fixed point, as shown below, has angular momentum and rotational kinetic energy (“rigid rotor”) The classical kinetic energy is given by: If the particle is rotating about a fixed point at radius r with a frequency ʋ (s−1 or Hz), the velocity of the particle is given by: where ω is the angular frequency (rad s−1 or rad Hz). The rotational kinetic energy can be now expressed as: Also where

  2. Consider a classical rigid rotor corresponding to a diatomic molecule. Here we consider only rotation restricted to a 2-D plane where the two masses (i.e., the nuclei) rotate about their center of mass. The rotational kinetic energy for diatomic molecule in terms of angular momentum Note that there is no potential energy involved in free rotation.

  3. Momentum Summary Classical QM Linear Momentum Energy Rotational (Angular) Momentum Energy

  4. Angular Momentum

  5. Angular Momentum

  6. Angular Momentum

  7. Angular Momentum

  8. Two-Dimensional Rotational Motion Polar Coordinates y r f x

  9. Two-Dimensional Rotational Motion

  10. Two-Dimensional Rigid Rotor Assume ris rigid, ie. it is constant

  11. Two-Dimensional Rigid Rotor

  12. Solution of equation

  13. Energy and Momentum As the system is rotating about the z-axis

  14. Two-Dimensional Rigid Rotor m 18.0 12.5 E 8.0 4.5 2.0 0.5 Only 1 quantum number is require to determine the state of the system.

  15. Spherical coordinates

  16. Spherical polar coordinate

  17. Hamiltonian in spherical polar coordinate

  18. Rigid Rotor in Quantum Mechanics Transition from the above classical expression to quantum mechanics can be carried out by replacing the total angular momentum by the corresponding operator: Wave functions must contain both θand Φ dependence: are called spherical harmonics

  19. Schrondinger equation

  20. Two equations

  21. Solution of second equation

  22. Solution of First equation Associated Legendre Polynomial

  23. Associated Legendre Polynomial

  24. For l=0, m=0

  25. First spherical harmonics Spherical Harmonic, Y0,0

  26. l= 1, m=0

  27. l= 1, m=0

  28. l=2, m=0

  29. l = 1, m=±1 Complex Value?? If Ф1 and Ф2 are degenerateeigenfunctions, their linear combinations are also an eigenfunction with the same eigenvalue.

  30. l=1, m=±1 Along x-axis

  31. Three-Dimensional Rigid Rotor States l m 3 2 6.0 1 0 -1 -2 -3 E 2 1 3.0 0 -1 -2 1 1.0 0 -1 0 0.5 Only 2 quantum numbers are required to determine the state of the system.

  32. Rotational Spectroscopy J: Rotational quantum number Rotational Constant

  33. Rotational Spectroscopy Wavenumber (cm-1) Rotational Constant Line spacing v Dv Frequency (v)

  34. Bond length To a good approximation, the microwave spectrum of H35Cl consists of a series of equally spaced lines, separated by 6.26*1011 Hz. Calculate the bond length of H35Cl.

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