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  1. 18_12afig_PChem.jpg Motion of Two Bodies w Each type of motion is best represented in its own coordinate system best suited to solving the equations involved Rotational Motion Motion of the C.M. Center of Mass Cartesian r r2 k Translational Motion Internal motion (w.r.t CM) Vibrational Motion Rc Internal coordinates r1 Origin

  2. Motion of Two Bodies Centre of Mass Weighted average of all positions Internal Coordinates: In C.M. Coordinates:

  3. Kinetic Energy Terms ? ? ? ? ? ? ? ?

  4. Centre of Mass Coordinates

  5. Centre of Mass Coordinates

  6. Centre of Mass Coordinates Similarly

  7. Centre of Mass Coordinates

  8. Centre of Mass Coordinates Reduced mass

  9. Hamiltonian Separable! C.M. Motion 3-D P.I.B Internal Motion Rotation Vibration

  10. Rotational Motion and Angular Momentum We rotational motion to internal coordinates Linear momentum of a rotating Body p(t1) p(t2) Ds f Angular Velocity Parallel to moving body Always perpendicular to r Always changing direction with time???

  11. Angular Momentum p v f m r w Perpendicular to R and p L Orientation remains constant with time

  12. r R Center of mass Rotational Motion and Angular Momentum As p is always perpendicular to r Moment of inertia

  13. r R Center of mass Rotational Motion and Angular Momentum

  14. r R Center of mass Rotational Motion and Angular Momentum Classical Kinetic Energy

  15. r R Center of mass Rotational Motion and Angular Momentum Sincer and p are perpendicular

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

  17. Angular Momentum

  18. Angular Momentum

  19. Angular Momentum in QM

  20. Angular Momentum

  21. Angular Momentum

  22. Two-Dimensional Rotational Motion Polar Coordinates y r f How to we get: x

  23. Two-Dimensional Rotational Motion product rule

  24. Two-Dimensional Rotational Motion product rule

  25. Two-Dimensional Rotational Motion

  26. Two-Dimensional Rotational Motion

  27. Two-Dimensional Rigid Rotor Assume ris rigid, ie. it is constant As the system is rotating about the z-axis

  28. 18_05fig_PChem.jpg Two-Dimensional Rigid Rotor

  29. 18_05fig_PChem.jpg Two-Dimensional Rigid Rotor

  30. 18_05fig_PChem.jpg Two-Dimensional Rigid Rotor Periodic m = quantum number

  31. 18_05fig_PChem.jpg Two-Dimensional Rigid Rotor

  32. 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.

  33. Normalization

  34. Normalization

  35. Orthogonality m = m’ m ≠ m’ 18_06fig_PChem.jpg

  36. 14_01fig_PChem.jpg Spherical Polar Coordinates ?

  37. 14_01fig_PChem.jpg Spherical Polar Coordinates

  38. 14_01fig_PChem.jpg The Gradient in Spherical Polar Coordinates Gradient in Spherical Polar coordinates expressed in Cartesian Coordinates

  39. 14_01fig_PChem.jpg The Gradient in Spherical Polar Coordinates Gradient in Cartesian coordinates expressed in Spherical Polar Coordinates

  40. 14_01fig_PChem.jpg The Gradient in Spherical Polar Coordinates

  41. 14_01fig_PChem.jpg The Gradient in Spherical Polar Coordinates

  42. 14_01fig_PChem.jpg The Laplacian in Spherical Polar Coordinates Radial Term Angular Terms OR OR

  43. Three-Dimensional Rigid Rotor Assume ris rigid, ie. it is constant. Then all energy is from rotational motion only.

  44. 18_05fig_PChem.jpg Three-Dimensional Rigid Rotor Separable?

  45. Three-Dimensional Rigid Rotor k2= separation Constant Two separate independent equations

  46. 18_05fig_PChem.jpg Three-Dimensional Rigid Rotor Recall 2D Rigid Rotor

  47. 18_05fig_PChem.jpg Three-Dimensional Rigid Rotor This equation can be solving using a series expansion, using a Fourier Series: Legendre polynomials Where

  48. Three-Dimensional Rigid Rotor Spherical Harmonics

  49. The Spherical Harmonics For l=0, m=0

  50. The Spherical Harmonics For l=0, m=0 Everywhere on the surface of the sphere has value what is ro ? r = (ro, q, f)