1. Harmonic Oscillator and Rigid Rotator
Yao-Yuan Chuang

2. Outline Classical Harmonic Oscillator
Conservation of Energy of a Classical Oscillator
Harmonic Oscillator Model of a Diatomic Molecule
The Harmonic Oscillator Approximation
The Energy Levels of a Quantum Mechanical Harmonic Oscillator
Infrared Spectra of Diatomic Molecules

3. E. Bright Wilson, Jr. ( )
Studied with Linus Pauling. Wrote 3 famous books, Introduction to Q.M., Molecular Vibrations, and Introduction to Scientific research. His son Kenneth was awarded Nobel Prize in 1982.

4. Molecular Motion
Particle in the box is useful for understanding how translational motion in various potentials.
Vibration – Harmonic Oscillator
Rotation – Rigid Rotator

5. Harmonic Oscillator Consider a mass m connected to a wall by a spring
stretched
compressed

6. Example
Show the general solution can be written in the form
Solution:

7. Total Energy

8. Harmonic Oscillator of a Diatomic Molecule

9. Harmonic Oscillator of a Diatomic Molecule

10. Internuclear Potential
Only lowest one
or two vibrational
energy levels are
occupied for most
molecules for ~300K
harmonic
Anharmonic
i.e. Morse
Bond Length
V(x) = ½ k x2
k: force constant

11. Force Constants

12. Quantum Harmonic Oscillator
The Schrodinger Equation for a one-dimensional harmonic oscillator is

13. Wave Functions
The wave functions corresponding to the eigenvalues for a harmonic oscillator are nondegenerate

14. Quantum Harmonic Oscillator

15. Wave Functions

16. Wave Functions

17. Example Problem 18.1

18. Example Problem 18.2

19. Example
Show that Y0 and Y1 are normalized and orthogonal to each other

20. Useful Integrals

21. Spherical Coordinates

22. Spherical and Cartesian

23. Integration

24. Integration

25. Example

26. Solid Angle
The solid enclosed by the surface that connects the origin and the area DA is called solid angle

27. Example

28. Molecular Motion

29. Classical Rigid Rotor

30. Classical Rigid Rotator

31. Quantum Rigid Rotor in 2D

32. Exmaple Problem 18.4

33. Rigid rotor in 2D

34. Rigid rotor in 2D

35. Rigid Rotator in 3D

36. Rigid Rotor in 3D

37. Quantization of Angular Momentum

38. Quantization of Angular Momentum

39. Spherical Harmonic Functions

40. Spherical Harmonics
Y(0,0)
Y(1,0)
Y(1,1)
Y(1,-1)

41. Spherical Harmonics
Y(2,1)
Y(2,0)
Y(2,2)
Y(2,-2)
Y(2,-1)

42. Spherical Harmonics Y(3,2) Y(3,-2) Y(3,1) Y(3,-1) Y(3,0) Y(3,3)

43. Spherical Harmonic Functions

44. Spatial Quantization

45. Classical Mechanics
Assume a particle with mass (m) moving with velocity (v) in a circular path (radius r) with linear momentum p (=mv) v m  r
Conservation of angular momentum

46. Orbital Angular Momentum in Quantum Mechanics and Vector Model
md=0,=90
md=+1, =65.9
md=+2, =35.3
md=-1, =114
md=-2, =144.7

47. Total Angular Momentum

48. Addition and Conservation of Angular Momentum in Quantum Mechanics
d1 and d2 are not coupled, each
precesses about the z axis
independently. d1,d2,m1,m2 are
good quantum numbers, and they
are separately obseravable. d d2 m
d1 and d2 are coupled to form d and they
precess together in phase, no longer have
constant z component, d1,d2,m1,m2
are not good quantum numbers d1

49. Angular Momentum Operator

50. Notation Orbital angular momentum of an electron in an atom (l and ml)
Spin angular momentum of an electron (s and ms)
Total angular momentum of an electron (j and mj)
For more than one electron (L,ML,S,MS,J,MJ)
Nucleus (I and MI)
Diatomic molecules (J and MJ)
Molecule (R and MR)

51. Example

52. Example

53. Example

54. Angular momentum operator

55. Commutator of Angular momentum operator

56. Commutator of Angular momentum operator

57. Commutator of Angular momentum operator