1. r2 r1 r Motion of Two Bodies w k Rc
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
Translational Motion k
Internal motion (w.r.t CM)
Vibrational Motion Rc
Internal
coordinates r1
Origin
18_12afig_PChem.jpg

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

3. Kinetic Energy Terms
Tanslational Motion: In C.M. Coordinates: ? ? ?
Rotation and Vibration: Internal Coordinates: ? ? ?

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
rotational motion requires 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 w r L f m Perpendicular to R and p
Orientation remains constant with time

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

13. Moment of Inertia and Internal Coordinates
Center
of mass R

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

15. Rotational Motion and Angular Momentum
Center
of mass R
Since r 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
Consider
product rule
product rule

24. Two-Dimensional Rotational Motion
Consider
product rule
product rule

25. Two-Dimensional Rotational Motion

26. Two-Dimensional Rotational Motion

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

28. Two-Dimensional Rigid Rotor
18_05fig_PChem.jpg

29. Two-Dimensional Rigid Rotor
18_05fig_PChem.jpg

30. Two-Dimensional Rigid Rotor
Periodic - Like a particle in a circular box
m = quantum number
18_05fig_PChem.jpg

31. Two-Dimensional Rigid Rotor
18_05fig_PChem.jpg

32. Two-Dimensional Rigid Rotor
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
For m = m’
For m ≠ m’
18_06fig_PChem.jpg

36. Spherical Polar Coordinates?
14_01fig_PChem.jpg

37. Spherical Polar Coordinates
14_01fig_PChem.jpg

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

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

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

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

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

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

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

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

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

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

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)

51. The Spherical Harmonics
Normalization:
In Spherical Polar Coordinates Z
r = (1, q, f) Y X
The wavefunction is an angular
function which has a constant value
over the entire unit circle.

52. The Spherical Harmonics
For l =1, m = 0
Along z-axis Z
r = (1, q, f) Y
The spherical Harmonics are often plotted as a vector starting from the origin with orientation q and f and its length is Y(q,f) X
The wavefunction is an angular
function which has a value varying
as on the entire unit circle.

53. The Spherical Harmonics
For l=1, m =±1
Complex Valued??
Along x-axis
Along y-axis
18_05fig_PChem.jpg

54. The Spherical HarmonicsXZ YZ
18_05fig_PChem.jpg

55. The Spherical Harmonics Are Orthonormal
Example
ODD

56. Yl,m are Eigenfuncions of H, L2, Lz

57. Dirac Notation
Continuous Functions
is complete
Vectors
Dirac
Bra
Ket

58. Dirac Notation

59. Dirac Notation
Example
Degenerate

60. Dirac Notation
Example

61. Dirac Notation
Example

62. 3-D Rotational motion & The Angular Momentum Vector
Rotational motion is quantized not continuous. Only certain states of motion are allowed that are determined by quantum numbers l and m.
l determines the length of the angular momentum vector
m indicates the orientation of the angular momentum with respect to z-axis
18_16fig_PChem.jpg

63. Three-Dimensional Rigid Rotor States3 2 1
6.0 -1 -2 -3 E 2 1
3.0 -1 -2 1
1.0 -1
0.5
Only 2 quantum numbers are require to determine the state of the system.

64. Rotational Spectroscopy
19_01tbl_PChem.jpg

65. Rotational Spectroscopy
J : Rotational quantum number
19_13fig_PChem.jpg

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

67. Rotational Spectroscopy
Predict the line spacing for the 16O1H radical.
r = 0.97 A = 9.7 x m
mO = amu = x kg
mH = amu = x kg
1 amu = 1 g/mol = (0.001 kg/mol)/6.022 x mol-1
= x kg

68. Rotational Spectroscopy
The line spacing for 1H35Cl is cm-1,
determine its bond length .
mCl = amu = x kg
mH = amu = x kg

69. ? ? The Transverse Components of Angular Momentum
Ylm are eigenfunctions of L2 and Lz but not of Lx and Ly
Therefore Lx and Ly do not commute with either L2 or Lz!!!

70. Commutation of Angular Momentum Components
FOIL
product rule

71. Commutation of Angular Momentum Components
FOIL
product rule

72. Commutation of Angular Momentum Components

73. Commutation of Angular Momentum Components

74. Cyclic Commutation of Angular Momentum

75. Commutation with Total Angular Momentum

76. Commutation with Total Angular Momentum

77. Commutation with Total Angular Momentum
Therefore they have simultaneous eigen functions, Yl,m
Also note that:
Therefore the transverse components do not
share the same eigen function as L2 and Lz.
This means that only any one component of angular momentum can be determined at one time.

78. Ladder Operators Consider: Note: Super operator
Like an eigen equation but for an operator!
Super operator

79. ? ? Ladder Operators What do these ladder operators actually do???
Recall That:
Raising Operator
Similarly
Lowering Operator

80. Ladder Operators Note: Similarly: Consider: Therefore is an
Eigenfunction of
with eigen values l and m+1
Which implies that

81. ? Ladder Operators These are not an eigen relationships!!!!
is not an normalization constant!!!
These relationships indicate a change in state, by Dm=+/-1, is caused by L+ and L-
Can these operators be applied indefinitely??
Not allowed
Recall: There is a max & min value for m, as it represents a component of L, and therefore must be smaller than l. ie. ?
Why is

82. More Useful Properties of Ladder Operators
Recall
This is an eigen equation of a physical observable that is always greater than zero, as it represents the difference between the magnitude of L and the square of its smaller z-component, which are both positive.
This means that m is constrained by l, and since m can be changed by ±1

83. More Useful Properties of Ladder Operators
Knowing that:
Lets show that mmin & mmax are l & -l.
Consider
have to be determined in terms of

84. More Useful Properties of Ladder Operators
Also note that:
Similarly

85. Ladder Operators
Recall

86. Ladder Operators Recall
Since the minimum value cannot be larger than the maximum value, therefore .

87. Spin Angular Momentum
Intrinsic Angular Momentum is a fundamental property like mass,and charge.

88. Coupling of Spin Angular Momentum

92. Spin and Magnetic Fields
Paramagnetism
ESR (EPR),
NMR (NPMR)
NQR
Mossbauer
Precession
Zeeman Splitting

93. Nuclear Magnetic Resonance

94. Fig. 1. 19F-NMR spectrum (56.4 MHz, 26°C) of the XeF5
XeF5+SbF6-
Fig F-NMR spectrum (56.4 MHz, 26°C) of the XeF5
cation (4.87 M XeF5 SbF6 in HF solution): (A) axial fluorine and (a) 129Xe satellites; (X) equatorial fluorines and (x) 129Xe satellites [30].