
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
Motion of Two Bodies Centre of Mass Weighted average of all positions Internal Coordinates: In C.M. Coordinates:
Kinetic Energy Terms ? ? ? ? ? ? ? ?
Centre of Mass Coordinates Similarly
Centre of Mass Coordinates Reduced mass
Hamiltonian Separable! C.M. Motion 3-D P.I.B Internal Motion Rotation Vibration
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???
Angular Momentum p v f m r w Perpendicular to R and p L Orientation remains constant with time
r R Center of mass Rotational Motion and Angular Momentum As p is always perpendicular to r Moment of inertia
r R Center of mass Rotational Motion and Angular Momentum
r R Center of mass Rotational Motion and Angular Momentum Classical Kinetic Energy
r R Center of mass Rotational Motion and Angular Momentum Sincer and p are perpendicular
Momentum Summary Classical QM Linear Momentum Energy Rotational (Angular) Momentum Energy
Two-Dimensional Rotational Motion Polar Coordinates y r f How to we get: x
Two-Dimensional Rotational Motion product rule
Two-Dimensional Rotational Motion product rule
Two-Dimensional Rigid Rotor Assume ris rigid, ie. it is constant As the system is rotating about the z-axis
18_05fig_PChem.jpg Two-Dimensional Rigid Rotor
18_05fig_PChem.jpg Two-Dimensional Rigid Rotor
18_05fig_PChem.jpg Two-Dimensional Rigid Rotor Periodic m = quantum number
18_05fig_PChem.jpg Two-Dimensional Rigid Rotor
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.
Orthogonality m = m’ m ≠ m’ 18_06fig_PChem.jpg
14_01fig_PChem.jpg Spherical Polar Coordinates ?
14_01fig_PChem.jpg Spherical Polar Coordinates
14_01fig_PChem.jpg The Gradient in Spherical Polar Coordinates Gradient in Spherical Polar coordinates expressed in Cartesian Coordinates
14_01fig_PChem.jpg The Gradient in Spherical Polar Coordinates Gradient in Cartesian coordinates expressed in Spherical Polar Coordinates
14_01fig_PChem.jpg The Gradient in Spherical Polar Coordinates
14_01fig_PChem.jpg The Gradient in Spherical Polar Coordinates
14_01fig_PChem.jpg The Laplacian in Spherical Polar Coordinates Radial Term Angular Terms OR OR
Three-Dimensional Rigid Rotor Assume ris rigid, ie. it is constant. Then all energy is from rotational motion only.
18_05fig_PChem.jpg Three-Dimensional Rigid Rotor Separable?
Three-Dimensional Rigid Rotor k2= separation Constant Two separate independent equations
18_05fig_PChem.jpg Three-Dimensional Rigid Rotor Recall 2D Rigid Rotor
18_05fig_PChem.jpg Three-Dimensional Rigid Rotor This equation can be solving using a series expansion, using a Fourier Series: Legendre polynomials Where
Three-Dimensional Rigid Rotor Spherical Harmonics
The Spherical Harmonics For l=0, m=0
The Spherical Harmonics For l=0, m=0 Everywhere on the surface of the sphere has value what is ro ? r = (ro, q, f)