Presentation on theme: "r2 r1 r Motion of Two Bodies w k Rc"— Presentation transcript:
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 involvedRotational MotionMotion of the C.M.Center of MassCartesianrr2Translational MotionkInternal motion (w.r.t CM)Vibrational MotionRcInternalcoordinatesr1Origin18_12afig_PChem.jpg
2 Motion of Two Bodies Centre of Mass Internal Coordinates: Weighted average of all positionsInternal Coordinates:In C.M. Coordinates:
3 Kinetic Energy TermsTanslational Motion: In C.M. Coordinates:???Rotation and Vibration: Internal Coordinates:???
10 Rotational Motion and Angular Momentum rotational motion requires internal coordinatesLinear momentum of a rotating Bodyp(t1)p(t2)DsfAngular VelocityParallel to moving bodyAlways perpendicular to rAlways 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 Centerof massRAs p is always perpendicular to rMoment of inertiaProxy for mass in rotational motion
13 Moment of Inertia and Internal Coordinates Centerof massR
14 Angular Momentum and Kinetic Energy Classical Kinetic EnergyrCenterof massR
15 Rotational Motion and Angular Momentum Centerof massRSince r and p are perpendicular
16 Momentum Summary Classical QM Linear Momentum Energy Rotational (Angular)MomentumEnergy
50 The Spherical Harmonics For l = 0, m = 0Everywhere on the surface of the sphere has valuewhat is ro ?r = (ro, q, f)
51 The Spherical Harmonics Normalization:In Spherical Polar CoordinatesZr = (1, q, f)YXThe wavefunction is an angularfunction which has a constant valueover the entire unit circle.
52 The Spherical Harmonics For l =1, m = 0Along z-axisZr = (1, q, f)YThe spherical Harmonics are often plotted as a vector starting from the origin with orientation q and f and its length is Y(q,f)XThe wavefunction is an angularfunction which has a value varyingas on the entire unit circle.
53 The Spherical Harmonics For l=1, m =±1Complex Valued??Along x-axisAlong y-axis18_05fig_PChem.jpg
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 vectorm indicates the orientation of the angular momentum with respect to z-axis18_16fig_PChem.jpg
63 Three-Dimensional Rigid Rotor States 3216.0-1-2-3E213.0-1-211.0-10.5Only 2 quantum numbers are require to determine the state of the system.
77 Commutation with Total Angular Momentum Therefore they have simultaneous eigen functions, Yl,mAlso note that:Therefore the transverse components do notshare 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 OperatorSimilarlyLowering Operator
80 Ladder Operators Note: Similarly: Consider: Therefore is an Eigenfunction ofwith eigen values l and m+1Which 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 allowedRecall: 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 RecallThis 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.Considerhave to be determined in terms of
84 More Useful Properties of Ladder Operators Also note that:Similarly