18_12afig_PChem.jpg Rotational Motion Center of Mass Translational Motion r1r1 r2r2 Motion of Two Bodies Each type of motion is best represented in its own coordinate system best suited to solving the equations involved  k RcRc Internal coordinates Cartesian Internal motion (w.r.t CM) Motion of the C.M. Origin r Vibrational Motion

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

Kinetic Energy Terms ? ? ??? ???

Centre of Mass Coordinates

Similarly

Centre of Mass Coordinates

Reduced mass

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

Rotational Motion and Angular Momentum We rotational motion to internal coordinates Linear momentum of a rotating Body ss  Angular Velocity Parallel to moving body p(t 1 ) p(t 2 ) Always changing direction with time??? Always perpendicular to r

Angular Momentum v m r  p L  Perpendicular to R and p Orientation remains constant with time

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

Rotational Motion and Angular Momentum r Center of mass R

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

Rotational Motion and Angular Momentum Since r and p are perpendicular r Center of mass R

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

Angular Momentum

Angular Momentum in QM

Angular Momentum

Two-Dimensional Rotational Motion x y  r How to we get: Polar Coordinates

Two-Dimensional Rotational Motion product rule

Two-Dimensional Rotational Motion product rule

Two-Dimensional Rotational Motion

Two-Dimensional Rigid Rotor Assume r is 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

E m Only 1 quantum number is require to determine the state of the system.

Normalization

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

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 OR Radial Term Angular Terms

Three-Dimensional Rigid Rotor Assume r is 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 Two separate independent equations k 2 = separation Constant

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: Where Legendre polynomials

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 r o ? r = (r o, 

The Spherical Harmonics r = (1,  Normalization: In Spherical Polar Coordinates r is fixed at r o. The wavefunction is an angular function which has a constant value over the entire unit circle. X Y Z

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

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

18_05fig_PChem.jpg The Spherical Harmonics YZ XZ

The Spherical Harmonics Are Orthonormal Example

Y l,m are Eigenfuncions of H, L 2, L z

Dirac Notation is complete Continuous Functions Vectors Dirac Bra Ket

Dirac Notation Degenerate

Dirac Notation

18_16fig_PChem.jpg 3-D Rotational motion & The Angular Momentum Vector m indicates the orientation of the angular momentum with respect to z-axis l determines the length of 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.

Three-Dimensional Rigid Rotor States E l Only 2 quantum numbers are require to determine the state of the system. m

19_01tbl_PChem.jpg Rotational Spectroscopy

19_13fig_PChem.jpg Rotational Spectroscopy J : Rotational quantum number

Rotational Spectroscopy Wavenumber (cm -1 ) Rotational Constant Frequency ( v ) vv v Line spacing

Rotational Spectroscopy Predict the linespacing for the 16 O 1 H radical. m O = amu = x kg m H = amu = x kg r = 0.97 A = 9.7 x m 1 amu = 1 g/mol = (0.001 kg/mol)/6.022 x mol -1 = x kg

Rotational Spectroscopy The line spacing for 1 H 35 Cl is cm -1, determine its bond length. m Cl = amu = x kg m H = amu = x kg

? ? The Transverse Components of Angular Momentum Y lm are eigenfunctions of L 2 and L z but not of L x and L y Therefore L x and L y do not commute with either L 2 or L z !!!

Commutation of Angular Momentum Components product rule

Commutation of Angular Momentum Components product rule

Commutation of Angular Momentum Components

Cyclic Commutation of Angular Momentum

Commutation with Total Angular Momentum

This means that only any one component of angular momentum can be determined at one time.

Ladder Operators

What do these ladder operators actually do??? ?? Recall That: Raising Operator Lowering Operator Similarly

Therefore is an eigenfunction of with eigen values l and m+1 Ladder Operators Which implies that

Ladder Operators This is not an eigen relationship!!!! is not an normalization constant!!! These relationships indicates that a change in state, by  m=+/-1, is caused by L + and L - Can these operators be applied indefinitely?? Remember that there is a max and min value for m, as it represents a component of L, and therefore must be smaller than L. ie. Why is Not allowed ?

More Useful Properties of Ladder Operators 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

More Useful Properties of Ladder Operators Lets show that m min and m max are l and -l resp. Have to be determined in terms of

More Useful Properties of Ladder Operators Also note that: Similarly

Ladder Operators

Since the minimum value cannot be larger than the maximum value, therefore.