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**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)