Harmonic Oscillator and Rigid Rotator Yao-Yuan Chuang
Outline Classical Harmonic Oscillator Conservation of Energy of a Classical Oscillator Harmonic Oscillator Model of a Diatomic Molecule The Harmonic Oscillator Approximation The Energy Levels of a Quantum Mechanical Harmonic Oscillator Infrared Spectra of Diatomic Molecules
E. Bright Wilson, Jr. (1908-1992) Studied with Linus Pauling. Wrote 3 famous books, Introduction to Q.M., Molecular Vibrations, and Introduction to Scientific research. His son Kenneth was awarded Nobel Prize in 1982.
Molecular Motion Particle in the box is useful for understanding how translational motion in various potentials. Vibration – Harmonic Oscillator Rotation – Rigid Rotator
Harmonic Oscillator Consider a mass m connected to a wall by a spring stretched compressed
Example Show the general solution can be written in the form Solution:
Total Energy
Harmonic Oscillator of a Diatomic Molecule
Harmonic Oscillator of a Diatomic Molecule
Internuclear Potential Only lowest one or two vibrational energy levels are occupied for most molecules for ~300K harmonic Anharmonic i.e. Morse Bond Length V(x) = ½ k x2 k: force constant
Force Constants
Quantum Harmonic Oscillator The Schrodinger Equation for a one-dimensional harmonic oscillator is
Wave Functions The wave functions corresponding to the eigenvalues for a harmonic oscillator are nondegenerate
Quantum Harmonic Oscillator
Wave Functions
Wave Functions
Example Problem 18.1
Example Problem 18.2
Example Show that Y0 and Y1 are normalized and orthogonal to each other
Useful Integrals
Spherical Coordinates
Spherical and Cartesian
Integration
Integration
Example
Solid Angle The solid enclosed by the surface that connects the origin and the area DA is called solid angle
Example
Molecular Motion
Classical Rigid Rotor
Classical Rigid Rotator
Quantum Rigid Rotor in 2D
Exmaple Problem 18.4
Rigid rotor in 2D
Rigid rotor in 2D
Rigid Rotator in 3D
Rigid Rotor in 3D
Quantization of Angular Momentum
Quantization of Angular Momentum
Spherical Harmonic Functions
Spherical Harmonics Y(0,0) Y(1,0) Y(1,1) Y(1,-1)
Spherical Harmonics Y(2,1) Y(2,0) Y(2,2) Y(2,-2) Y(2,-1)
Spherical Harmonics Y(3,2) Y(3,-2) Y(3,1) Y(3,-1) Y(3,0) Y(3,3)
Spherical Harmonic Functions
Spatial Quantization
Classical Mechanics Assume a particle with mass (m) moving with velocity (v) in a circular path (radius r) with linear momentum p (=mv) v m r Conservation of angular momentum
Orbital Angular Momentum in Quantum Mechanics and Vector Model md=0,=90 md=+1, =65.9 md=+2, =35.3 md=-1, =114 md=-2, =144.7
Total Angular Momentum
Addition and Conservation of Angular Momentum in Quantum Mechanics d1 and d2 are not coupled, each precesses about the z axis independently. d1,d2,m1,m2 are good quantum numbers, and they are separately obseravable. d d2 m d1 and d2 are coupled to form d and they precess together in phase, no longer have constant z component, d1,d2,m1,m2 are not good quantum numbers d1
Angular Momentum Operator
Notation Orbital angular momentum of an electron in an atom (l and ml) Spin angular momentum of an electron (s and ms) Total angular momentum of an electron (j and mj) For more than one electron (L,ML,S,MS,J,MJ) Nucleus (I and MI) Diatomic molecules (J and MJ) Molecule (R and MR)
Example
Example
Example
Angular momentum operator
Commutator of Angular momentum operator
Commutator of Angular momentum operator
Commutator of Angular momentum operator