H = ½ω(p2 + q2)

The Harmonic Oscillator QM

Recap of the Rotational and Vibrational Energy Level Expressions for a Rigid

Diatomic Molecule Vibrating with Simple Harmonic Motion

Recap Rot & Vib Energy Level

y = ax2

The Quadratic Curve

Harmonic Oscillator

A Classical Description E = T + V E = ½mv2 + ½kx2

B QM description - the Hamiltonian H v = E(v) v

C Solve the Hamiltonian - Energy Levels G(v) = ω(v+ ½) (cm-1)

D Selection Rules - Allowed Transitions v = ±1

E Transition Frequencies > G = ω

F Intensities - THE SPECTRUM

J Analysis - Pattern recognition; assign quantum numbers

H Experimental Details - spectrometers, lasers

I More Advanced Details: anharmonicity

J Information: potential, force constants, group identification

Harry Kroto 2004

Hooke

F = -kx

Anharmonic Oscillator

Born and Oppenheimer

Born-Oppenheimer Theory

E= i Ei

Born Oppenheimer Separation

Separation Vibration Rotation

Born Oppenheimer Separation Vib - Rot

Harry Kroto 2004

Vibration Rotation Spectroscopy

CO Infra Red Spectrum (Colin)

ABC Rotation of a Diatomic Molecule

CO Rotational Spectrum PROBLEM

Hamilton