H = ½ ω (p 2 + q 2 ) The Harmonic Oscillator QM.

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Presentation transcript:

H = ½ ω (p 2 + q 2 ) 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 = ax 2 The Quadratic Curve

Harmonic Oscillator

AClassical Description E = T + V E = ½mv 2 + ½kx 2 B QM description - the Hamiltonian H  v  = E(v)  v  CSolve the Hamiltonian - Energy Levels G(v) = ω(v+ ½) (cm -1 ) DSelection Rules - Allowed Transitions  v = ±1 ETransition Frequencies >  G = ω FIntensities - THE SPECTRUM J Analysis - Pattern recognition; assign quantum numbers HExperimental Details - spectrometers, lasers IMore Advanced Details: anharmonicity JInformation: potential, force constants, group identification Harry Kroto 2004

Hooke F = -kx

Anharmonic Oscillator

Born and Oppenheimer

Born-Oppenheimer Theory E=  i E i

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