Lecture # 8 Quantum Mechanical Solution for Harmonic Oscillators CHEM 515 Spectroscopy Lecture # 8 Quantum Mechanical Solution for Harmonic Oscillators

Harmonic Oscillator Model This kind of motion is called simple harmonic motion and the system a simple harmonic oscillator.

Potential Energy for Harmonic Oscillator The oscillator has total energy equal to kinetic energy + potential energy. when the oscillator is at A, it is momentarily at rest, so has no kinetic energy U=0

Energy Levels for a Quantum Mechanical Harmonic Oscillator

Harmonic Oscillator Potential Curves Morse potential

Morse Potential It is a better approximation for the vibrational structure of the molecule than the quantum harmonic oscillator because it explicitly includes the effects of bond breaking, such as the existence of unbound states. It also accounts for the anharmonicity of real bonds.

Morse Potential The dissociation energy De is larger than the true energy required for dissociation D0 due to the zero point energy of the lowest (v = 0) vibrational level.

Vibrational Wave Functions (ψvib)

Vibrational Wave Functions (ψvib)

Vibrational Wave Functions (ψvib)

Probability Distributions for the Quantum Oscillator (ψ2vib) The square of the wave function gives the probability of finding the oscillator at a particular value of x.

Probability Distributions for the Quantum Oscillator (ψ2vib) there is a finite probability that the oscillator will be found outside the "potential well" indicated by the smooth curve. This is forbidden in classical physics.

Vibrational-Rotational Energy Levels

Various Types of Infrared Transition