CHEM%20515%20Spectroscopy

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CHEM 515Spectroscopy

• Lecture 8
• Quantum Mechanical Solution for Harmonic
  Oscillators

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Harmonic Oscillator Model

• This kind of motion is called simple harmonic
  motion and the system a simple harmonic
  oscillator.

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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

U0
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Energy Levels for a Quantum Mechanical Harmonic
Oscillator
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Harmonic Oscillator Potential Curves
Harmonic oscillator
Morse potential
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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.

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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.

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Vibrational Wave Functions (?vib)
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Vibrational Wave Functions (?vib)
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Vibrational Wave Functions (?vib)
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Probability Distributions for the Quantum
Oscillator (?2vib)

• The square of the wave function gives the
  probability of finding the oscillator at a
  particular value of x.

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Probability Distributions for the Quantum
Oscillator (?2vib)

• 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.

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Vibrational-Rotational Energy Levels
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Various Types of Infrared Transition