Skip to main content
Chemistry LibreTexts

6.3: Quantum-Mechanical Description of the Harmonic Oscillator

  • Page ID
  • alt

    Figure \(\PageIndex{3}\).6: The harmonic oscillator wavefunctions describing the four lowest energy states.

    Exercise \(\PageIndex{21}\)

    Figure \(\PageIndex{7}\) in terms of the magnitude of the normal coordinate Q. Couch your discussion in terms of the HCl molecule. How would you describe the location of the atoms in each of the states? How does the oscillator position correspond to the energy of a particular level?

    Exercise \(\PageIndex{23}\)

    Plot the probability density for energy level 10 of the harmonic oscillator. How many nodes are present? Plot the probability density for energy level 20. Compare the plot for level 20 with that of level 10 and level 1. Compare these quantum mechanical probability distributions to those expected for a classical oscillator. What conclusion can you draw about the probability of the location of the oscillator and the length of a chemical bond in a vibrating molecule? Extend your analysis to include a very high level, like level 50.

    In completing Exercise \(\PageIndex{23}\), you should have noticed that as the quantum number increases and becomes very large, the probability distribution approaches that of a classical oscillator. This observation is very general. It was first noticed by Bohr, and is called the Bohr Correspondence Principle. This principle states that classical behavior is approached in the limit of large values for a quantum number. A classical oscillator is most likely to be found in the region of space where its velocity is the smallest. This situation is similar to walking through one room and running through another. In which room do you spend more time? Where is it more likely that you will be found?