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Problem 12.1: Compare classical and quantum harmonic oscillator probability distributions

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The normalized classical probability distributions and quantum-mechanical probability densities for the harmonic oscillator are shown in position and momentum space.

  1. Click on Position Graph below the right-hand graph. The graph shows the probability that a particle is in the ground state at some position x.  You may vary n to see higher energy states. Under the left-hand graph, a ball is attached to a spring and the spring is initially stretched. What does the classical probability distribution of finding the particle as a function of x look like?   Briefly discuss your reasoning.  After you answer, click  Position Graph below the left-hand graph and check yourself.   Did your answer agree with the given answer. Explain why or why not.
  2. Under what conditions would the right-hand graph look like the left-hand graph?  In other words, what is the correspondence between the classical and quantum position probabilities of a particle in a harmonic oscillator potential energy function?  Check your answer using the above "Position Graph" buttons.
  3. Click on Momentum Graph on the right-hand graph. Displayed is a graph of the probability density in momentum space as a function of p. The box <p> gives the expectation value of the momentum of the particle. Now click on Velocity Graph on the left-hand graph. What is the difference you see?  Why does this difference exist?
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