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Problem 10.1: Compare classical and quantum infinite square well probability distributions

n =

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The normalized classical probability distributions and quantum-mechanical probability densities for the infinite square well are shown in position and momentum space. Restart.

  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 bouncing back and forth between the two walls. What does the classical probability distribution as a function of x look like?  Briefly discuss your reasoning.  After you answer, click "Position Graph" below the left-hand graph and check your answer. Explain why your answer agreed with, or disagreed with, the given answer.
  2. Under what conditions could the right-hand graph look like the left-hand graph?  In other words, what is the correspondence between the classical probability distribution and quantum position probability of a particle in a 1-d box?  Check your answer using the "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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