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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.
- 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.
- 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.
- 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?