## 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?

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