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Visualizing the Probability Density for a Particle Confined to a Ring: Instructor's Guide

Main Ideas

  • Students use Maple to visualize eigenfunctions and linear combinations of eigenfunctions in one dimension on a ring.
  • Students use Maple to observe the time evolution of linear combinations of eigenfunctions
  • Students can review the relationship between visual and algebraic representations of wavefunctions in a simple context before working with the more complicated functions in the hydrogen atom solution.

Students' Task

Estimated Time: 30 minutes + 10 minute wrap-up

Students work through a Maple worksheet, observing the time-dependent probability density for different combinations of eigenstates on the ring. They spend the class time trying to understand the worksheet and what the plots mean. They also look for patterns and try to understand how the mathematical representation of the wave function corresponds to the visual representations in maple and the meaning of this in a physical context.

Prerequisite Knowledge

  • Students should have some familiarity with Maple. In particular, they need to know how to navigate a worksheet, execute exiting commands, edit commands, and manipulate animation plots.
  • Students should be familiar with the eigenfunctions and time evolution for the motion of a particle confined to a 1-D ring.


Activity: Introduction

A common student mistake is to apply the time dependence to the wavefunction as a whole rather than applying the time dependence to each energy eigenfunction. The introduction of this activity provides a useful context in which to remind students of this common mistake.

When students first open up Maple it is useful to walk them through the worksheet explaining the basic purpose of each step in the worksheet. It is not important to focus on the details at this point, but going through the steps in the worksheet helps keep some groups from getting stuck or just blasting through the worksheet without understanding what Maple is doing.

It may be helpful to review with students the meaning of the probability density since this is the primary quantity plotted in this worksheet.

Activity: Student Conversations

After students have some time to play with the worksheet and understand what is being calculated and what value is being plotted, it is useful to ask some of the following questions.

  • What is being plotted?
  • What causes the time dependence? How is that related to what you've seen in the case of spins?
  • What do you expect to happen if you change the amplitudes in front of one of the components of the wave function?
  • What do you expect to happen if you make one of the coefficients complex or imaginary?
  • The fourth function down seems to oscillate from one side of the x-axis to the other. That is, the largest peaks appear along the x-axis. How would you change the wave function to maintain the shape of this sloshing, but move these peaks to the y-axis or somewhere else?
  • When/How do you get standing waves? Traveling waves?
  • What is the difference between those wave functions that appear to be sloshing verses those that appear to be rotating?

It it is sometimes helpful to have a follow up discussion after students have had some time to play with the worksheet outside of class. This gives each student some time to be in control of the worksheet and gives them time to play without the distractions and pressures of class.

Activity: Wrap-up

Discussing what patterns students observed is a good start to a wrapup. This conversation can often lead to discussions of which wavefunctions result in observable time dependence and which do not. It is useful to look at the case of two wave functions added together, since this shows the simplest time dependence.

This observation can be connected back to the time evolution of the two state system.

$$|\Psi\rangle = a_m e^{-i {E_m\over\hbar} t} |m\rangle +a_n e^{-i {E_n\over\hbar}t} |n\rangle$$

$$\langle\phi|\Psi\rangle = a_m e^{-i {E_m\over\hbar}t} \langle\phi|m\rangle +a_n e^{-i {E_n\over\hbar}t} \langle\phi|n\rangle$$

$$\langle\phi|\Psi\rangle = e^{-i {E_m\over\hbar}t} (a_m \Phi_m(\phi) +a_n e^{-i {(E_n-E_m)\over\hbar}t} \Phi_n(\phi))$$

$$|\langle\phi|\Psi\rangle|^2 = e^{i {E_m\over\hbar}t} (a_m^* \Phi_m^*(\phi) +a_n^* e^{i {(E_n-E_m)\over\hbar}t} \Phi_n^*(\phi)) e^{-i {E_m\over\hbar}t} (a_m \Phi_m(\phi) +a_n e^{-i {(E_n-E_m)\over\hbar}t} \Phi_n(\phi))$$

$$|\langle\phi|\Psi\rangle|^2 = e^{i {E_m\over\hbar}t} e^{-i {E_m\over\hbar}t} (a_m^*a_m \Phi_m^*(\phi)\Phi_m(\phi)) + a_m^*a_n e^{i {(E_n-E_m)\over\hbar}t} \Phi_m^*(\phi)\Phi_n(\phi)) + a_n^*a_m e^{-i {(E_n-E_m)\over\hbar}t} \Phi_n^*(\phi)\Phi_m(\phi) + a_n^*a_n e^{i{(E_n-E_m)\over\hbar}t}e^{-i {(E_n-E_m)\over\hbar}t} \Phi_n^*(\phi)\Phi_n(\phi))$$

If $a_m$ and $a_n$ are real then

$$|\langle\phi|\Psi\rangle|^2 = a_m^2 |\Phi_m(\phi)|^2 + 2a_m a_n \cos\bigl(\frac{\Delta E_{mn}}{\hbar} t\bigr) \Phi_m(\phi)\Phi_n(\phi) + a_n^2 |\Phi_n(\phi)|^2$$


This activity can be used as part of a sequence of Maple activities that allows one to explore probability densities for a particle first confined to a ring, then to the surface of a sphere, and finally for the entire three-dimensional hydrogen atom.

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