Shown is a ramped potential and six trial wave functions.
You may choose a trial wave function by clicking any Trial Wave
Function link. Then you may choose a level n from 1 to 7 by
using the slider. Two questions further test the understanding of the relationship between the wave
function and the potential.
Questions
1. Which Trial Wave Function(s) could represent the energy
eigenstates of the green potential?
2. How do you know? Be as explicit and as complete as
possible in your explanation.
Answers
1. Trial Wave Function C.
2. Notice the shape of the potential. The potential is
deeper at x = -1 and shallower at x = 1. We therefore expect that
for a given energy the wavelength should be smaller (greater KE) towards x = -1,
and the wavelength should be larger (smaller KE) towards x = +1. As the
energy gets larger, the wave function will have a non-zero value closer to x =
1. Finally, the amplitude must be larger where the well is shallower, as
the probability of finding the particle there is greater than in the deeper part of
the well.
Features of this Script
Two DataGraph applets are embedded on the same page. Unique applet name/id
necessary for each instance. The EnergyEigenvalue applet calculates the wave
functions but is absent from the screen as it's data are sent to the DataGraph
on the right.
Required Resources
Jar files: DataGraph4_.jar, EnergyEigenvalue4_.jar, STools4.jar
References
This problem is inspired by one of the best quantum mechanics
problems ever posed [D. Styer, Quantum Mechanics: See it Now, AAPT
Kissimmee, FL Jan 2000 and http://www.oberlin.edu/physics/dstyer/TeachQM/see.html.],
Problem 3-17 (Exposing an unsuccessful plot, p. 152) in An
Introduction to Quantum Physics, A. P. French and Edwin F. Taylor, Norton,
New York, 1978.
Credits
Script by Mario Belloni and Wolfgang Christian.
Questions by Mario Belloni and Larry Cain.
Java applets by Wolfgang Christian.