## A Potential and Possible Wave Functions

#### Description

Shown is a ramped potential and six trial wave functions.
You may choose an n from 1 to 10 then click the **Trial Wave
Function** link to see that state. Two questions further test the understanding of the relationship between the wave
function and the potential.

#### Questions

1. Which Trial Wave Function could represent the energy
eigenstate(s) of the orange potential?

2. How do you know? Be as explicit and as complete as
possible.

#### 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 can 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 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 French's *An
Introduction to Quantum Physics*.

#### Credits

Script by Mario Belloni.

Questions by Mario Belloni.

Java applets by Wolfgang Christian.