A Potential and Possible Wave Functions

 Choose a Wave Function, then a Level:  
Trial Wave Function A Trial Wave Function D
Trial Wave Function B Trial Wave Function E
Trial Wave Function C Trial Wave Function F

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.