Physics 214b-2008 Walter F. Smith Assignment 8

Due: 4 pm Wednesday 4-2-08

Reading: Townsend Ch. 5 and/or Griffiths Ch. 3

Exercises

All problems are group problems, unless otherwise marked; you are encouraged to work in small groups on these.. For the individual problems, you are not allowed to consult any other students, but you may ask me for help.

4.28 part c only

8A. a. Using Mathematica, find the values of the coefficients for problem 4.28. If you used complex exponentials for your solution in the region 0<x<a, you should redo problem 4.28 using sine functions for that region instead. (If you use complex exponentials in a region for which the solutions are of the bound state form, the coefficients must be complex. If instead you use sines and cosines in such a region, the coefficients are real. Do you see why we only need the sine in this case?)

b. Using Mathematica, show explicitly that your answer for the reflection coefficient is correct. Don’t forget to use FullSimplify.

4.2 (individual problem)

8B. Look at the potential described in problem 4.9.

a. Sketch the two lowest energy eigenfunctions (assuming the well is deep enough that there are at least two energy eigenfunctions).

b. What wavefunctions that you have previously studied are these identical to?

c. Show that there will be no bound state unless .

d. Write the mathematical form for the ground state eigenfunction (assuming the well is deep enough for one to exist), and solve for the values of all constants that appear. (You will probably want to use Mathematica for this.)

8C. Consider the finite square well with, where is the ground state energy for the case . Let the particle bound in this well be an electron, and let the width of the well be 2 Å.

a. Explain why, using the infinite square well as a naïve guide, one might expect that there would be only one bound state for this finite square well.

b. In fact, there are two bound states. Use Mathematica to solve graphically for the energy of the higher state (i.e. the lowest energy odd state). You will not be able to get an exact number (unless you can make Mathematica work better than I can) – you will need to read the energy off your graph.