Phy 360

Weeks 6 - 7:  The Time Independent Schrödinger Equation II
Scattering States

Readings

Griffiths Chapter 2.

Physlet Quantum Physics and Ejs modeling
Chapters 6, 7, and 8

A. P. French and E. F. Taylor, “Qualitative plots of bound state wave functions,” Am. J. Phys. 39, 961-962 (1971).

Demos

Homework

Due Tuesday October 27 at 10:30.

Physlet Quantum Physics (available in the PCC)

Theory

Start with a normalized Gaussian f(x) and compute its Fourier transform F(k).  Compute the uncertainties Δx*Δk for this transform pair.  You may use Mathematica for this problem.

EJS Simulation

First read the French and Taylor paper on bound state wave functions.  Then use the Ejs shooting method in the ejs_qm_ShootMethod.jar archive from the download directory for this course to answer the following questions:

  1. Run the simulation with the default values.  Find the first few energy eigenvalues for the default potential V(x)=x*x/2 and sketch the eigenfunctions. At what quantum number do you notice the effect of the hard walls in the simulation?

  2. Set the x-minimum value to zero to create a hard wall at the origin.  Keep the x-maximum at 5 and keep the default potential.  How does this hard wall at the origin change the energy eigenvalues and eigenfunctions that were found in question 1?  Give values to three significant figures for the eigenvalues and sketch the eigenfunctions.

  3. Set x-minimum to -5, x-maximum to 5, and the potential to V(x) = 0.  Find the first few energy eigenvalues and sketch the eigenfunctions.  Repeat with x-minimum to -1 and x-maximum to 1.  How did the energy eigenvalues change when you narrowed the well?

  4. Set x-minimum to zero, x-maximum to 10, and the potential to V(x) = x.  Give values for the first three eigenvalues and sketch the eigenfunctions.

  5. Set x-minimum to -5, x-maximum to 5, and the potential to V(x) = x.  Give values for the first three eigenvalues and sketch the eigenfunctions.  How are the energy eigenvalues and energy eigenfunctions related to the values you found in question 4?

  6. Set x-minimum to -5 and x-maximum to 5 and set the potential to V(x) = abs(x).  Give values for the first three eigenvalues and sketch the eigenfunctions.  How are the energy eigenvalues and energy eigenfunctions related to the values you found in question 4?

Note that the simulation requires hard walls at x-minimum and x-maximum.  Only the region between the two extremum has the given potential.  Do you notice the effect of the hard wall in the above simulations?

A. P. French and E. F. Taylor, “Qualitative plots of bound state wave functions,” Am. J. Phys. 39, 961-962 (1971).

Griffiths

The entire class may work together (as a group) on Griffiths problem 2.27.  Placing your name on the class solution is your affirmation that you understand he solution and that you have contributed to the group project.

Optional challenge problem:

Build an EJS model that shows free particle E>0 energy eigenfunctions for the double delta potential.