PHYCS 498CQM: Homework 2

The numerical problems below can be done with F90 codes that can be downloaded from the class Web pages.

1. Thijssen Exercise A.3 (part b and c) Work the parts of the problem to carry out the error analysis of the the Numerov algorithm.
2. Scattering Cross section for the H-Kr problem described in Thijssen, Chapter 2.

Formulate the problem as a differential equation with one-parameter (i.e. show that only one combination of the parameters in the problem is sufficient to specify the differential equation).
The radial Schrödinger equation can be integrated using the F90 class codes, which already have the Lennard-Jones potential as an option. The output is the radial equation at the last 5 points.
Use this program to calculate the wavefunction for a range of energies and for various angular momenta.
Write a program to calculate the phase shifts and the contribution of each angular momentum to the total scattering cross section (Eq. 2.8). Plot the results as a function of energy, and compare with the description in the text for real scattering of H-Kr.
3. Eigenvalues of the one-dimensional Schrödinger equation:

You may use the class programs that solve with the numerov method and matching at the classical turning point (or at a point you choose).
1. Test on a harmonic oscillator to see that the results are correct.
2. For the harmonic oscillator change the matching point until you find a case where the numerical solution is very bad.
3. Is there a bound state in the H-Kr problem for any l?
4. If not vary the parameters to an example where there is is a bound state.

Hand in (or email a tar file)
1. Your solution for problem 1 and the first question in problem 2 (hand-written is OK).
2. Your code (in any language) for problem 2 and plots for the cross scetion as a function of energy.
3. Output for your results for problem 3.