Exercise 2: Eigenstates and eigenvalues of single-particle problems

The program for finding the eigenvalues of the harmonic oscillator are in the github folder https://github.com/NuclearStructure/PHY981/tree/master/doc/pub/spdata/programs.

You can use this program to solve the exercise below, or write your own using your preferred programming language, be it python, fortran or c++ or other languages. Here I will mainly provide fortran, python and c++.

a) Compute the eigenvalues of the three lowest states with a given orbital momentum and oscillator frequency \( \omega \). Study these results as functions of the the maximum value of \( r \) and the number of integration points \( n \), starting with \( r_{\mathrm{max}}=10 \). Compare the computed ones with the exact values and comment your results.

b) Plot thereafter the eigenfunctions as functions of \( r \) for the lowest-lying state with a given orbital momentum \( l \).

c) Replace thereafter the harmonic oscillator potential with a Woods-Saxon potential using the parameters discussed above. Compute the lowest five eigenvalues and plot the eigenfunction of the lowest-lying state. How does this compare with the harmonic oscillator? Comment your results and possible implications for nuclear physics studies.