6810: 1094 Activities 5

Online handouts: eigen_tridiagonal.cpp printout, eigen_basis.cpp printout, harmonic_oscillator.cpp printout, nan_test.cpp printout, and square_well.nb Mathematica notebook.

Your goals for these activities:

• Take a first look at your shell start-up file.
• See some examples of nan's and inf's.
• Use the eigen_tridiagonal program to look at eigenvalues of the harmonic oscillator potential.
• Find the lowest bound-state eigenvalues of two familiar potentials using the eigen_basis program.
• Examine (and try to understand) how the accuracy of your results depends on the size of the harmonic oscillator basis and the choice of the basis parameter b.

Optional: Aliases in Your Start-up Shell

The shell is the program you type to at the command line. If you are using Linux on a Department account, you are most likely using tcsh. If you are using Cygwin or Ubuntu or a Mac, you are probably using bash. There are many good things to learn about the shells. Today we'll just learn about aliases. Just follow the corresponding instructions for bash (tcsh).

• Bash (tcsh) aliases. The .bashrc (or .cshrc.more) file sits in your home directory and is "sourced" when you start an interactive terminal. Take a look at the sample .bashrc (or .cshrc.more) file printout. These files, without the ".", are in the session 5 zip file. Copy any aliases of interest to your own .bashrc (or .cshrc.more) or the whole thing using cp bashrc ~/.bashrc  (or cp cshrc.more ~/.cshrc.more) if you don't have one already. Activate the changes with the command: source ~/.bashrc (or source ~/.cshrc). Try some aliases such as ll (for "long listing") and df, which shows info about the disk file system. Do they work? Questions?
  
  
  
  
• Create your own alias. E.g., you could make an alias to change from your start-up (login) directory to your 6810 working directory. Be creative! What line did you add to the start-up file?
  
  
  
  

Nan's and Inf's

Just a quickie: Take a look at nan_test.cpp, use make_nan_test to create nan_test and then run it.

• What do you think the result of 0.*(1./0.) will be? Predict and then modify the code to check it out. [Note: Use the variables "numerator" and "denominator" to calculate (1./0.).]
  
  
  
  

Bound States by Matrix Diagonalization in Coordinate Representation

The program eigen_tridiagonal.cpp uses the GSL library routines explored in eigen_test.cpp (session 4) to find the eigenvalues and lowest eigenvector of the harmonic oscillator using a method described in the Session 5 notes. With the units here, the lowest eigenvalue should be (3/2)hbar-omega = 1.5 (read comments in code!).

1. Using make_eigen_tridiagonal, compile and run the code a few times to see how it works. Try various values of Rmax and N such as Rmax=3, N=20 or 50 (make a chart here of Rmax, N and the two lowest eigenvalues for five pairs of Rmax,N). How can you verify that the code is working?
   
   
   
   
   
   
   
   
2. Change the code so that only the lowest few eigenvalues are printed out. Look at the output file eigen_tridiagonal.dat and plot it with gnuplot. What is this function? The value at r=0 is not given; what should it be?
   
   
   
   
   
   
3. You need to pick a reasonable value of Rmax. Justify your choice based on the eigen_tridiagonal.dat plot:
   
   
   
   
   
   
4. For your choice of Rmax, try N = 4,8,16,32,64,128,256,512,1024 (you could add a loop to calculate these). How does the relative error for the lowest eigenvalue scale with N? Attach an appropriate plot to validate your answer.
   
   
   
   
   
5. Explain the slope you found based on the approximation to the second derivative.
   
   
   
   
   
   
6. Bonus: repeat with Rmax = 4 and explain what happens to your graph.
   
   
   
   

Bound States from Diagonalizing the Hamiltonian in a Basis

The program in eigen_basis.cpp uses the GSL library routines to diagonalize (i.e., to find the eigenvalues and eigenvectors) a Hamiltonian matrix in a basis of harmonic oscillator wave functions. You may want to refer to the GSL handout on eigensystems (there is also a printout of eigen_basis.cpp).

The eigen_basis program uses units with the particle mass=1 and hbar=1. The program asks you to choose

• a potential (Coulomb or square well);
• the parameter b for the harmonic oscillator basis (see harmonic_oscillator.cpp for the definition);
• the number of basis states to use.

The Coulomb potential is defined with Ze²=1, which means that the Bohr radius is also unity. This means that the exact bound energy levels are given by E_n = -1/2n², with n=1,2,...
The square well potential is defined with radius R=1 and depth V₀ = 50. You should find that there are three bound states.

Here are some subgoals. There won't be time for everything (as usual) but you'll have a chance to finish them as part of a future assignment.

1. Run the Mathematica notebook square_well.nb (make sure you understand what it is doing; e.g, look up FindRoot in the Help Browser). Find the bound-state energies for the square well parameters used here (you need to change the notebook parameters!).
   
   
   
   
   
   
2. Compile and link the code eigen_basis using make_eigen_basis. This also compiles harmonic_oscillator.cpp. Run it a few times with each of the potentials to get familiar with it. If you try too large a basis size, the run time may be too long (so start small!). Look through the printout to see the basic idea of how the code works and find where the equation for the matrix element is implemented.
3. Based on the "exact" results from Mathematica, which of the approximate eigenvalue(s) for the square well are most reliable? Why do you think this is?
   
   
   
   
   
   
4. Considering all three of the lowest eigenvalues, which are calculated most effectively, those of the Coulomb potential or the square well potential? Can you explain your observation?
   
   
   
   
   
   
5. You have under your control the size of the basis (i.e., the dimension of the matrix) and the harmonic oscillator parameter b (see harmonic_oscillator.cpp for the definition). For a fixed basis size (pick one that reproduces the ground state reasonably), how do you find the optimum b? (Hint: think gnuplot!) Can you qualitatively (or semi-quantitatively) account for your result? (Think about the potentials and guess what the lowest wave functions will look like and what changes about the basis when the harmonic oscillator parameter b is changed.)
   
   
   
   
   
   
6. If you now fix b (if you have time you can consider two or three different values in turn), how can you find how the accuracy of the ground state energy scales with the basis size? Make an appropriate plot.
   
   
   
   
   
   
   
   
7. Look at the code. How could you make it more efficient? (What do you think is the limiting factor based on the scaling of the time with the size of the basis?) For example, could you speed it up by almost a factor of two? (Hint, hint!)
   
   
   
   
   
   
8. (For PS#3) How would you find the wave function that corresponds to a given state (e.g., the ground state)? Add code to generate the lowest wave function for the lowest bound state (hint: it involves the eigenvector).