University of the Witwatersrand
Computational Physics : 2004 Project 1 :
Semi-Classical Quantisation of Molecular Vibrations
in a Lennard - Jones Potential
String together modules from Numerical Recipes in a ``driver program''
to perform the elementary mathematical operations discussed in
Lecture 1. You can the calculate , the energies of
the quantised vibration states.
You will have to think quite carefully about the structure
of your calculation in order to achieve the correct relationships
between the various numerical components.
Usually, if this is your first excursion into Computational
Physics, your calculation will initially be unstable and subject
to run time errors and crashing, because of numerical
issues related to external errors, internal errors and stability.
At this stage, try to visualise the program flow and the values of
variables with debugging software.
Once you have the program running, devise methods to explore
the reliability of the results from a purely numerical point
of view. Document these.
Now you can study the reliability of the results using your physics
insight. Here, you would consider analgous problems, which may be
simpler, or analytically solvable, or problems where your
intuition about the behaviour is better informed.
For example, if you have designed your code to be
modular, then switch the Lennard-Jones potential for the
Harmonic Oscillator potential. Note that it is sensible to
tune the parameters of the Harmonic Oscillator potential by matching it
to the Taylor expansion of the Lennard-Jones potential up to
second order. Do you get evenly spaced energy levels ? Does the
Lennard-Jones potential correspond to the Harmonic Oscillator potential
at least near the minimum ? Now you can try the Square Well potential.
This is also analytic.
Derive the result
Now run your program for different values of . To be
realistic, consider that for H, and for
O, . Interpret the
result. Does it conform to your intuition ?
Finally, visualise the energy levels within the potential and
sketch the corresponding phase space trajectories. The simplest way is to
generate output files of your results and read these into a scientific