Physics 115/242, Computational Physics (Spring 2008)

Instructor: Peter Young, ISB 212, Tel: 9-4151
e-mail:
Time and Place: MWF 9:30 - 10:40, Earth & Marine Sciences B214.

Office Hour: Will be decided at the first class.

Computational Physics is intended to be of interest to students in other science and engineering departments as well as physics. Two aspects of the course should be particularly noted:

1. In addition to requiring students to write code in one of the standard programming languages, C, C++, or fortran, to study such topics as errors, integration, and solution of differential equations, a substantial part of the course will involve using the powerful features of MATHEMATICA, including its graphics capabilities, to study some more advanced topics such as chaos, period doubling, fractals, and quantum mechanics problems with non-trivial potentials.
2. It will also be offered at the GRADUATE level as Physics 242. Students taking the course at the graduate level will be required to solve some additional and harder problems, and do some more advanced projects.

Prerequisites

If you have trouble with the prerequisites, then either talk to me, or send me an e-mail at or see me at the end of the first class.

Books

No books are required. An best (optional) text for the C/fortran part is:

• Computational Physics: Problem Solving with Computers, by R. H. Landau and M. J. Paez, Wiley. This has quite a bit of the early material but not some of the later material of the C/fortran part (e.g. Monte Carlo and Molecular Dynamics simulations). There is a web site: http://www.physics.orst.edu/~rubin/Books/CPbook/index.html which includes programs in C and fortran.

• Computational Physics by Tao Pang, Cambridge University Press. Has quite a bit on simulations, but doesn't have the early material such as sources of error.
• Numerical Methods for Physics by A. L. Garcia, Prentice Hall. Emphasis on differential equations, ordinary and partial.
• Computational Physics by N. J. Giardino and H. Nakanishi, Prentice Hall. Lots of material on differential equations and simulations.
• Numerical Recipes in C (also exists in versions for Fortran and C++) by Press et al. Cambridge University Press. This is the ``bible'' for numerical methods. It is far more thorough and detailed than the material to be covered in the course, but no serious student who use computational methods in science should be without it. There is a web site with all the routines available on-line at http://www.nr.com

The best book for the Mathematica part is:

• A Physicists Guide to Mathematica by P. T. Tam, Academic Press. This is a few years old, and discusses mainly versions 2.2 and 3 of Mathematica, though the current version is version 5. However, virtually all the text on version 3 also applies to version 4, see the author's web site http://www.humboldt.edu/~ptt1/PTHome.html, especially http://www.humboldt.edu/~ptt1/tam0300V4.htm. Most of it should also work in the current version, version 5. However, as we will discuss in class, version 5 has some new "features", which means that old code may need slight modification.

• The Mathematica Book by S. Wolfram, Cambridge University Press. A huge volume written by the creator of Mathematica.
• Mathematica for Physicists by R.L. Zimmermann and F.L. Olness, Addison Wesley. Has a useful concise introduction followed by lots of examples of using Mathematica to solve problems in physics.
• Mathematica for Scientists and Engineers by R. Gass, Prentice Hall.
• A crash course in Mathematica by S. Kaufmann, Birkhauser. A useful concise introduction.
• Mathematica for Calculus--Based Physics by M. de Jong, Addison-Wesley.
• Mastering Mathematica by John W. Gray, Academic Press.

• Introduction to Scientific Programming by J. L. Zachary, Telos. This book is too elementary for the course, but does provide some simple examples, and an introduction to Mathematica.
• Physics by Computer by W. Kinzel and G. Reents, Springer. A good source of ideas for numerical problems to work on. Requires a high degree of sophistication and indepedence from the student.

These books are all available on reserve in the library. It is probably not necessary to buy both a C-based and an Mathematica-based book, and you may wish to discuss with me before buying a book.

Software

Topics

• Representation of numbers on the computer.
• Errors; roundoff and approximation.
• Numerical Differentation; use of midpoint and error-extrapolation methods to improve accuracy.
• Numerical Integration; trapezoidal rule, Simpson's rule, Romberg integration, treatment of singularities at the endpoints, midpoint rule. Monte Carlo integration.
• Root finding; bisection, secant method, Newton-Raphson, and fixed point iteration.
• Numerical Solution of Ordinary Differential Equations; Euler method, Runge-Kutta. leapfrog, discussion of symplectic algorithms. Application to the Kepler problem. Molecular dynamics simulations.
• Least squares fitting.
• Introduction to sorting algorithms.
• Stochastic (i.e. random) processes; random numbers, random walks, Monte Carlo simulations in statistical physics.
• Introduction to Mathematica, including miscellaneous problems.
• Zeroes of the Riemann zeta function (an example of Mathematica's knowledge of Mathematical functions)
• Projectiles with air resistance.
• Logistic Map--Period Doubling.
• Chaos in differential equations; e.g. transition to chaos in the Duffing equation.
• Fractals--Mandelbrot set.
• Quantum Mechanics; energy levels in quantum wells--coordinate representation.
• Quantum Mechanics; energy levels in quantum wells--matrix formulation.
• Solitons in the sine-Gordon and Korteweg de Vries equations.

Evaluation of Performance