Computational Physics :

a Physics Honours / MSc by Coursework module

presented by

Dr. S. H. Connell :

Physics Dept. and Schonland Research Centre for Nuclear Sciences

 

"The art of computing is insight, not numbers"

 

Description: An advanced course in the use of computers to model problems in physics. The course material is presented in 9 topics of two lectures each. There are 3 tutorials which explore and extend the lecture material. There are 3 projects, requiring investigation, the development of a computer model, and then the verification and careful use of the model leading to insight into the problem studied.

 

Assumptions: A basic competence in quantum mechanics, nuclear physics, solid state physics, numerical analysis and any programming language. In the past, students with verly little or no experience of computing, or previous exposure to numerical analysis have been accepted. Extra tuition and mentoring is available in this case.

 

Evaluation : Year Mark 40 %

(4 Tutorials 5% each)

(2 Projects 10% each)

Final Exam 60%

(Theory part 15%)

(Take Home Project 45%)

 

Philosophy: Modern high performance computers have allowed the development of a new branch of physics - computational physics. Physical systems can be either solved or simulated mathematically using the computer to an astonishingly high degree of complexity. It is important that this process goes beyond the application of blind numbercrunching algorithms. This course develops insight into numerical analysis, computational methods and physical systems in a way that is unique to physicists. The tools of physical intuition, extrapolation to known behaviours / extremes, substitution with / application to familiar systems, comparison to other numerical or analytical approximations or even nature itself are some of the procedures which give insight and confidence in the computer based exploration of new physics territory.

 

Resources

1 Lecture Notes

2 Computational Physics - Fortran Version, SE Koonin and DC Meredith, Addison-Wesley (1990)

3 Numerical Analysis by Ralston and Robinowitz

4 Numerical recipes in C (Fortrtan) : the art of scientific computing. Cambridge University Press, 1992.

5 Computational Physics, John Wiley & Sons, RH Landau and MJ Pez

6 Specialised materials (Research papers, Program Libraries, WWW etc )


LECTURE # 0: Introduction

* C++, starter tutorials

* Linux, editors, compilers, makefiles

* Numerical Recipes

 

LECTURE # 1: Elementary Mathematical Operations

* numerical differentiation

* numerical integration

* determining roots

* Example - scattering from a Central Potential

* Project 1 - Semi-Classical Quantisation of molecular vibrations, using the Lennard - Jones potential.

 

LECTURE # 2: Ordinary Differential Equations

 

* simple methods of solution

* multi-step procedures and implicit methods

* Runge-Kutta methods

* stability

* Example - Order and Chaos in 2-D motion

* Example - Structure of white dwarfs

 

LECTURE # 3: Boundary- and Eigen-Value Problems

 

* the Numerow algorithms

* direct integration of boundary value problems

* solving eigen-value problems using Greens functions

* Example - Eigenvalues and stationary states of the Schrodinger Wave Equation.

* Project 2 - Bound States - QM treatment of the deuteron

 

LECTURE # 4: Numerical Integration

 

* special functions

* Gaussian quadratures

* Example - Solution of a QM scattering problem using Born Approximation and the DWBA.

* Example - Scattering states using Partial Waves

 

 

LECTURE # 5: Matrix Operations

 

* inverting a matrix

* eigen-values of a tridiagonal matrix

* Gauss-Jordan elimination

* LU decomposition

* Example - Many body sytems; a schematic shell model

 


LECTURE # 6: Partial Differential Equations

 

* Elliptic, parabolic, hyperbolic PDE's; initial value problems, boundary value problems.

* Finite Differencing

* Simultaneous Over Relaxation (SOR)

* Operator splitting (ADI)

* Example - Parallel plate capacitor.

* Example - Field calculations and particle trajectories.

 

LECTURE # 7: Modelling of Data

 

* Statistics; max. likelihood parameter estimation

* Linear Least Squares fitting

* Non-linear methods; Simplex, Levenberg-Marquardt, Variable metric.

* Example - Nuclear Form Factors.

* Errors, distributions and significance

testing

* Example - 3 families of Leptons from LEP data.

 

LECTURE # 8: Monte-Carlo Simulations

 

* basis of the Monte-Carlo strategy

* Uniform random deviates.

* Sampling methods for other distributions (discrete and continuous); direct solution, tabular and Neumann Rejection methods.

* Specific cases for deviates; Poisson, Binomial, Gaussian, Exponential and Gamma.

* Example - Particle transport; simulation of high energy lepton and photon Cherenkov spectra in lead glass detectors.

* Example GEANT4 : particle - particle and particle - matter interactions from 10 keV to 10 Tev.