COMPUTATIONAL PHYSICS
CONTENTS I INTRODUCTION TO COMPUTATIONAL PHYSICS INTRODUCTION CHOICE OF PROGRAMMING LANGUAGE DESIGNING PROGRAMS GETTING STARTED REPRESENTATION OF INTEGER NUMBERS REAL NUMBERS AND NUMERICAL PRECISION REPRESENTATION OF REAL NUMBERS FURTHER EXAMPLES LOSS OF PRECISION MACHINE NUMBERS FLOATING-POINT ERROR ANALYSIS ADDITIONAL FEATURES OF C/C++ AND FORTRAN 90/95 OPERATORS IN C/C++ POINTERS AND ARRAYS IN C/C++ MACROS IN C/C++ NUMERICAL DIFFERENTIATION THE SECOND DERIVATIVE OF E X ERROR ANALYSIS HOW TO MAKE ?GURES WITH GNUPLOT RICHARDSON’S DEFERRED EXTRAPOLATION METHOD CLASSES, TEMPLATES AND MODULES A ?RST ENCOUNTER, THE VECTOR CLASS CLASSES AND TEMPLATES IN C++ USING BLITZ++ WITH VECTORS AND MATRICES BUILDING NEW CLASSES MODULE AND TYPE DECLARATIONS IN FORTRAN 90/95 OBJECT ORIENTING IN FORTRAN 90/95 AN EXAMPLE OF USE OF CLASSES IN C++ AND MODULES IN FORTRAN 90/95 LINEAR ALGEBRA PROGRAMMING DETAILS DECLARATION OF ?XED-SIZED VECTORS AND MATRICES RUNTIME DECLARATIONS OF VECTORS AND MATRICES FORTRAN FEATURES OF MATRIX HANDLING LU DECOMPOSITION OF A MATRIX SOLUTION OF LINEAR SYSTEMS OF EQUATIONS INVERSE OF A MATRIX AND THE DETERMINANT PROJECT: MATRIX OPERATIONS NON-LINEAR EQUATIONS AND ROOTS OF POLYNOMIALS ITERATION METHODS BISECTION METHOD NEWTON-RAPHSON’S METHOD THE SECANT METHOD AND OTHER METHODS CALLING THE VARIOUS FUNCTIONS ROOTS OF POLYNOMIALS POLYNOMIALS DIVISION ROOT ?NDING BY NEWTON-RAPHSON’S METHOD ROOT ?NDING BY DE?ATION BAIRSTOW’S METHOD NUMERICAL INTERPOLATION, EXTRAPOLATION AND ?TTING OF DATA INTERPOLATION AND EXTRAPOLATION POLYNOMIAL INTERPOLATION AND EXTRAPOLATION QUBIC SPLINE INTERPOLATION NUMERICAL INTEGRATION EQUAL STEP METHODS GAUSSIAN QUADRATURE ORTHOGONAL POLYNOMIALS, LEGENDRE MESH POINTS AND WEIGHTS WITH ORTHOGONAL POLYNOMIALS APPLICATION TO THE CASE N = 2 GENERAL INTEGRATION INTERVALS FOR GAUSS-LEGENDRE OTHER ORTHOGONAL POLYNOMIALS APPLICATIONS TO SELECTED INTEGRALS TREATMENT OF SINGULAR INTEGRALS OUTLINE OF THE MONTE-CARLO STRATEGY FIRST ILLUSTRATION OF THE USE OF MONTE-CARLO METHODS, CRUDE INTEGRATION SECOND ILLUSTRATION, PARTICLES IN A BOX RADIOACTIVE DECAY PROGRAM EXAMPLE FOR RADIOACTIVE DECAY OF ONE TYPE OF NUCLEUS BRIEF SUMMARY PHYSICS PROJECT: DECAY OF 210BI AND 210PO RANDOM NUMBERS PROPERTIES OF SELECTED RANDOM NUMBER GENERATORS PROBABILITY DISTRIBUTION FUNCTIONS THE CENTRAL LIMIT THEOREM IMPROVED MONTE CARLO INTEGRATION CHANGE OF VARIABLES IMPORTANCE SAMPLING ACCEPTANCE-REJECTION METHOD MONTE CARLO INTEGRATION OF MULTIDIMENSIONAL INTEGRALS BRUTE FORCE INTEGRATION 10 RANDOM WALKS AND THE METROPOLIS ALGORITHM 10 1 MOTIVATION 10 2 DIFFUSION EQUATION AND RANDOM WALKS 10 2 1 DIFFUSION EQUATION 10 2 2 RANDOM WALKS 10 3 MICROSCOPIC DERIVATION OF THE DIFFUSION EQUATION 10 3 1 DISCRETIZED DIFFUSION EQUATION AND MARKOV CHAINS 10 3 2 CONTINUOUS EQUATIONS 10 3 3 NUMERICAL SIMULATION 10 4 THE METROPOLIS ALGORITHM AND DETAILED BALANCE 10 5 PHYSICS PROJECT: SIMULATION OF THE BOLTZMANN DISTRIBUTION 11 MONTE CARLO METHODS IN STATISTICAL PHYSICS 11 1 PHASE TRANSITIONS IN MAGNETIC SYSTEMS 11 1 1 THEORETICAL BACKGROUND 11 1 2 THE METROPOLIS ALGORITHM 11 2 PROGRAM EXAMPLE 11 2 1 PROGRAM FOR THE TWO-DIMENSIONAL ISING MODEL 11 3 SELECTED RESULTS FOR THE ISING MODEL 11 3 1 PHASE TRANSITIONS 11 3 2 HEAT CAPACITY AND SUSCEPTIBILITY AS FUNCTIONS OF NUMBER OF SPINS 11 3 3 THERMALIZATION 11 4 OTHER SPIN MODELS 11 4 1 POTTS MODEL 11 4 2 XY-MODEL 11 5 PHYSICS PROJECT: SIMULATION OF THE ISING MODEL 12 QUANTUM MONTE CARLO METHODS 12 1 INTRODUCTION 12 2 VARIATIONAL MONTE CARLO FOR QUANTUM MECHANICAL SYSTEMS 12 2 2 THE HYDROGEN ATOM 12 2 3 METROPOLIS SAMPLING FOR THE HYDROGEN ATOM AND THE HARMONIC OSCILLATOR 12 2 4 A NUCLEON IN A GAUSSIAN POTENTIAL 12 2 5 THE HELIUM ATOM 12 2 6 PROGRAM EXAMPLE FOR ATOMIC SYSTEMS 12 3 SIMULATION OF MOLECULAR SYSTEMS 12 3 1 THE H + 2 MOLECULE 12 3 2 PHYSICS PROJECT: THE H 2 MOLECULE 12 4 MANY-BODY SYSTEMS 12 4 1 LIQUID 4HE 12 4 2 BOSE-EINSTEIN CONDENSATION 12 4 3 QUANTUM DOTS 12 4 4 MULTI-ELECTRON ATOMS 13 EIGENSYSTEMS 13 1 INTRODUCTION 13 2 EIGENVALUE PROBLEMS 13 2 1 SIMILARITY TRANSFORMATIONS 13 2 2 JACOBI’S METHOD 13 2 3 DIAGONALIZATION THROUGH THE HOUSEHOLDER’S METHOD FOR TRI-DIAGONALIZATION 13 3 SCHRÖDINGER’S EQUATION (SE) THROUGH DIAGONALIZATION 13 4 PHYSICS PROJECTS: BOUND STATES IN MOMENTUM SPACE 13 5 PHYSICS PROJECTS: QUANTUM MECHANICAL SCATTERING 14 DIFFERENTIAL EQUATIONS 14 1 INTRODUCTION 14 2 ORDINARY DIFFERENTIAL EQUATIONS (ODE) 14 3 FINITE DIFFERENCE METHODS 14 3 1 IMPROVEMENTS TO EULER’S ALGORITHM, HIGHER-ORDER METHODS 14 4 MORE ON ?NITE DIFFERENCE METHODS, RUNGE-KUTTA METHODS 14 5 ADAPTIVE RUNGE-KUTTA AND MULTISTEP METHODS 14 6 PHYSICS EXAMPLES 14 6 1 IDEAL HARMONIC OSCILLATIONS 14 6 2 DAMPING OF HARMONIC OSCILLATIONS AND EXTERNAL FORCES 14 6 3 THE PENDULUM, A NONLINEAR DIFFERENTIAL EQUATION 14 6 4 SPINNING MAGNET 14 7 PHYSICS PROJECT: THE PENDULUM 14 7 1 ANALYTIC RESULTS FOR THE PENDULUM 14 7 2 THE PENDULUM CODE 14 8 PHYSICS PROJECT: PERIOD DOUBLING AND CHAOS 14 9 PHYSICS PROJECT: STUDIES OF NEUTRON STARS 14 9 1 THE EQUATIONS FOR A NEUTRON STAR 14 9 2 EQUILIBRIUM EQUATIONS 14 9 3 DIMENSIONLESS EQUATIONS 14 9 4 PROGRAM AND SELECTED RESULTS 14 10PHYSICS PROJECT: SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS 15 TWO POINT BOUNDARY VALUE PROBLEMS 15 1 INTRODUCTION 15 2 SCHRÖDINGER EQUATION 15 3 NUMEROV’S METHOD 15 4 SCHRÖDINGER EQUATION FOR A SPHERICAL BOX POTENTIAL 15 4 1 ANALYSIS OF U() AT = 0 15 4 2 ANALYSIS OF U() FOR ! 1 15 5 NUMERICAL PROCEDURE 15 6 ALGORITHM FOR SOLVING SCHRÖDINGER’S EQUATION 16 PARTIAL DIFFERENTIAL EQUATIONS 16 1 INTRODUCTION 16 2 DIFFUSION EQUATION 16 2 1 EXPLICIT SCHEME 16 2 2 IMPLICIT SCHEME 16 2 3 PROGRAM EXAMPLE 16 2 4 CRANK-NICOLSON SCHEME 16 2 5 NON-LINEAR TERMS AND IMPLEMENTATION OF THE CRANK-NICOLOSON SCHEME 16 3 LAPLACE’S AND POISSON’S EQUATIONS 16 4 WAVE EQUATION IN TWO DIMENSIONS 16 4 1 PROGRAM FOR THE + 1 WAVE EQUATION AND APPLICATIONS 16 5 INCLUSION OF NON-LINEAR TERMS IN THE WAVE EQUATION II ADVANCED TOPICS 17 MODELLING PHASE TRANSITIONS 17 1 METHODS TO CLASSIFY PHASE TRANSITION 17 1 1 THE HISTOGRAM METHOD 17 1 2 MULTI-HISTOGRAM METHOD 17 2 RENORMALIZATION GROUP APPROACH 18 HYDRODYNAMIC MODELS 19 DIFFUSION MONTE CARLO METHODS 19 1 DIFFUSION MONTE CARLO 19 2 OTHER QUANTUM MONTE CARLO TECHNIQUES AND SYSTEMS 20 FINITE ELEMENT METHOD 21 STOCHASTIC METHODS IN FINANCE 22 QUANTUM INFORMATION THEORY AND QUANTUM ALGORITHMS