This module will introduce a range of computational and analytic methods that can be used to model the properties and dynamics of physical systems. The module is divided into three parts: Molecular Simulation, Computational Quantum Mechanics and Mathematical Physics.
In Molecular Simulation, we will consider the molecular dynamics (MD) and Monte Carlo (MC) simulation methods. These methods may be used to simulate the behaviour of systems at the molecular level. We will investigate first the molecular dynamics method, which is deterministic, and then the Monte Carlo method, which is stochastic. Particular attention will be paid to the physical basis of the algorithms used and their efficient and reliable implementation. We will then focus on how to extract physical properties from the results of the simulation and assess the errors in them. A range of applications will be introduced.
In Computational Quantum Mechanics, we will explore computation methods for modelling systems at the quantum mechanical level. Whilst simple problems can be solved analytically, numerical/computational methods have to be used for anything more complex than a few electrons. The theoretical approximations used and their consequences will be explored by lectures and supported by practical sessions
In Mathematical Physics, the concept of a Green’s Function will be introduced and applied to describe the general response of linear systems analytically. Examples including the harmonic oscillator and the wave equation will be considered and their behaviour contrasted with non-linear regimes that either show unusual coherence (solitons) or chaotic behaviour.
Module learning outcomes
Describe and apply the physical principles underlying MD and MC simulation methods.
Assess the advantages and disadvantages of the MD and MC methods.
Write an MD or MC program.
Write a program to extract physical properties from an MD or MC simulation and calculate the errors in them.
Discuss a variety of applications of MD and MC methods.
Computational Quantum Mechanics
Derive integration schemes for TISE and TDSE problems.
Use numerical integration to solve single electron problems.
Analyse the results of numerical experiments in terms of QM principles.
Calculate approximate analytical solutions with the Rayleigh-Ritz method.
Explain the usefulness of Total Energy calculations.
Recognise the various approximations used in different Total Energy schemes and their consequences.
Use pre-written Total Energy packages for ab initio calculations of material properties.
Define the Green’s Function for linear differential equations subject to boundary conditions.
Find the Green’s Functions for simple 1D systems e.g. the harmonic oscillator and the wave equation and calculate the response of such systems to general forces.
Derive Green’s Functions in terms of the eigenvalues and eigenvectors of the so called homogeneous equation.
Give simple physical interpretations of Green’s Functions in more complex systems.
Extend the D’Alembert Solution of the wave equation to incorporate dispersion and non-linear soliton behaviour.
Investigate simple computer models e.g. the logistic model and non-linear oscillators to illustrate the principles of deterministic chaos.
Overview of particle simulation.
Introduction to molecular dynamics (MD) simulations.
Equations of motion for atomic systems; phase space and trajectories.
Finite difference methods. Assessing reliability.
Interaction potentials: Lennard-Jones and Coulomb potentials.
Computational techniques: efficient calculation of forces; periodic boundary conditions and the minimum image criterion; potential truncation, shifted potentials and other methods of calculating long-range forces; neighbour lists.
Initialisation, equilibration and production processes. Recording the results from the production run.
Static properties of fluids: internal energy, temperature, pressure, mean-square force; heat capacity; the pair-distribution function.
Dynamic properties of fluids: diffusion, viscosity, dynamic structure. Overview of time correlation functions and the calculation of transport coefficients.
Estimation of errors.
Molecular dynamics in other ensembles and with other potentials.
Introduction to Monte Carlo (MC) simulations.
The Metropolis algorithm; the underlying principles and basic statistical mechanics; generation of points in configuration space with probability proportional to the Boltzmann factor. The principle of detailed balance. Analysing the results.
Monte Carlo in other ensembles.
Computational Quantum Mechanics:
The Single Electron Problem.
Time Independent Schrödinger Equation.
Time Dependent Schrödinger Equation.
Total Energy Methods.
Basis Sets & Pseudopotentials.
Density Functional Theory.
Quantum Monte Carlo.
Revision of differential equations and delta functions.
Application to the harmonic oscillator, the wave equation and the diffusion equation.
Dispersive wave equations – examples of deep and shallow water waves.
Molecular Simulation: The module will be assessed by an assignment, which will involve writing and testing a program that implements an application of the methods to a simple physical system (for example, use MD to find the constant volume heat capacity of a Lennard-Jones system as a function of temperature and present the outcome of this experiment in a formal report, similar in style to those you have written for the Computational Physics Laboratory).
Computational Quantum Mechanics: In the PPQs for this part of the module you will practise the different skills required to complete the assignment successfully. The major component of the assignment will require writing and testing a computer program to solve a particular QM problem.
Mathematical Physics: In the PPQs you will practise applying the theoretical methods to solve problems. A final one-hour exam at the end of the course will be used to assess whether you have achieved the learning outcomes.