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2010/1 Module Catalogue
 Module Code: PHY2056 Module Title: MATHEMATICAL, QUANTUM AND COMPUTATIONAL PHYSICS
Module Provider: Physics Short Name: PH2M-MQCP
Level: HE2 Module Co-ordinator: SEAR RP Dr (Physics)
Number of credits: 30 Number of ECTS credits: 15
 
Module Availability

Semester 1 and Semester 2.

Assessment Pattern

Assessment Pattern

 

Unit(s) of Assessment

 

Assessment Weighting

 

Mathematics IV Examination

 

25%

 

Quantum Physics Examination

 

25%

 

Computational Mathematics Coursework (Semester 1)

 

17%

 

Mathematics IV Class Test (Semester 1)

 

8%

 

Quantum Physics Coursework  (Semester 2)

 

8%

 

Computational Modelling Coursework (Semester 2)

 

17%

 

Qualifying Condition(s):

 

University general regulations refer.

 

 

Assessment Schedule

 

Examination Paper 1 (June):

 

2.5 hour examination consisting of;

 

Answer 2 from 3 questions on Mathematics IV

 

Answer 2 from 3 questions on Quantum Physics

 

(weighted at 75% of the MCQP examination Unit of Assessment for each of Mathematics IV and Quantum Physics)

 

 

Examination Paper 5 (June):

 

3 hour examination consisting of sections on Mathematical Quantum and Computational Physics (PHY2056), Classical Physics (PHY2015) and Modern Physics (PHY2017);

 

Answer 4 questions on Mathematics IV and Quantum Physics, in total

 

(weighted at 25% of the MCQP examination Unit of Assessment for each of Mathematics IV and Quantum Physics)

 

Semester 1 Coursework:

 

Computational Mathematics – 1 feedback and 3 assessed computational assignments,

 

Mathematics IV Class Test during week 15

 

Semester 2 Coursework:

 

Quantum Physics Coursework

 

Computational Modelling – report on modelling project

 

 

 

 

 

Module Overview

Mathematics IV and Quantum Physics:

 

Mathematical and Quantum Physics are delivered by lecture and tutorial periods. The mathematics element deals with the vector calculus, the structure and methods of solution, both analytical and numerical, of homogeneous and inhomogeneous second order ordinary differential equations. Mathematics IV also introduces partial differential equations in Cartesian coordinates and involving time. The Quantum Physics elements form a first course in the basic formalism of quantum mechanics, its physical interpretation and its application to simple problems. The emphasis is on elementary (one-dimensional) quantum physics, eigenfunctions in one-dimensional step potentials and barrier potentials and their interpretation in terms of transmission and reflection coefficients. Tunnelling and the concept of a bound state are discussed. 

 

 

Computational Mathematics and Computational Modelling:

 

Through the Level HE1 Computational Laboratory work or equivalent, students will be assumed to have acquired a working knowledge of the required computing facilities, the use of editors and some experience in writing shorter programs in the FORTRAN 90/95 programming language. The Computational Mathematics component (autumn) includes 10 one hour computing laboratory sessions in which the student will carry out four (one formative, three summative) assessed Computational Mathematics assignments to implement the taught numerical methods. These include applications that are relevant to the quantum physics component.

 

 

The Computational Modelling component (spring) comprises a six week computational physics/IT based project. This component uses the students’ previous experience in computer literacy to extend their experiences through a computer-based investigation of a topic of particular relevance to their chosen field of study or specialist degree programme.

 

Prerequisites/Co-requisites

PH1031 - Waves, Particles and Quanta Module or equivalent

 

PH1012 – Mathematics Module

 

PH1011 - Computational Laboratory component, or equivalent

 

Module Aims

Mathematics IV and Quantum Physics:

 

To introduce the physical significance and properties of the gradient, divergence and curl operators and to give some practice in their use. To introduce Ñ2 in preparation for a discussion of the equations of the mathematical physics. To familiarise and through worked examples provide expertise in the analytical and numerical solution of differential equations required for modelling quantum mechanical and other physical systems in one and more dimensions and involving time.

 

 

To introduce the concept of a complex probability amplitude and how to calculate with it and make physical predictions.  To introduce the role of the Schrödinger equation in quantum dynamics.  To develop the properties of a linear operator, its eigenvalue spectrum and properties of its eigenfunctions.  To provide methods to calculate bound state eigenfunctions in an infinite square well potential. To explore one-dimensional quantum systems such as finite barriers and finite wells, their applications, and to introduce concepts such as orthogonality, parity, Hermiticity and completeness. To develop proficiency in the application of mathematical methods to these problems.

 

 

Computational Mathematics and Computational Modelling:

 

The aims of the Computational Mathematics component are to further develop computational skills and give practical experience in implementing some of the key algorithms used for modelling in quantum physics and related fields.

 

 

The aims of the Computational Modelling component are two-fold:

 

1.                  To write a moderate-sized computer program to model a given physical process.

 

2.                  To use the program to investigate the underlying physics of the given process.

 

Learning Outcomes

Mathematics IV and Quantum Physics:

 

Mathematics: 

 

Students will recognise and will be able to use the operators of vector calculus, to classify homogeneous and inhomogeneous systems of linear differential equations, and be able to apply several methods of solution in simple cases.  Students will have an appreciation of, and an ability to solve, partial differential equations in Cartesian coordinates and involving time, and will have gained a familiarity in the forms of the solutions in physically interesting cases.

 

 

Quantum Physics

 

i.                Recognition of leading role of the wave function in quantum mechanics

 

ii.              To calculate probability densities, probabilities, means, uncertainties (standard deviations)

 

iii.             To compare and contrast time evolution in quantum and classical mechanics, and the role of the Hamiltonian.

 

iv.            To use operators, operator expressions, commutators; to find eigenvalues and eigenvectors of common operators; to use the relation between eigensolutions and results of measurements

 

v.              To find and interpret QM eigensolutions of an infinite square well

 

vi.            To use superpositions of energy eigenstates: to find their time evolution and interpret their probability densities.

 

vii.           .To solve Schrodinger's equation for step and barrier potentials; to find transmission and reflection coefficients; to compare quantum and classical results

 

viii.         To solve Schrodinger's equation for a square well potential after parity separation; to find and interpret bound states

 

ix.            To find, interpret and use eigenfunction expansions

 

x.              To revise and integrate the module material in solving problems.

 

 

Computational Mathematics and Computational Modelling:

 

Through the Computational Mathematics component, students will be able to solve linear and non┐linear differential equations numerically using simple finite difference algorithms (in FORTRAN 90/95) and will have an appreciation of the accuracy of the methods used.  They should be able to apply these methods to related problems.

 

 

Through the Computational Modelling component, students should have:

 

-          developed their ability to write moderate-sized computer programs to model physical processes;

 

-          An improved confidence in developing and writing computer programs for scientific applications

 

-          an awareness of the value and also of the limitations of numerical methods in the simulation of physical systems

 

They will have gained a deeper understanding of the physical processes and principles underlying the particular system they have modelled.

 

Module Content

Mathematics IV and Quantum Physics:

 

Mathematics IV

 

Introduction to the physical significance and properties of the gradient, divergence and curl operators and to Ñ2 for use in equations of the mathematical physics.

 

 

Review of the solution of first-order ordinary differential equations: use of integrating factors.

 

 

Homogeneous and inhomogeneous ordinary second order differential equations: the source term and its physical interpretation, arbitrary constants of solution and boundary conditions.

 

 

The solution of equations with constant coefficients: the complementary function, the particular integral; the general solution, development of the D operator technique of solution, the characteristic equation, detailed solution of second order equations with constant coefficients.

 

 

Equations with functional coefficients: equations of Cauchy form, solution by series, the method of Frobenius, indicial equations; recurrence relations, convergence, the method of variation of parameters.

 

 

Introduction to equations of more than one variable: The equations of mathematical physics, Laplace's equation, the wave equation, the diffusion equation, Poisson's equation; coordinate systems, Ñ2 in Cartesian systems, arbitrary functions of solution and boundary conditions.

 

 

Discussion of the method of separable solutions: introduction to separable solutions in Cartesian coordinates and in time, the use of Fourier series.

 

 

Quantum Physics

 

The Schrödinger equation for a point particle (electron) moving in 1-D in a potential field V(x).  Ψ as a position probability amplitude.  Calculations of ‌&╚Ψ&╚‌‌2 for various complex forms of Ψ.

 

Probability interpretation.  Probability density.  de Broglie waves and the invention of the Schrödinger equation.  Uncertainty principle.  Connection between Schrödinger equation and classical expression for total energy of a particle.

 

 

Contrast between Newtonian and Schrödinger description of time evolution of state.  Comparison of role of Newton 's Laws and Schrödinger equation.

 

 

Calculation of average values of functions of x using probabilistic interpretation of ≥Ψ≥2.  Definition of <f(x)>,Δx.

 

 

Basic ideas about linear operators.  Algebra of operators.  The commutator and its properties.  The momentum operator and its eigenfunctions and eigenvalues.  The Hamiltonian operator.  Energy as eigenvalue of the Hamiltonian.  The Schrödinger equation for complex systems with a classical Hamiltonian.  Interpretation of Ψ for systems with many degrees of freedom.  Physical interpretation of eigenfunctions and eigenvalues in terms of measurements.

 

 

The one-dimensional box (or infinite square well).  Dependence of the eigenvalue spectrum on the size of the box.  Degeneracy.  Eigenfunctions of the position operator.  Connection between eigenfunctions of the Hamiltonian and solutions of the Schrödinger equation.  Stationary states.  General solution of the Schrödinger equation as a superposition of energy eigenstates.  Physical meaning of the expansion coefficients.  Simple examples of time dependent probability amplitudes, e.g., particle initially in one half of a box.  Completeness.  Expectation value in terms of expansion coefficients.

 

 

Energy eigenfunctions for step potentials: Classically forbidden regions. Comparison with Newtonian predictions. Reflections and transmission coefficients. Applications to neutron scattering from a nucleus. Penetration in depth. The barrier potential. Tunnelling. Solutions with definite parity. Approximation for tunnelling probability for a barrier of arbitrary shape. a - decay.

 

 

Energy eigenfunctions for step potentials: Classically forbidden regions.  Comparison with Newtonian predictions.  Reflections and transmission coefficients.  Applications to neutron scattering from a nucleus.  Penetration depth.  The barrier potential.  Tunnelling.  Solutions with definite parity.  Approximation for tunnelling probability for a wide barrier.  Ramsauer effect and neutron size resonances.  Comparison with optics. α-decay.

 

 

Approximate formula for tunnelling probability for a barrier of arbitrary shape and for a Coulomb barrier.  Deuteron fusion at room temperature and in the sun.

 

 

The square well potential.  Bound states.  Parity.  Dependence of number of bound states on depth and width of the hole.

 

 

General expansion of Ψ in complete sets of eigenfunctions and physical interpretation of the coefficients.  Expression for expectation value in terms of coefficients.  Comparison with Fourier series.  Orthogonality of eigenfunctions.  Formula for expansion coefficients as an overlap integral.  Deduction of formula for expectation value in terms of Ψ.  Applications to simple Ψ.  The Correspondence Principle.  Newtonian mechanics as a limiting case of quantum mechanics.  Condition for validity of Newtonian description.

 

 

Computational Mathematics and Computational Modelling:

 

Computational Mathematics

 

Discussion of algorithms for the solution of ordinary differential equations: discussion of the methods of Euler (simple, modified, improved) and Runge-Kutta and numerical studies of their accuracy for solution of first order equations.  Second order equations expressed as a pair of coupled first order equations, solution of second order differential equations using the methods of Euler and Runge-Kutta, treatment of one point and two point boundary conditions. Elementary discussion of finite difference methods for the solution of partial differential equations: application to the solution of Laplace 's equation in 2D (Cartesian coordinates), the treatment of Dirichlet (constant) and Neumann (derivative) type boundary conditions.

 

 

Computational Modelling

 

A six half-day computational modelling project on a topic of particular relevance to the chosen field of study or degree course. The precise nature of the projects will vary from year to year, and the student will have a degree of choice. Typical project topics include Waves in an Annular Drum, Schrödinger Equation for the Harmonic Oscillator, Three-Body Interactions, Fourier Analysis of an EEG, Neural Networks, Chaotic Billiards and The Travelling Salesman Problem.

 

Methods of Teaching/Learning

Mathematics IV and Quantum Physics:

 

Mathematics IV and Quantum Physics are each delivered as 44 hours of lectures, tutorial or workshop periods taught throughout the academic year.

 

 

Computational Mathematics and Computational Modelling:

 

Computational Mathematics is delivered as 5 lectures on numerical methods and 10 hours of computing sessions. Computational Modelling is delivered as 24 hours in computer laboratory format, timetabled as 4 hours per week.

 

 

The computing laboratories are available to students outside timetabled periods.

 

Selected Texts/Journals

1.            Mary L Boas, Mathematical Methods for the Physical Sciences, Wiley, 1983.

 

2.            G Arfken and H Webber, Mathematical Methods for Physicists, [5th Edition], Academic Press.

 

3.             K F Riley, M P Hobson and S J Bence, Mathematical Methods for the Physical Sciences, Cambridge University Press.

 

4.             B H Bransden and C J Joachain, Introduction to Quantum Mechanics, Longman

 

5.             P T Matthews, Introduction to Quantum Mechanics, McGraw-Hill.

 

6.             P C W Davies and D Betts, Quantum Mechanics.

 

7.             I S Sokolnikoff and R M Redheffer, Mathematics of Physics and Modern Engineering, McGraw Hill, 1965.

 

 

It is strongly advised that students buy one of 1, 2 and 3 for the Mathematical Physics part of the course. Text 3. is more advanced than 1 and 2.  Reference 4 is the recommended book for the Quantum Physics component and will also serve Level HE3 Quantum Physics.  Texts 5, 6 and 7 are recommended reading.

 

 

The following are provided for further reading:

 

1.      M Chester , Primer of Quantum Mechanics, Wiley.

 

2.      Greenhow, Introductory Quantum Mechanics.

 

  1. R Eisberg and R Resnick, Quantum Physics, Wiley

     

 

Computational Modelling:

 

Texts/Journals are specific to the individual project undertaken.

 

Last Updated

August 2010.