PHYS2B21 - Mathematical Methods in Physics and Astronomy

Prerequisites

In order to take this course, students should have studied the material in the precursor PHYS1B21 mathematics course.

Aims

This course aims to:

• provide the remaining mathematical foundations for all the second and third year compulsory Physics and Astronomy courses;
• prepare students for the second semester Mathematics option MATHB8;
• give students practice in mathematical manipulation and problem solving at second-year level.

The PHYS1B21 and PHYS2B21 syllabuses together cover all the mathematical requirements of the compulsory Physics and Astronomy courses. The five major areas treated in 2B21 are of special relevance to Quantum Mechanics and Electromagnetism, and the applications of these subjects to many other topics, including plasma, condensed matter, atomic, molecular, and particle physics. At the end of each section of the course, students should be able to appreciate when to use a particular technique to solve a given problem and be able to carry out the relevant calculations. Specifically,

For Matrices, students should be able to:

• manipulate vectors in a complex n-dimensional space and represent linear transformations in this space by matrices;
• perform matrix algebra, including multiplication and inversion, using a wide variety of matrices including unitary, Hermitian, and orthogonal matrices;
• solve linear simultaneous equations through the use of matrices and determinants;
• find the eigenvalues and eigenvectors of a matrix up to 3x3;
• diagonalise a matrix up to 3x3 and apply the technique to physical and mathematical problems.

For Differential Equations, students should be able to:

• solve a variety of second order linear partial differential equations, including the Laplace and wave equations, by the method of separation of variables, using both Cartesian and polar coordinates, and impose boundary conditions;
• know what type of series solution to seek when solving an ordinary second-order linear and homogeneous differential equation by examining its singularity structure and derive a solution consistent with the boundary conditions;
• investigate the convergence of the series solution and understand how and why polynomial solutions arise.

For Legendre Functions, students should be able to:

• solve the Legendre differential equation by series method and find the conditions necessary for a polynomial solution;
• derive and apply the generating function and recurrence relations for Legendre polynomials;
• employ the orthogonality relation of Legendre polynomials to develop functions as series of such polynomials;
• manipulate spherical harmonics up to l=2.

In Fourier Analysis, students should be able to:

• derive the formulae for the expansion coefficients for real and complex Fourier series;
• make analyses using sinusoidal and complex functions for both periodic and non-periodic functions and be aware of possible convergence problems;
• understand the possible complications when differentiating or integrating Fourier series;
• use Parseval's identity to deduce the values of some infinite series;
• derive the formulae for the expansion coefficients for real and complex Fourier transforms;
• perform Fourier transforms of a variety of functions and derive and use Dirac delta functions;
• apply the convolution theorem to physical problems.

In Vector Calculus, students should be able to:

• understand the concepts of scalar and vector fields;
• carry out algebraic manipulations with the div, grad, curl, and Laplacian operators in Cartesian coordinates;
• derive and apply the divergence and Stokes' theorems in physical situations, and deduce coordinate-independent expressions for the vector operators;
• derive and use expressions for the vector operators in cylindrical and spherical polar coordinates.

Methodology and Assessment

The 33 lectures in this half-unit course are reinforced by approximately 11 discussion periods, where the lecturer goes over examples of relevant problems without introducing any new examinable material. In addition there are 2 revision lectures in Term-3. The end-of-session written examination counts for 90% of the assessment. The 10% continuous assessment component is based primarily on the best eight out of ten homework sheets (8%). Up to 2% credit will be derived from the results of the mid-sessional test examination held just before the Christmas break.

Textbooks

A book which covers essentially everything in both this and the first-year 1B21 mathematics course is Mathematical Methods in the Physical Sciences, by Mary Boas (Wiley). This book will also be of use in the B8 option given in the second semester. A more specialised second-year book is Mathematical Methods for Physics and Engineering, by K.F. Riley, M.P. Hobson and S.J. Bence (Cambridge University Press). Mathematical Methods for Physicists, by G.B. Arfken and H.-J. Weber (Academic Press), is more challenging but is rewarding for the mathematically-inclined students.

(The approximate allocation of lectures to topics is shown in brackets below.)

Linear Vector Spaces and Matrices [12.5]

Definition and properties of determinants, especially 3 x 3. [1.5]

Revision of real 3-dimensional vectors, Complex linear vector spaces, Linear transformations and their representation in terms of matrices, Multiple transformations and matrix multiplication. [3]

Properties of matrices, Special matrices, Matrix inversion, Solution of linear simultaneous equations. [4]

Eigenvalues and eigenvectors, Eigenvalues of unitary and Hermitian matrices, Real quadratic forms, Normal modes of oscillation. [4]

Partial Differential Equations [3.5]

Superposition principle for linear homogeneous partial differential equations, Separation of variables in Cartesian coordinates, Boundary conditions, One-dimensional wave equation, Derivation of Laplace's equation in spherical polar coordinates, Separation of variables in spherical polar coordinates, the Legendre differential equation, Solutions of degree zero.

Series Solution of Ordinary Differential Equations [2]

Derivation of the Frobenius method, Application to linear first order equations, Singular points and convergence, Application to second order equations.

Legendre Functions [4]

Application of the Frobenius method to the Legendre equation, Range of convergence, Quantisation of the l index, Generating function for Legendre polynomials, Recurrence relations, Orthogonality of Legendre functions, Expansion in series of Legendre polynomials, Solution of Laplace's equation for a conducting sphere, Associated Legendre functions, Spherical harmonics.

Fourier Analysis [5]

Fourier series, Periodic functions, Derivation of basic formulae, Simple applications, Gibbs phenomenon (empirical), Differentiation and integration of Fourier series, Parseval's identity, Complex Fourier series. [2.5]

Fourier transforms, Derivation of basic formulae and simple application, Dirac delta function, Convolution theorem. [2.5]

Vector Operators [6]

Gradient, divergence, curl and Laplacian operators in Cartesian coordinates, Flux of a vector field, Divergence theorem, Stokes' theorem, Coordinate-independent definitions of vector operators. Derivation of vector operators in spherical and cylindrical polar coordinates.