Mathematical Methods - ?· 1 Mathematical preliminaries This section provides some basic mathematical…

Download Mathematical Methods - ?· 1 Mathematical preliminaries This section provides some basic mathematical…

Post on 02-Mar-2019

214 views

Category:

Documents

0 download

Embed Size (px)

TRANSCRIPT

Mathematical Methods

Prof Andre Lukas

Rudolf Peierls Centre for Theoretical PhysicsUniversity of Oxford

MT 2018

Contents

1 Mathematical preliminaries 41.1 Vector spaces: (mostly) a reminder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Topology and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.3 Measures and integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Banach and Hilbert spaces 312.1 Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3 Linear operators on Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Fourier analysis 423.1 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Orthogonal polynomials 614.1 General theory of ortho-normal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 The Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.3 The Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.4 The Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 Ordinary linear differential equations 725.1 Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.3 Bessel differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.4 The operator perspective - Sturm-Liouville operators . . . . . . . . . . . . . . . . . . . . . 83

6 Laplace equation 876.1 Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.2 Laplace equation in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.3 Laplace equation on the two-sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.4 Laplace equation in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7 Distributions 1077.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.2 Convolution of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.3 Fundamental solutions - Green functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.4 Fourier transform for distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

8 Other linear partial differential equations 1168.1 The Helmholz equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1168.2 Eigenfunctions and time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.3 The heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1188.4 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

1

9 Groups and representations 1239.1 Groups: some basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1279.3 Lie groups and Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319.4 The groups SU(2) and SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1369.5 The Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Appendices 149

A Manifolds in Rn 149A.1 Definition of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149A.2 Tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150A.3 Integration over sub-manifolds of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151A.4 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

B Differential forms 155B.1 Differential one-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155B.2 Alternating k-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158B.3 Higher-order differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160B.4 Integration of differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

C Literature 166

2

Foreword: Lecturing a Mathematical Methods course to physicists can be a tricky affair and followingsuch a course as a second year student may be even trickier. The traditional material for this course consistsof the classical differential equations and associated special function solutions of Mathematical Physics. Ina modern context, both in Mathematics and in Physics, these subjects are increasingly approached in theappropriate algebraic setting of Banach and Hilbert Spaces. The correct setting for Quantum Mechanicsis provided by Hilbert Spaces and for this reason alone they are a mandatory subject and should atleast receive a rudimentary treatment in a course on Mathematical Methods for Physicists. However, theassociated mathematical discipline of Functional Analysis merits a lecture course in its own right andcannot possibly be treated comprehensively in a course which also needs to cover a range of applications.What is more, physics students may not yet have come across some of the requisite mathematics, suchas the notion of convergence and the definition of integrals. All of this places an additional overhead onintroducing mathematical key ideas, such as the idea of a Hilbert Space.

As a result of these various difficulties and requirements Mathematical Methods courses often end up ascollections of various bits of Mathematical Physics, seemingly unconnected and without any guiding ideas,other than the apparent usefulness for solving some problems in Physics. Sometimes, ideas developed inthe context of finite-dimensional vector spaces are used as guiding principles but this ignores the crucialdifferences between finite and infinite-dimensional vector spaces, to do with issues of convergence.

These lecture notes reflect the attempt to provide a modern Mathematical Physics course whichpresents the underlying mathematical ideas as well as their applications and provides students with anintellectual framework, rather than just a how-to-do toolkit. We begin by introducing the relevantmathematical ideas, including Banach and Hilbert Spaces but keep this at a relatively low level of for-mality and quite stream-lined. On the other hand, we will cover the traditional subjects related todifferential equations and special functions but attempt to place these into the general mathematical con-text. Sections with predominantly mathematical background material are indicated with a star. Whilethey are important for a deep understanding of the material they are less essential for the relatively basicpractical tasks required to pass an exam. I believe the ambitious, mathematically interested student canbenefit from the combination of mathematical foundation and applications in these notes. Students whowant to focus on the practical tasks may concentrate on the un-starred sections.

Two somewhat non-traditional topics, distributions and groups, have beed added. Distributions areso widely used in physics - and physicists tend to discuss important ideas such Green functions usingdistributions - that they shouldnt be omitted from a Mathematical Physics course. Symmetries havebecome one of the central ideas in physics and they are underlying practically all fundamental theoriesof physics. It would, therefore, be negligent, in a course on Mathematical Methods, not to introduce theassociated mathematical ideas of groups and representations.

The two appendices are pure bonus material. The first one is a simple account of sub-manifolds in Rn,including curves and surfaces in R3, as encountered in vector calculus. Inevitably, it also does some of thegroundwork for General Relativity - so certainly worthwhile for anyone who would like to learn Einsteinstheory of gravity. The second appendix introduces differential forms, a classical topic in mathematicalphysics, at an elementary level. Read (or ignore) at your own leisure.

Andre LukasOxford, 2018

3

1 Mathematical preliminaries

This section provides some basic mathematical background which is essential for the lecture and can alsobe considered as part of the general mathematical language every physicist should be familiar with. Thepart on vector spaces is (mainly) review and will be dealt with quite quickly - a more detailed treatmentcan be found in the first year lecture notes on Linear Algebra. The main mathematical theme of thiscourse is the study of infinite-dimensional vector spaces and practically every topic we cover can (andshould) be understood in this context. While the first year course on Linear Algebra dealt with finite-dimensional vector spaces many of the concepts were introduced without any reference to dimension andstraightforwardly generalise to the infinite-dimensional case. These include the definitions of vector space,sub-vector space, linear maps, scalar products and norms and we begin by briefly reviewing those ideas.

One of the concepts which does not straightforwardly generalise to the infinite-dimensional case is thatof a basis. We know that a finite-di