MPhys Physics with Astrophysics

Mathematics of Waves and Fields

To introduce and develop the mathematical skills and knowledge needed to understand classical fields and quantum mechanics.

On completion successful students will be able to:

1. Solve partial differential equations using the method of separation of variables.
2. Define the term “orthogonality” as applied to functions, and recognise sets of orthogonal functions which are important in physics (e.g. trigonometric functions and complex exponentials on appropriate intervals, Legendre polynomials, and spherical harmonics).
3. Represent a given function as a linear superposition of orthogonal basis functions (e.g. a Fourier series) using orthogonality to determine the coefficients,
4. State how a Fourier transform differs from a Fourier series, and calculate Fourier transforms of simple functions.
5. Solve eigenvalue problems (differential equations subject to boundary conditions) either in terms of standard functions or as power series.
6. Use partial differential equations to model wave, heat flow and related phenomena.
7. Make basic use of Dirac notation.

1.  Wave problems in one dimension

Separation of variables
Normal modes of a string:  eigenfunctions and eigenvalues
General motion of a string

2.  Fourier series

Orthogonality and completeness of sines and cosines
Complex exponential form of Fourier series

3.  Other PDE’s 

Laplace’s equation
The heat-flow equation

4.  Integral transforms 

Fourier transform
Convolutions
Wave packets and dispersion

5.  Special functions  

Orthogonal sets of eigenfunctions
Series solution of differential equations
Legendre polynomials and related functions
Bessel functions

6.  Problems in two and three dimensions  

Normal modes of a square membrane; degeneracy
Wave guide
Normal modes of circular and spherical systems
Heat flow in circular and spherical systems
Laplace’s equation:  examples in cartesian and polar coordinates

7.  Dirac notation 
Vector spaces
Ket notation
Inner products and Bras
Hilbert spaces
 

* Other 10% Tutorial Work/attendance 

Students will receive feedback on their work and performance in this module as a component of their weekly tutorial meeting with their academic tutor. 

Boas, M.L. Mathematical Methods for Physical Sciences, 3rd edn. (Wiley, 2006)
Martin, B.R. & Shaw, G, Mathematics for Physicists. (Wiley 2015)
Riley, K.F. Hobson, M.P. & Bence, S.J. Mathematical Methods for Physics and Engineering, 3rd edn (Cambridge 2006) [Chapters 12 to 19]
Stephenson, G. Partial differential equations for scientists and engineers (Imperial College 1996)