Physics 501: Mathematical Physics I
Fall 2011
[Figures following each major topic indicate the approximate number of
lecture hours devoted to it.]
- Introduction and course overview [1]
- Linear vector spaces and matrices [4]
[Riley & Hobson, Chapter 1]
- vector spaces; linear independence
- inner products; Gram-Schmidt orthogonalization
- linear operators
- matrix representation
- coordinate transformations
- eigenvalue problems
- matrix diagonalization
- examples and applications
- Numerical methods for linear systems [2]
[Numerical Recipes, Sections 2.1-2.6, 2.10; 11.1-11.3]
- Gauss and Gauss-Jordan elimination (gaussj)
- LDU decomposition (ludcmp, lubksb)
- tridiagonal systems (tridag)
- iterative methods
- singular value decomposition (svdcmp, svbksb)
- Jacobi and Householder transformations
- diagonalization of matrices (jacobi, tred2, tqli)
- applications
- least-squares fits
- basis expansion in quantum systems
- Complex analysis [7]
[Riley & Hobson, Chapters 14 and 15]
- functions in the complex plane
- multivalued functions
- analytic functions
- Cauchy's theorem
- Cauchy's integral formula
- Taylor and Laurent series
- zeros and singularities
- the residue theorem
- contour integration
- examples and applications
- Ordinary differential equations [5]
[Riley & Hobson, Chapters 6 - 8]
- first-order systems
- second-order systems: series solutions
- second solutions
- Green's functions (1-D)
- Sturm-Liouville theory
- eigenfunctions of Hermitian operators
- completeness and mean-square convergence of Fourier series
- uniform, pointwise, and mean-square convergence
- Bessel's inequality and Parseval's identity
- expansion of delta function and Green's functions
- Monte Carlo Methods [1]
[Numerical Recipes, Chapter 7]
- Fourier series [1]
[Riley & Hobson, Chapter 4]
- Euler-Fourier formulae
- the Gibbs phenomenon
- Numerical solution of ordinary
differential equations [5]
[Numerical Recipes, Chapters 16 and 17]
- initial-value problems (NR Chapter 16)
- Runge-Kutta methods
-
predictor-corrector methods
- time-reversible methods
- timestep control
- boundary-value problems (NR Chapter 17)
- shooting
- (aside: solution of algebraic equations)
- shooting to a fitting point
- relaxation (brief)
- Integral transforms [4]
[Riley & Hobson, Chapter 5]
- Fourier transforms
- relation to Fourier series
- the Fourier integral and the inverse transform
- transforms of derivatives and integrals
- transforms of convolutions
- applications to linear systems
- Laplace transforms
- derivation from Fourier transforms
- inversion integral
- convolution theorem
- applications to ODEs and PDEs
- Discrete and Fast fourier transforms [2]
[Numerical Recipes, Chapters 12, 13.0-13.4]
- discrete orthogonality relations
- the DFT and its inverse
- sampling and aliasing
- convolution and filtering
- numerical determination of the DFT
- the FFT
- applications