Duke Physics

Mathematical Methods of Physics (PHY 301)

Course description:

Includes topics in probability theory, complex analysis, asymptotic expansions, group theory, Fourier analysis, Green functions, ordinary and partial differential equations; and use of Mathematica.

Possible principal texts:

1. K.F. Riley, M.P. Hobson and S.J. Bence Mathematical Methods for Physics and Engineering.
2. D.A. McQuarrie Mathematical Methods for Scientists and Engineers.

Other texts to consider:

1. G.B. Arfken and H.J. Weber Mathematical Methods for Physicists.
2. J.J. Kelly Graduate Mathematical Physics.

Prerequisites:

Undergraduate courses in intermediate calculus (such as MTH 103 or equivalent) are required. Students must have some familiarity with partial differentiation, multiple integrals, infinite series, differential vector calculus (grad, div, curl, Laplacian), integral vector calculus (divergence and Stokes theorems), coordinates (spherical and cylindrical), matrices and determinants, simultaneous linear equations, Fourier series, and complex numbers.

Syllabus

• Probability: distributions, generating functions, central limit theorem, stochastic processes.
• Complex Variables: analytic functions, complex integrals, residues, contour integration asymptotic expansions.
• Group Theory: definitions, examples, applications, representations, characters, product reps, Clebsch-Gordan coefficients, irreducible representations, irreducible Tensors, Wigner-Eckart.
• Fourier transforms, delta functions, convolution, correlation, power spectrum density.
• ODEs: exact solutions, series solutions, Legendre polynomials and functions, Frobenius method, Bessel functions, eigenfunctions, orthogonal functions, Sturm-Liouville theory, Green's functions, qualitative methods, numerical methods.
• PDEs: separation of variables, cylindrical coordinates (Bessel), spherical coordinates (Legendre and Spherical Harmonics), Green's functions, boundary problems.