Mathematical Methods in the Core Graduate Courses
Physics P506 (Electricity and Magnetism I)
- Calculus: Cylindrical, spherical, coordinate systems, transformations of basis vectors, Jacobians, differential operators in orthogonal curvilinear coordinates.
- Delta Functions: Symbolic rules, properties, changes of variable, redundant coordinates.
- Green's Functions: Gauss and Stokes Theorems, Green's Theorems, Green's functions for the Laplacean on R^{n}, n=1,2,3 directly and by Fourier transform, Green's functions on bounded regions for Dirichlet and Neumann problems.
- Separation of variables for the Laplace operator: (a) Two dimensional boundary value problems and Fourier series, Fourier series expansions for Green's functions. (b) Spherical coordinates, Gamma Function, Legendre polynomials and expansions, orthogonality relations, associated Legendre functions, orthogonality relations, spherical harmonics, orthogonality relations, addition theorem for spherical harmonics, spherical harmonic expansions for Green's functions. (c) Cylindrical coordinates, Bessel functions (J in some detail as a power series solution to the ordinary differential equation, regular singular points, other Bessel functions and their asymptotic properties near the origin and at infinity), Fourier-Bessel series, Wronskians for second order ordinary differential equations, Fourier-Bessel expansions of Green's functions.
- Numerical Methods: Iterative solutions to Laplace's equation, comparison between analytical and numerical results, treatment of problems with less symmetry than the usual analytical exercises.
Physics P507 (Electricity and Magnetism II)
- Calculus: Longitudinal and transverse decomposition of vector fields (Coulomb gauge) with differential operators and in momentum space.
- Green's Functions: Retarded Green's functions for the wave operator on R^{s+1} for s=1,2,3, spherical harmonic - Bessel expansion.
- Special Relativity: Lorentz transformations, translations, Lorentz group, parity, time reversal, infinitesimal generators, Poincaré group, infinitesimal generators, vectors, tensors, transformation properties and covariance. Covariance of Maxwell's equations.
Physics P511 (Quantum Mechanics I)
- Finite Dimensional Vector Spaces: Vector spaces over the complex numbers, linear independence, basis vectors, inner products, expansions, matrix representation of operators, change of bases, eigenvalues, eigenvectors, diagonalization of matrices, simultaneous diagonalization of hermitean commuting operators.
- Hilbert Space: Generalization of material in (i) to infinite dimensions, L_{2}(R^{n},dx),L_{2}(S^{n-1},dΩ, differential operators and Fourier transform, space translations and the linear momentum operator, discrete and continuous spectrum for hermitean operators, spectral theorem and eigenfunction expansions, the harmonic oscillator in the Schrodinger and number operator representation, Hermite functions, differential and raising and lowering operator (Schwinger) representations of the angular momentum operators, spherical harmonics as eigenfunctions, Laguerre functions and the Hydrogen atom.
Physics P512 (Quantum Mechanics II)
- Rotation Group: Concepts of group theory, three dimensional rotation group SO(3), Euler angles, helicity representation of rotations, Poncaré sphere, double connectedness and SU(2), infintesimal generators and commutation rules, finite dimensional representations in terms of angular momentum, decomposition of reducible representations and the addition of angular momentum, irreducible tensors.
- Angular Momentum Theory: Transformation of states and operators under rotations and translations, Clebsch-Gordon coefficients, Wigner-Eckert Theorem.
- Symmetric Group: Permutations, symmetric/ anti-symmetric wave functions, representations, Young tableaux.
Physics P521 (Classical Mechanics)
- Linear Algebra: Diagonalization of matrices, normal modes, small oscillation expansions.
- Variational Calculus: Stationary problems for functionals, Euler - Lagrange equations, Hamilton's Principle.
- Contour Integration: Applications to Green's functions.
- Euler angles: Used for rigid body motions.
- Sturm-Liouville theory for strings. Bessel functions for drums, Rayleigh - Ritz variational methods for strings, membranes.
Physics P556 (Statistical Physics)
- Combinatiorial probability: Axioms for probabilities on finite sets, permutations, combinations, counting problems, Cauchy formula.
- Stationary Phase Methods: Estimates of multiple integrals and sums using the methods of stationary phase.