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 Rn,
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.
(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 Rs+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, L2(Rn,dx),L2(Sn-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.
(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,
Symmetric Group: Permutations, symmetric/
anti-symmetric wave functions, representations, Young tableaux.
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
Euler angles: Used for rigid body motions.
Sturm-Liouville theory for strings. Bessel functions
for drums, Rayleigh - Ritz variational methods for strings, membranes.
Combinatiorial probability: Axioms for probabilities
on finite sets, permutations, combinations, counting problems, Cauchy
Stationary Phase Methods: Estimates of multiple
integrals and sums using the methods of stationary phase.