Here are some of the videos of the lectures
of fall 2009
Lecture
1: linear algebra up to linear independence.
Lecture
2: dimension of a vector space, inner and outer
products, and Dirac notation.
Lecture
3: laser demo on polarization vectors, identity
operators, vectors and their components, linear operators
and their matrix elements, determinants.
Lecture
4: determinants, systems of linear equations,
eigenvectors and eigenvalues, the adjoint of a linear
operator, hermitian operators, unitary operators,
anti-unitary and anti-linear operators, Wigner's theorem
on symmetry in quantum mechanics, eigenvalues of a square
matrix.
Lecture
5: eigenvalues of a square matrix, functions of
matrices, hermitian matrices.
Lecture
6: hermitian matrices, normal matrices, determinant
of a normal matrix, tricks with Dirac notation, compatible
normal matrices.
Lecture
7: compatible normal matrices, a matrix satisfies
its characteristic equation, the singular-value
decomposition.
Lecture 8: Part
1Part
2: applications of the singular-value decomposition,
LAPACK, the rank of a matrix, the tensor or direct
product, density operators.
Lecture
9: on tensor products, density operators,
correlation functions, groups, and complex Fourier series.
Lecture
10: on the Fourier series for \( \exp(-m |x|) \),
real Fourier series, the Fourier series for x, complex and
real Fourier series for an interval of length L.
Lecture
11: on Fourier series in several variables, the
convergence of Fourier series, and quantum-mechanical
examples.
Lecture
12: on quantum-mechanical examples, the harmonic
oscillator, non-relativistic strings, periodic boundary
conditions, and the transition to the Fourier transform.
Lecture
13: on the transition to the Fourier transform, the
Fourier transform of a gaussian and of a real function, a
representation of the delta-function, Parseval's identity,
derivatives of a Fourier transform, and momentum and
momentum space.
Lecture
14: on the uncertainty principle, Fourier transforms
in several variables, application to differential
equations, Fick's law, and diffusion.
Lecture
15: on the wave equation, diffusion, convolutions,
Gauss's law, and the magnetic vector potential.
Lecture
16: on the Fourier transform of a convolution,
finding Green's functions, Laplace transforms, examples of
Laplace transforms, how to measure the lifetime of a
fluorophore, differentiation and integration of Laplace
transforms, inverting Laplace transforms, convergence of
infinite series, tests of convergence, series of
functions, uniform convergence, the Riemann zeta function,
and power series.
Lecture
17: on power series, the geometric series, the
exponential series, factorials and the Gamma function,
Taylor series, Fourier series, the binomial theorem, the
binomial coefficient, double factorials, the logarithmic
series, Bernoulli numbers and polynomials, the Lerch
transcendent, asymptotic series, and the exponential
integrals.
Lecture
18: on dielectrics, analytic functions, Cauchy's
integral theorem, Cauchy's integral formula, the
Cauchy-Riemann conditions, and harmonic functions.
Lecture
19: on the Cauchy-Riemann conditions and the contour
integral of a general function, harmonics functions,
applications to two-dimensional electrostatics, Earnshaw's
theorem, Taylor series, and Cauchy's inequality.
Lecture
20: on Cauchy's inequality, Liouville's theorem, the
fundamental theorem of algebra, Laurent series, poles,
essential singularities, and the calculus of residues.
Lecture 21 Part
1Part
2: on the calculus of residues, ghost contours,
third-harmonic microscopy, and several examples of the use
of ghost contours.
Lecture
22: on logarithms, cuts, roots, contour integrals
around cuts, Cauchy's principal value, i-epsilon rules,
and application to Feynman's propagator.