Lecture Notes on Methods of Mathematical Physics (PHY503)

Fall 2020, Jacobus Verbaarschot

Lecture 1
Functionals, Functional Derivatives, Euler-Lagrange equations, Constraints

Lecture 2

Lecture 3
Action principle, Noether's theorem, Linear Vector spaces, linear maps

Lecture 4
Dual space, Orthonormal basis, Scalar product

Lecture 5
Direct Sum, Qoutient space, Co-kernel

Lecture 6
Projection operator, Linear equation, Fredholm alternative

Lecture 7
Determinant, Adjugate matrix, Inverse, Cailey's theorem

Lecture 8
Derivative, Diagonalization, Jordan Canonical form

Lecture 9
Quadratic Form, Symplectic Form

Lecture 10
Function spaces, Banach space, Hilbert space, Cauchy sequence, Complete space, Cauchy-Schwarz inequality

Lecture 11
Parseval's theorem, Orhtogonal polynmials

Lecture 12
Three step recurrence relations, Legendre, Hermite and Tchebychev polynomials

Lecture 13

Lecture 14
Axioms of roup theory

Lecture 15
Subgroups, Cosets, Normal subgroup, Quotient group, Examples

Lecture 16
Conjugacy classes, Permutation group

Lecture 17

Lecture 18
Group action on sets, tansitive, faithful, free

Lecture 19
Representations of groups, real, pseudo-real

Lecture 20
Direct sum, Direc product

Lecture 21
Schur's Lemma

Lecture 22
Unitarity of representations, Orthogonality

Lecture 23
Characters, orthogonality relations, character table of S3

Lecture 24
Completeness of characters, character table of S4

Lecture 25
Group Algebra

Lecture 26
Group theory in Quantum Mechanics

Lecture 27
Vibrational Spectrum of H_2 O

Lecture 28

Lecture 29
Vibrational Spectrum of H_2 O

Lecture 30
Vibrational Spectrum of H_2 O

Lecture 31
Lie Groups

Lecture 32
Symplectic Group

Lecture 33
Time reversal invariance, SU(2)

Lecture 34
Invariant vector fields

Lecture 35
Maurer-Cartan form

Lecture 36
Euler angles, group integration, Haar measure

Lecture 37
Relation between O(3) and SU(2)

Lecture 38
SO(N) spinor representation of SU(2), Spin(N)

Lecture 39
Adjoint representation, Peter Weyl theorem, character of SU(2)

Lecture 40
Lie Algebras, Killing form

Lecture 41
Killing Metric, Casimir Operator, SU(2)

Lecture 42
SU(3), General Semi-Simple Lie Algebras