## Mathematical Physics

## Course Description

This is the first semester of a two semester sequence on Concepts of Mathematical Physics. This sequence is part of the core curriculum for graduate students in Physics. It is impossible to neither study nor do research in Physics without the use of Mathematical Methods and Concepts. This course is designed to give an introduction (or review) to some of those concepts.

It is not a “mathematical course” in a sense of rigorousness and completeness. In fact, any of the “topics” mentioned below can easily take one or two semesters in the form of a regular mathematical course. Instead: motivation will be given for some of the methods showing its necessity to solve problems, and secondly technicalities of the method will be introduced, and then applied to solve more problems.

To introduce Physics graduate students to fundamental techniques of classical and quantum mechanics at a theoretically sophisticated level

To demonstrate usefulness of this techniques in real-life application problems, thereby helping students to master other core subjects (classical mechanics, electrodynamics, optics, quantum mechanics, etc.)

To develop problem solving skills and research attitude.

To prepare students for the Ph.D. qualifying exams at some of the University.

Undergraduate Linear Algebra, Calculus, Physics 01, Physics 02.

Topic covered in this section will revolve around the concept of Vector Space in Physics.

- Introduction: From Pitagora to Supersymmetry — physics, mathematics, and the meaning of mathematical physics.

**Vector, Operators, Matrices, Tensors:**

- Vectors and Tensors in Classical Physics. Linear superpositions. Scalar, vector and inner products. Vector spaces (finite dimensional).
- Hilbert space. Quantum states as vectors in Hilbert spaces. Linear functional and dual vector space <bralket notation>. Orthonormal basis and Gram-Schmidt orthogonalization procedure.
- Linear operators. Commutator, anticommutator, Poisson brackets in classical mechanics. Identity, adjoint, self-adjoint, and Hermitian operators. Outer products and projection operators. Operator Algebra.
- Eigenproblem: eigenvalues and eigenstates. Discrete and continuous spectra. Delta function (and other distributions). Inverse operator and Green functions.
- Change of basis. Matrices and matrix operations. Matrix diagonalization.
- Orthogonal function sets. Fourier expansion.
- Vectors and Matrices with computer Algebra packages and with LAPACK.
- Tensors product of vector spaces. Separable and non-separable states. Tensor product of operators and Kronecker (direct) product of matrices. Statistical operator (density matrix), entanglement, von Neumann entropy. Schmidt decomposition. Quantum information science: entanglement, quantum teleportation, quantum dense coding. Multilinear mappings and algebra of tensors.

__Symmetries in Physics — Group Theory for Physical Problems__:

- Symmetries in classical and quantum physics.
- Group theory concepts and general theorems. Conjugacy classes.

**Condensed Matter Physics:**

- Discrete symmetry

(point, translation, space) groups. Representation of finite groups in

Hilbert spaces. Equivalent representations and Characters. Reducible and

irreducible representations. Group projectors. Subduction and Induction. - Direct products and

Clebsch-Gordan expansion. - Symmetries in

Quantum Mechanics. Wigner Theorems. Symmetries in Molecular Vibrations. - Vector and Tensor

Operators in Quantum Mechanics. Wigner-Eckhart Theorem and selection

rules. Spinors.

**High Energy Physics:**

- Lie Groups and Lie

Algebras. Casimir operators. Unitary representations of Lie Groups.

Representations by Young Diagrams, irreducible tensors. - Lorentz and

Poincare group. Internal Symmetries and Gauge Theories. - Coleman-Mandula

Theorem. A Glimpse into Supersymmetry.

- Coursework: at the end of the course.
- Exams: 1 mid-course and final course in each courses (open books).

In addition to the homework assignments, there will be a one hour problem sets (as your exams) in each end of the chapter and final report (as your final exams, about two hours exams).

Your final grade is determined using the following approximate formula:

The homework is 40% of your grade,

The midterm exam is 20%, and

The final is 40%.

Here is a guideline for your final grade, as a percentage of the total number of points (scaled as above):

90 – 100, some type of A

70 – 89, some type of B

50 – 69, some type of C

49, and below some type of D

These numbers may be lowered, depending upon numerous factors, but will not be raised (i.e., if you have an 90 average you are assured of at least an A-). The course grades are not curved.

There will be approximately one practical assignment, due on the first lecture the following days. As a rule, late assignments will not be accepted without the prior consent of the instructor. You may collaborate with others on the problems, but you must make a note of your collaborators (just as if you were writing a scientific paper). Noting your collaborators does not in any way detract from your grade. However, each problem set must be written individually-do not simply copy your collaborator’s solutions verbatim (this will be considered a form of plagiarism). Please have mercy on your grader and make your solutions neat, concise, and intelligible. Solutions which are seriously lacking in any of these categories will be marked down, even if they are ostensibly “correct”.

Usage of Computer Algebra Software (Maple or Wolfram), as well as numerical computation (if necessary) is strongly encouraged. Maple or Wolfram worksheets will be accepted as solutions to the homework assignments.