Graduate School in Science – Theoretical Physics Syllabus

(Astrophysics, Field Theory, Mathematical Physics, Modeling and Simulation)

Specialization of theoretical physics offers a sound preparation in fundamental theoretical subjects like quantum mechanics and field theory, with a range of other possibilities. With a focus on mathematical methods of theoretical physics, their practical applications and their interdisciplinary character it opens a wide set of possibilities for further career in science and industry.

This is a postgraduate course in theoretical physics consisting of both lecture courses and a research project. The level is suitable for students who have completed their first degree.

Prerequisites of admission:

· Bachelor's degree with a major in physics

· Bachelor's degree with courses in the following subjects or their equivalents: general physics, electricity and magnetism, classical mechanics, optics, thermodynamics, calculus, linear algebra, elements of differential equations

There are three main suggested paths of graduate studies:

· astrophysics

· mathematical physics and field theory

· modeling and simulations

In each path the student takes the obligatory courses (GSS and THP1, THP2, THP4 in the first year) and courses of his/her choice. For example: a student taking the astrophysics options takes in the first year obligatory courses (25 credits) and courses THP10 and THP11 (8 credits).

Required number of credits to complete 1^st year:
36 (courses) + 10 (research project assigned by tutor)

In the second year he/she chooses (in addition to the obligatory GSS) and THP12, THP14, THP15, THP19, THP21 and THP22 obtaining 28 credits.

Required number of credits to complete 2^nd year: 28 (courses) + 16 (research project)

Details of the courses may vary. The minimal number of students in each course shall be set by the lecturer and the Director of the Graduate School. If there is no sufficient number of student in the course, the course may be canceled, joined with other courses or take a different form (seminar, with regular assignment for the students).

1^st year courses:

title	activity type	hours/ week	hours/ year	form of crediting	credits
General School Seminar GSS	Seminar	2	60	Participation	4
Elements of Quantum Mechanics THP1	Lectures Tutorials	2 2	30 30	Exam Test	4 3
Quantum Mechanics II THP2	Lectures Tutorials	2 2	30 30	Exam Test	4 3
Mathematical Methods and Differential Equations THP3	Lectures Tutorials	2 2	30 30	Exam	4 3
Introduction to Electrodynamics THP4	Lectures Tutorials	2 2	30 30	Exam Test	4 3
Introduction to Field Theory THP5	Lectures	2	30	Exam/test	3
Simulations in Physics (Monte Carlo Methods) THP6	Lectures, Tutorials	2 2	30 30	Exam/essay/test	3 2
Doing Science with Mathematica and Maple THP7	Lecture/ Tutorials	2	30	Project	3
Introduction to Theoretical Methods in Biophysics THP8	Lectures	2	30	Arranged with Lecturer	3
Statistical Physics THP9	Lectures Tutorials	2 2	30 30	Exam Test	3
Introduction to Special and General Relativity THP10	Lectures Tutorials	2 2	30 30	Exam Test	3 2
Introduction to Observational Astronomy THP11	Lectures	2	30	Arranged with Lecturer	3

2^nd year courses

General School Seminar GSS	Seminar	2	60	Participation	4
Introduction to Nonlinear Physics THP12	Lectures	2	30	Arranged with lecturer	4
Phase Transitions and Critical Phenomena THP13	Lectures	2 2	30 30	Arranged with lecturer	4 2
Introduction to Particle Physics THP14	Lectures	2	30	Arranged with lecturer	4
Modern Theoretical Physics Seminar THP15	Seminar	2	30	Project/essay	4
Doing Science with Mathematica and Maple THP16	Lectures/ tutorial	2	30	Project	4
Advanced Quantum Field Theory THP17	Lectures Tutorials	2 2	30 30	Exam/essay Test/	4 3
Introduction to Econophysics THP18	Lectures	2	45	Arranged with lecturer	4
Geometry in Theoretical Physics THP19	Lectures	2	30	Arranged with lecturer	4
Computer Simulations and Modeling THP20	Lectures Tutorials	2 2	30 30	Arranged with lecturer	4 3
Astrophysics: Physics of Stars THP21	Lectures	2	30	Arranged with lecturer	4
Cosmology: guide to the Universe THP22	Lectures	2	30	Arranged with lecturer	4
Time Series Analysis THP23	Lectures	2	30	Arranged with lecturer	4
Practical Quantum Mechanics THP24	Lectures Tutorials	2 1	30 15	Arranged with lecturer	4 3

Obligatory courses are marked gray,
courses joined with other specialization (partially) are underlined.

Courses

Graduate School Seminar

General Seminar aiming at improving interdisciplinary background of School students.

Diploma Project

One-year project, supervised by a professor, fulfills requirements for MSc Diploma Thesis of the Jagiellonian University and most European Universities.

Elements of Quantum Mechanics

Revision of basic QM: principles, the Schrödinger equation, wave functions, and physical interpretation. Bound and continuum states in one-dimensional systems. Hydrogen atom. (joined course with other specializations)

Quantum Mechanics II

Perturbation theory. Dirac equation. Relativistic hydrogen atom. Quantum many-body theory. (part of the course may be joined with a similar course on other specializations)

Mathematical Methods and Differential Equations

Distributions, representations of Lie groups and Lie algebras, angular momentum, elementary differential equations in physics, biology, chemistry and engineering. Initial and boundary problems, discretization of differential problems. Direct and iterative solutions, Linear and nonlinear problems.

Introduction to Electrodynamics

Maxwell equations, symmetries, equations of motions, Noether theorem, application of electrodynamics, elements of quantum electrodynamics, tests of quantum electrodynamics

Introduction to Field Theory

Lagrange function and action, equations of motion, symmetries, gauge theories, abelian and nonabelian gauge theories, gauge fixing, methods of quantization of field theories and gauge theories.

Geometry in Theoretical Physics

Manifolds, vector fields, differential forms, instantons and monopoles, Dirac monopole,
t’Hoft-Polyakov monopole, topological defects in physics, vortices.

Introduction to Particle Physics

Basics of fundamental particles and their interactions: quark, gluons, leptons, Z,W bosons, methods of detections, electroweak and strong interactions.

Introduction to Theoretical Methods in Biophysics

Introduction to the methods of theoretical physics used in biology.

Modern Theoretical Physics Seminar
Seminar with students presentations on modern topics in theoretical physics.

Introduction to Nonlinear Physics

Introduction to theoretical description of nonlinear phenomena, physical problems with nonlinear differential equations, examples. Solitons.

Practical Quantum Mechanics

Many body problem, dipole moment of the water molecule, relativistic quantum mechanics:
the Klein paradox in many body sectors. Quantum computers: quantum mechanics of NOT
and CNOT gates.

Introduction to Special and General Relativity

Basic concepts, Minkowski space, Lorentz transformations, twin paradox, elementary solutions; gravity; equivalence principle applications in astrophysics, singularity theorems, Hawking radiation, gravitational waves.

Basics of Observational Astronomy

Basis of observational astronomy: telescopes, radioastronomy, X-ray and gamma-ray astronomy.

Astrophysics: Physics of Stars

Introduction to the theory of the evolution of stars, white dwarfs, neutron stars, supernovae.

Cosmology: Guide to the Universe

Mathematical basics of cosmology, microwave background radiation, black holes,
matter in the Universe, formation of structures, gravitational waves in cosmology

Simulations in Physics: Monte Carlo Methods

Mathematical foundation for Monte Carlo (MC) methods; Random number generators;
statistical tests. Adaptive MC techniques; MC simulation; random walks and their applications. Metropolis algorithm.

Doing Science with Mathematica and Maple

Modeling in physics, biology, medicine and chemistry with mathematical software. Unification of computing, simulation and visualization. Examples: Chaos with a heart, synchronization processes in biology, discrete maps and chaos, binary collisions, dimensional analysis. Duffing equation, chemical oscillators, frequency analysis of time series, etc.

Computer Simulations and modeling: from theory to practice

Description: Advanced Monte Carlo Projects, Molecular Dynamics simulation (course may be joined with similar a course from other specializations)

Econophysics - modeling of financial markets

Elements of contemporary financial markets. Time series analysis (random walks, heavy tails phenomena, GARCH and HARCH modeling) Derivatives - Black Scholes model for options, Contemporary methods of risk assessment (VaR – Value-at-Risk) Methods of portfolios construction Genetic and adaptive algorithms, "minority game".

Time Series Analysis
Discrete Fourier Transform; Fast Fourier Transform; Shannon sampling theorem; linear filters, stochastic models AR, MA, ARMA, ARIMA; wavelets Takens embedding theorem, phase space reconstruction and nonlinear time series analysis; nonlinear prediction and denoising; statistical properties of chaotic attractors.

Details of the courses may vary. The minimal number of students in each
course shall be set by the lecturer and the Director of the Graduate School.

If there is no sufficient number of student in the course the course may be
canceled, joined with other courses or take a different form (seminar, with
regular assignment for the students).