Matematical and Computational Modelling in Physics

Master study programme Matematical and Computational Modelling in Physics is designed for students of physics. (The programme for students of mathematics is called Mathematical Modelling in Physics and Technology, details concerning this study programme can be found here.)

Contacts

If you have any questions concerning the study of mathematical modelling, please contact Martin Čížek (cizek@mbox.troja.mff.cuni.cz) or Vít Průša (prusv@karlin.mff.cuni.cz).

Admission

Concerning the general overview of the admission procedure, please see the webpages of the Faculty of Mathematics and Physics. If you are not yet familiar with advanced subjects such as functional analysis but you are still strongly motivated to study mathematical modelling, please contact us, and we will try to recommend you textbooks and other material for self-study.

Note also, that if you have a sufficiently strong academic record, you can apply for the exemption from the entrance examination.

It is expected that you have sufficient knowledge of the topics listed below. Note that most of the topics are at a sufficient level also covered by courses Linear Algebra I, Linear Algebra II, Mathematical Analysis I, Mathematical Analysis II, Mathematics for Physicists I, Mathematics for Physicists II and Mathematics for Physicists III, which are a part of the standard curriculum in the bachelor study programme Physics at the Faculty of Mathematics and Physics.

  • Differential and integral calculus (single variable and multivariable). Limits, derivatives, line, surface and volume integrals. Foundations of calculus of variations.
  • Foundations of measure theory, Lebesgue integral. These topics are taught, for example, in the undergraduate course Measure and Integration Theory I, see the course syllabus for what is in this case meant by “foundations”.
  • Foundations of linear algebra. Vector spaces, matrices, determinants, Jordan canonical form, orthogonalization, eigenfunctions and eigenvalues, multilinear algebra, quadratic forms.
  • Foundations of numerical solution of systems of linear algebraic equations. Schur theorem, QR decomposition, LU decomposition, singular value decomposition, least squares method, conjugate gradient method, GMRES, backward error, sensitivity and numerical stability. These topics are taught, for example, in the undergraduate course Analysis of Matrix Calculations I, see the course syllabus for what is in this case meant by “foundations”.
  • Foundations of complex analysis. Cauchy theorem, residue theorem, conformal mapping, Laplace transform. These topics are taught, for example, in the undergraduate course Introduction to Complex Analysis, see the course syllabus for what is in this case meant by “foundations”.
  • Foundations of functional analysis and theory of metric spaces. Banach and Hilbert spaces, operators and functionals, Hahn-Banach theorem, dual spaces, bounded operators, compact operators, theory of distributions. These topics are taught, for example, in the undergraduate course Introduction to Functional Analysis, see the course syllabus for what is in this case meant by “foundations”. (If you have studied the standard bachelor study programme Physics, then Introduction to Functional Analysis will be anyway your compulsory subject in the first year of your master study. The topics listed above are at a sufficient level covered by the undergraduate course Mathematics for Physicists III.)
  • Foundation of theory of ordinary differential equations. Existence of solution, maximal solution, systems of linear ordinary differential equations, elements of stability theory. These topics are taught, for example, in the undergraduate course Ordinary Differential Equations, see the course syllabus for what is in this case meant by “foundations”.
  • Foundations of the classical theory of partial differential equations. Quasilinear equations of the first order, Laplace equation and heat equation – fundamental solutions and maximum principle, wave equation – fundamental solution, finite differences method. These topics are taught, for example, in the undergraduate course Introduction to Partial Differential Equations, see the course syllabus for what is in this case meant by “foundations”.
  • Foundations of classical mechanics. Newton laws of motion, Lagrange equation, Hamilton equations, variational principles, rigid body dynamics. These topics are taught, for example, in the undergraduate course Theoretical Mechanics, see the course syllabus for what is in this case meant by “foundations”.

Curriculum

The expected length of the study is two years. You are expected to choose the topic of your thesis during the first year of your study. There are three sets of courses: compulsory, compulsory elective and optional.

Concerning the compulsory courses you must complete all of them. The strange term compulsory elective means that you must complete some of these courses. (The term “some” means that you need certain number of credits from these subjects, see faculty webpages for details.) The choice is yours, but it should reflect the topic of your thesis. Talk to your thesis supervisor if you want to check which courses are suitable for you! The choice of the optional courses is completely up to you. Ask your thesis supervisor, classmates or the people responsible for the study programme, see contacts, for a hint.

The (new) study programme Computational and Mathematical Methods in Physics is expected to start in 2015/2016. Detailed curriculum should appear on this page soon, in the mean time please see the faculty webpages.

Compulsory courses

You must complete all these courses. The next to the last column indicates the recommended year of study. Besides the courses listed below you must also complete three “virtual” one semester courses called “Diploma Thesis Seminar”. You will get the credit for this course provided that your thesis supervisor confirms that you work on the thesis.

Course Annotation Year Semester
Introduction to Functional Analysis normed linear spaces, Banach and Hilbert spaces, linear functionals and operators, Hahn-Banach theorem, uniform boundedness principle, open mapping theorem, duality and reflexive spaces, Riesz-Schauder theory of compact operators, Fredholm theorems, convolutions and basic properties of L_p, distributions and tempered distributions 1st winter
Partial Differential Equations 1 notion of the weak solution, Lebesgue spaces, Sobolev spaces, linear elliptic equations 1st winter
Analysis of Matrix Calculations 1 fundamentals of numerical linear algebra, solving of linear algebraic equations, least squares method, eigenvalue problems 1st winter
Finite Element Method 1 foundations of the theory of finite element method, linear elliptic equations 1st winter
Numerical Solution of Ordinary Differential Equations one-step and multi-step methods (algorithms, analysis, convergence), discrete and continuous dynamical systems 1st winter
Computer Simulations in Many-particle Physics principles of the Monte Carlo and molecular dynamics methods 1st winter
Partial Differential Equations 2 Bochner spaces, parabolic and linear hyperbolic equations of the second order 1st summer
Matrix Iterative Methods iterative methods, Krylov subspace methods 2nd winter

Compulsory elective courses

The courses are grouped into five categories — continuum mechanics, molecular dynamics, quantum systems, general relativity and particle physics. Once you have chosen the particular branch of physics you want to specialize in, you should focus on the courses in the corresponding category. You must complete some of these courses. (The term “some” means that you need certain number of credits from the given category, see faculty webpages for details.) The choice is yours, but it should reflect the topic of your thesis. Talk to your thesis supervisor if you want to check which courses are suitable for you!

Continuum mechanics

Course Annotation Year Semester
Continuum Mechanics introduction to continuum mechanics, kinematics, balance equations, linearized elasticity, Navier–Stokes fluid 1st/2nd winter
Mixture Theory mechanics and thermodynamics of mixtures 2nd winter
Computer Solutions of Problems in Continuum Mechanics basic commercial/open source software for numerical computation (Matlab, Femlab, FEniCS) and its application to solution of partial differential equations, basic numerical libraries (Blas, Lapack, Petsc), finite element libraries (Feat, Featflow), libraries for parallel computation (MPI, OpenMP) 1st summer
Thermodynamics and Mechanics of non-Newtonian Fluids non-Newtonian phenomena, constitutive relations for non-Newtonian fluids, thermodynamics of non-Newtonian fluids 1st/2nd summer
Thermodynamics and Mechanics of Solids mechanics and thermodynamics of nonlinear solids, constitutive relations for non-linear solids 1st/2nd summer
Nonlinear Functional Analysis nonlinear operators in Banach and Hilbert spaces, monotone, pseudomonotone and potential operators 2nd summer
Numerical Solution of Evolutionary Equations theoretical and practical aspects of the numerical solution of evolutionary differential equations 2nd summer
Solution of Nonlinear Algebraic Equations theoretical and practical aspects of the numerical solution of nonlinear equations and their systems 2nd winter

Molecular dynamics

Course Annotation Year Semester
Methods of Mathematical Statistics principles of estimation and hypothesis testing 2nd summer
Advanced Simulations in Many-particle Physics advanced methods of Monte Carlo and molecular dynamics and their applications to various problems (critical phenomena, complex molecules, non-equilibrium phenomena, transport coefficients, kinetic MC, growth processes, optimalization problems, quantum MC, ab initio simulations) 2nd summer
Introduction to Plasma and Computational Physics I basics of computational physics, introduction to plasma physics 2nd winter
Introduction to Plasma and Computational Physics II elementary processes in plasma, transport effects in plasma, advanced techniques of particle modelling, particle and fluid modelling in plasma physics and in plasma chemistry 2nd summer
Computer Modelling of Biomolecules drug design, searching and visualization of biomolecular structures 2nd summer
Thermodynamics and Statistical Physics II thermodynamic limit, Gibbs paradox, identical particles, quantum statistical ensembles, the classical limit, fluctuation theory, equivalence of statistical ensembles, ideal Bose and Fermi gas, interacting systems (virial expansion, critical phenomena, mean field approximation, the scaling hypothesis), transport phenomena, Boltzmann kinetic equation 2nd summer

Quantum systems

Course Annotation Year Semester
Quantum Theory of Molecules Born-Oppenheimer and adiabatic approximation, Hückel method, Hartree, Hartree-Fock and Roothaan equations, semiempirical and ab initio methods, correlation energy, symmetry of molecules, intermolecular interactions, polarizability, vibrations of molecules, chemical reactivity 2nd summer
Quantum Scattering Theory theory of atomic processes with applications in non-relativistic astrophysics and chemical physics, formal collision theory and methods for solving scattering problems, single-particle approach in atomic physics 2nd winter
Theory of Collisions of Atoms and Molecules advanced theory of atomic processes with applications in non-relativistic astrophysics and chemical physics, introduction to many-particle atomic and molecular theory, computational methods for collision processes and applications to collisions of electrons with atoms and molecules 2nd summer
Group Theory and its Applications in Physics group theory and representation theory for both finite and continuous (Lie) groups 2nd winter
Quantum Mechanics II identical particles and many-particle systems, angular momentum in quantum theory, mean field and theory of atoms and molecules, symmetry and conservation laws, time-dependent perturbation theory, density matrix and open systems 2nd summer

Relativistic physics

Course Annotation Year Semester
General Theory of Relativity general relativity and its applications in astrophysics and cosmology 2nd summer
Geometrical Methods of Theoretical Physics I elements of topology, differentiable manifolds, tangent bundles, vector and tensor fields, affine connection, parallel transfer and geodesic curves, torsion and curvature, Riemann and pseudo-Riemann manifolds, Riemann connection, Gauss theory of surfaces, Gauss formula, Lie derivative, Killing vectors, exterior calculus, integration on manifolds, integrable densities 2nd winter
Relativistic Physics I tensor analysis, curvature and Einstein gravitational law, Schwarzschild solution of Einstein equations, black holes and gravitational collapse, astrophysics of black holes, general relativity in other branches of physics, linearised theory of gravitation, gravitational waves 2nd winter
Foundations of Numerical Study of Spacetimes basic numerical methods for solving Einstein equations of general relativity 2nd winter
Geometrical Methods of Theoretical Physics II Riemann geometry in terms of forms, Hodge theory, Lie groups and algebras, fibre bundles, geometry of gauge fields, two-component spinors 2nd summer
Introduction to Analysis on Manifolds foundations of differential geometry 2nd winter

Particle physics

Course Annotation Year Semester
Elementary Particle Physics basic properties of elementary particles, models (SU(3), the eightfold way, the quark model, interactions (strong, electromagnetic, weak) and their unification 2nd winter
Fundamentals of Electroweak Theory phenomenological V-A theory of the weak interactions, idea of unification of weak and electromagnetic interactions, nonabelian gauge fields and the Higgs mechanism, Glashow-Weinberg-Salam standard model of electroweak interactions 2nd summer
Quarks, Partons and Quantum Chromodynamics quark model of hadrons, parton model and the deep inelastic scattering of leptons off nucleons, synthesis of mentioned models in the framework of the field theory 2nd summer
Particles and Fields I gauge theories 2nd winter
Selected Topics on Quantum Field Theory I path integral in quantum mechanics, functional methods and Green functions, Wick rotation and the partition sum, Berezin integral 2nd winter
Software and data processing in particle physics I overview of software used in particle physics 1st summer
Software and data processing in particle physics II simulations of particle collisions and thier passage through detector, statistics in evaluation of data acquired in modern detectors 2nd winter
Neural Nets in Particle Physics neural nets, training, error backpropagation, the Hessian matrix, regularization in neural network, Bayesian neural networks
dual use of neural networks – approximations and decision
2nd winter

Other

Course Annotation Year Semester
Functional Analysis 1 topological spaces, weak topology, vector integration, spectral theory 2nd winter
Ordinary Differential Equations 2 dynamical systems, Poincaré-Bendixson theory, Carathéodory theory, optimal control, Pontryagin maximum principle, bifurcations, stable, unstable and central manifolds 2nd winter
Partial Differential Equations 3 linear and nonlinear evolution equations, semigroup theory, asymptotic behaviour of the solutions to differential equations, pptimal control of evolution equations 2nd winter
Theory of Nanoscale Systems I independent fermion and boson models, Hartree-Fock theory for fermions and bosons, Brueckner-Hartree-Fock theory, density functional theory, quantum dots in a magnetic field, Monte Carlo methods 2nd winter
Modern Computational Physics I genetic algorithms, wavelet transform 2nd winter
Modern Computational Physics II neural networks, advanced algoritms for molecular dynamics 2nd summer
Statistical Methods in High Energy Physics basic statistical methods frequently used in analysis of experimental data in high energy physics 2nd summer
Theory of Nanoscale Systems I independent fermion and boson models, Hartree-Fock theory for fermions and bosons, Brueckner-Hartree-Fock theory, density functional theory, quantum dots in a magnetic field, Monte Carlo methods 2nd winter

State exam

After two years the study is completed by the state exam that has two parts — thesis defense and oral part.

Thesis defense

The student presents, usually in twenty minutes, the results of his/her thesis. After that he/she must discuss the issues pointed out in the referee reports. (The referee reports are available in the Study Information System beforehand. It is absolutely crucial to be ready to discuss referee’s comments in a highly qualified manner.) At the end the student answers question by the committee members and the audience.

Oral part

The oral part of state exam consists of answering several question concerning the topics specific for the given study programme. In principle the student will be answering questions regarding the topics studied in the compulsory courses listed in the curriculum above.

The oral part of the state exam proceeds as follows. The student answers, after a preparation, in total six questions regarding

  • theory of partial differential equations (one question),
  • functional analysis (one question),
  • finite element method (one question),
  • theory of solution of systems of algebraic equations (one question),

the remaining two questions are chosen according to student’s specialisation — continuum mechanics, molecular dynamics, quantum systems, general relativity and particle physics.

Answering one question is expected to take ten minutes. The students is expected to demonstrate deeper insight into the discussed topics and the ability to see the subject matter in a broad context. (The student is not expected to give extreme details of each proof. He/she should be able to describe the basic ideas behind the proofs and explain why the definitions/notions/theorems are stated as they are.) Detailed list of the topics for the state exam is listed below. You can also download the list of topics as a PDF file. (In preparation, the link is not active now.)

Partial differential equations

The topics are covered in courses Partial differential equations 1 a Partial differential equations 2. It is expected that the student is also familiar with the classical theory of partial differential equations at the level of the course Introduction to partial differential equations. (See Admission.)

Sobolev spaces

Weak derivative, definition and basic properties of Sobolev spaces Wk,p — reflexivity, separability, density of smooth function, extension operator for W1,p functions and domain with Lipschitz boundary. Theorems concerning the continuous and compact embedding of Sobolev spaces into Lebesgue and Hölder spaces. Definition of the trace operator for functions in Sobolev spaces, trace theorem, inverse trace theorem.

Weak solution of linear elliptic partial differential equations in bounded domain

Definition of the weak solution to linear elliptic partial differential equation with various boundary conditions. Existence of a solution via Riesz representation theorem (symmetric operator), via Lax–Milgram lemma and via Galerkin method. Compactness of the solution operator, eigenvalues and eigenvectors. Fredholm alternative and its applications. Maximum principle for the weak solution. W2,2 regularity via finite differences technique. Selfadjoint operator, equivalence with the minimization problem for a quadratic functional.

Weak solution of nonlinear elliptic partial differential equations in bounded domain

Fundamental of calculus of variations, fundamental theorem of calculus of variations, dual formulation, relation to convexity. Existence and uniqueness of the solution to nonlinear problems via fixed point theorems (nonlinear Lax–Milgram). Existence via Galerkin method and Minty trick — monotone operator and semilinear term.

Second order linear parabolic partial differential equations

Bochner spaces and their basic properties, Gelfand triple, Aubin–Lions lemma. Definition of the weak solution. Initial conditions. Existence of a solution via Galerkin method, uniqueness and regularity of the solution (spatial and temporal), smoothing property, maximum principle.

Second order linear hyperbolic partial differential equations

Definition of the weak solution. Initial conditions. Existence of a solution via Galerkin method, uniqueness, regularity (spatial and temporal), finite propagation speed.

Numerical mathematics

The topics are covered in courses Finite element method and Matrix iterative methods 1. It is expected that the student is also familiar with fundamentals of numerical mathematics at the level of the course Analysis of matrix calculations 1 and Fundamentals of numerical mathematics. (See Admission.)

Finite element method for solution of linear elliptic partial differential equations

Galerkin and Ritz methods for solution of abstract linear elliptic equations. Estimate on the discretization error, Céa lemma. Defintion of the abstract finite element, simple examples of finite elements of Lagrange and Hermite type. Approximation theory in Sobolev spaces, approximation properties of polynomial preserving operators. Application of these results to Lagrange and Hermite type finite elements. Rate of convergence of approximate solutions to linear elliptic partial differential equations. Estimate of the rate of convergence in L2 norm, Nitsche lemma.

Fundamentals of numerical integration in finite element method.

Solution of systems of algebraic equations and eigenvalues computation

Methods for solution of systems of linear algebraic equations and eigenvalues computation. Spectral decomposition of operators and matrices. Invariant subspaces and spectral information, normality. Comparison of direct and iterative methods for solution of systems of linear algebraic equations. Projection process and the problem of moments. Description of the convergence of the iterative methods. Relation between iterative methods for linear equations and methods for eigenvalues computation. Comparison of methods for solution of linear and nonlinear systems of algebraic equations. Numerical stability and description of the algebraic error with respect to the problems in mathematical modelling.

Functional analysis

The topics are partially covered in course Functional analysis I. Theory of function spaces is also partially discussed in courses Partial differential equations 1 and Partial differential equations 2. It is expected that the student is also familiar with fundamental of functional analysis at the level of the course Inroduction to functional analysis. (See Admission.)

Hilbert and Banach spaces

Definition, norm, scalar product, examples. Linear functionals. Hahn-Banach theorem. Dual space, representations of some dual spaces (Hilbert spaces, Lebesgue spaces). Riesz representation theorem. Weak and weak-* topology. Banach-Alaoglu theorem. Weak compactness. Reflexivity.

Continuous linear operators

Definition, basic properties, norm, space of linear operators, adjoint operator. Spectrum and its basic properties, spectral radius, Gelfand-Mazur theorem. Compact operators, symmetric operator, selfadjoint operator, closure, closed operator, definition and properties of adjoint operator. Eigenvalues and eigenvectors of symmetric elliptic operators.

Fixed point theorems

Banach theorem, Brouwer theorem, Schauder theorem.

Integral transformations and fundamentals of the theory of distributions

Definition of Fourier transform in L1 and its basic properties, Fourier inversion theorem, Fourier transform of convolution and derivative. Space of test functions, characterization of a distribution, order of a distribution, operations with distributions (derivative, multiplication), Schwarz space and tempered distributions, Fourier transformation for functions in Schwarz space and tempered distributions, its basic properties. Fourier transform in L2.

Continuum mechanics

The topics are covered in courses Continuum mechanics, Thermodynamics and mechanics of nonnewtonian fluids and Thermodynamics and mechanics of solids.

Kinematics

Description of the motion of continuous media. Deformation of line, surface and volume elements, deformation, deformation gradient, polar decomposition of deformation gradient and its interpretation, right and left Cauchy–Green tensor, Green–Saint-Venant tensor. Rate of deformation of line, surface and volume elements. Velocity, velocity gradient, symmetric velocity gradient, material time derivative. Streamlines and pathlines. Kinematic condition for material surface. Lagrange and Euler description. Compatibility conditions. Isotropic tensor functions, representation theorem for isotropic tensor functions.

Dynamics

Balance equations (mass, momentum, angular momentum, total energy) in Euler and Lagrange description. Integral form for the balance equations, localization principle. Cauchy stress tensor, first Piola–Kirchhoff stress tensor. Balance equations in non-inertial reference frame.

Simple constitutive relations

Compressible and incompressible Navier–Stokes model, linearized elasticity. Boundary conditions. Geometric linearization.

Nonnewtonian fluids

Continuum thermodynamics, balance equations. Entropy. Rate of entropy production. Incompressibility. Maximization of the rate of entropy production and its applicability in theory of constitutive relations. Nonnewtonian phenomena. Viscosity as a function of symmetric part of the velocity gradient and the pressure, normal stress differences, activation/deactivation criteria, stress relaxation, non-linear creep. Principle of frame indifference and its consequences. Frame-indifferent time derivatives. Representation theorem for isotropic tensorial functions. Classical constitutive relations. Differential, integral and rate type models. Power law fluid, fluid with pressure dependent viscosity, Bingham type fluids.

Solids

Principle of frame indifference and its consequences. Finite strain theory, linearized elasticity, incompressible materials in finite elasticity and linearized elasticity, hyperelasticity, properties with respect to the determinant of the deformation gradient. First Piola–Kirchhoff tensor for hyperelastic material. Material models in finite elasticity, elastic constants for hyperelastic materials, homogenoeus/inhomogeneous material.

Rheological models. Kelvin–Voigt, Maxwell, viscous heat conducting materials, thermoelastic materials, adiabatic materials. Clausius–Duhem inequality and its consequences for constitutive relations.

Molecular dynamics

Základy statistické fyziky

Statistický popis termodynamiky. Liouvilleova rovnice. Základní statistické soubory, mikrokanonický, kanonický a velký kanonický soubor. Kvantová statistická mechanika. Maxwellovo-Boltzmannovo, Fermiho-Diracovo a Boseovo-Einsteinovo rozdělení. Základní vztahy statistické mechaniky pro simulace (výpočet termodynamických veličin, egodický teorém) a role numerických simulací.

Základy simulace fyzikálních systémů metodou Monte Carlo

Základy metody Monte Carlo (integrace metodou MC, Markovovy řetězce, chyba MC integrace). Realizace Monte Carlo kroku (prosté a prefreční vzorkování, Metropolisova metoda). Metody generování pseudonáhodných čísel. MC simulace diskrétních modelů (Isingův model, práh perkolace). MC simulace jednoduchých modelů kapalin.

Základy molekulární dynamiky

Princip metody molekulární dynamiky. Pohybové rovnice klasického mnohočásticového systému. Interakční potenciály a okrajové podmínky. Simulace v různých statistických souborech —- NVE, NVT, NPT, grandkanonický.

Určování termodynamických a strukturních vlastností ze simulací

Výpočet měrného tepla a susceptibily pomoci fluktuací. Radiální distribuční funkce.

Pokročilé metody simulace mnoha částic

Fázové přechody a kritické jevy. Fázové a reakční rovnováhy. Simulace procesů růstu. Kvantové simulace. Výpočet kinetických koeficientů. Kinetické Monte Carlo. Určování energetických bariér užitím molekulární statiky. Optimalizační metody.

Základy modelování fyziky plazmatu

Charakteristika a typy plazmatu. Kvazineutralita plazmatu, Debyeova stínící vzdálenost. Teoretický popis plazmatu, kinetický popis, Boltzmannova rovnice, zákony zachování, magnetohydrodynamický popis.

Quantum systems

Základy kvantové mechaniky

Popis stavů a měřitelných veličin. Operátory, komutační relace. Časový vývoj v kvanové mechanice. Popis měření. Stacionární stavy a integrály pohybu.

Řešitelné systémy

Částice v potenciálové jámě, lineární harmonický oscilátor, coulombické pole.

Moment hybnosti a spin

Definice momentu hybnosti, spektrum a vlastní funkce. Skládání momentů hybnosti, Clebschovy-Gordanovy koeficienty. Vektorové a tenzorové operátory, ireducibilní složky a Wignerova-Eckartova věta.

Základní přibližné metody

Variační metoda a poruchový počet. Systémy mnoha částic: symetrizační postulát, bosony, fermiony, Slaterů determinant, vliv spinu.

Teorie rozptylu

Mollerovy operátory a S-matice. Účinný průřez. Časově nezávislá formulace rozptylu, Lippmannova-Schwingerova rovnice. Póly S-matice a vlastní fáze. Základy mnohokanálové teorie rozptylu.

Základní metody mnohočásticové kvantové fyziky

Metoda středního pole, korelační energie a metody pro její výpočet, druhé kvantování. Základy teorie atomů a molekul: elektronová struktura, vibrační a rotační stavy molekul, použití teorie grup, optické přechody.

Výpočetní metody teorie rozptylu

Rozvoj do parciálních vln. Bornova řada. Variační principy. Teorie R-matice.

Relativistic physics

Výchozí principy speciální a obecné teorie relativity

Prostoročas, čtyřrozměrný formalismus, transformace souřadnic. Metrika, afinní konexe, kovariantní derivace. Paralelní přenos a rovnice geodetiky. Posun frekvence v gravitačním poli. Lieova derivace a Killingovy vektory, tenzorové hustoty. Integrování na varietách (hustoty, integrální věty). Křivost prostoročasu.

Einsteinův gravitační zákon a jeho důsledky

Tenzor energie a hybnosti, zákony zachování a pohybové rovnice. Einsteinovy rovnice gravitačního pole. Schwarzschildova a Reissnerova-Nordströmova metrika. Kerrova a Kerrova-Newmanova metrika.

Relativistická astrofyzika a kosmologie

Relativistické modely hvězd. Gravitační kolaps a černé díry. Kritické chování gravitačního kolapsu. Relativistická kosmologie, FLRW modely.

Vlastnosti Einsteinových rovnic

Linearizovaná teorie gravitace a rovinné gravitační vlny. Lagrangeovský formalismus v obecné relativitě, zákony zachování. Hamiltonovský formalismus v obecné relativitě, počáteční problém. Konformní rozklad rovnic vazeb, počáteční data. Einsteinovy rovnice jako hyperbolický systém parciálních diferenciálních rovnic.

Particle physics

Základní představy a metody kvantové teorie pole

Rovnice relativistické kvantové mechaniky. Kvantování volných polí. Interakce polí, Feynmanovy diagramy.

Klasifikace a vlastnosti elementárních částic

Leptony, hadrony a nositelé interakcí. Spin, parita, nábojová parita, podivnost, izospin. Zákony zachování.

Struktura hadronů

Kvarkový model, barva, partony, distribuční funkce.

Základy standardního modelu elementárních částic

Elektroslabé interakce. Higgsův mechanismus. Kvantová chromodynamika.

Interakce částic s prostředím a metody měření částic v experimentech

Měření energie, hybnosti a doby letu částic. Identifikace částic. Monte Carlo simulace průchodu částic detektorem.

Metody analýzy dat v experimentech fyziky částic

Softwarové nástroje. Výběrová pravidla a multivariační analýza. Neuronové sítě.