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P Practice
B Laboratory
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C ECTS credits
E Exam
M Midsemester mark
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Mathematics
(28 credits)
Code  Subject  L  P  B  R  C  I  II  III  IV  V  VI  Responsible 

BMETE92AF35  Mathematical Methods in Physics 1  4  2  0  E  6  6  Dr. Tasnádi Tamás Péter  
The course gives an introduction to mathematical tools used in the Experimental Physics 12 courses without giving precise proofs. The aim of the course is to develop the calculation facility of the students and enable them to use mathematical methods in physical problems. One fourth of the lectures and the practices are devoted to practice the subject on specific problems. The course is jointly held by the Institutes of Mathematics and Physics. Themes: Complex numbers, basic laws of algebra, algebraic, trigonometric, and exponential forms of complex numbers, complex operations. Vectors, matrices: operations (scalar, cross, diadic product), determinant and its properties, trace, LeviCivita symbol, linear system of equations, inverse matrix, Gauss elimination, eigenvalue, eigenvector, characteristic polynomial. Differentiation: definition, basic rules, higher order derivatives, Taylor series, partial derivative, total derivative, Young's theorem, differentiation of vectors, divergence, gradient, curl, nabla symbol, Jacobian matrix. Integration: definitions, definite, indefinite, partial, usubstitution, multiple integral, path, surface, volume integrals, Gauss, Stokes theorem. Literature: – E. Kreyszig: Advanced Engeneering Mathematics, Wiley, 2011. – B.R. Martin, G. Shaw: Mathematics for Physicists, Wiley, 2015. – K. Weltner, S. John, W.J. Weber, P. Schuster, J. Grosjean: Mathematics for Physicists and Engineers, Fundamentals and Interactive Study guide, Springer, 2014. – K.F. Riley, M.P. Hobson, S.J. Bence: Mathematical Methods for Physics and Engineering: A Comprehensive Guide, 3rd ed., Cambridge Univ. Press, 2006. 

BMETE92AF36  Mathematical Methods in Physics 2  4  2  0  E  6  6  Dr. Tasnádi Tamás Péter  
The course gives an introduction to mathematical tools used in the Experimental Physics 12 courses without giving precise proofs. The aim of the course is to develop the calculation facility of the students and enable them to use mathematical methods in physical problems. One fourth of the lectures and the practices are devoted to practice the subject on specific problems. The course is jointly held by the Institutes of Mathematics and Physics. Themes: curvilinear coordinates, covariant, contravariant operations, transformation, cylindrical, spherical coordinates, derivatives. Linear algebra: basis, dual vector space, symmetric operators, similarity transformations, invariants, matrix polynomial, matrix functions, spectral decomposition. Complex analysis: poles, residue theorem, contour integral. Distributions: Dirac delta, operations. Fourier transformation: applications: Fourierseries, convolution, Green's theorem. Literature: – E. Kreyszig: Advanced Engeneering Mathematics, Wiley, 2011. – B.R. Martin, G. Shaw: Mathematics for Physicists, Wiley, 2015. – K. Weltner, S. John, W.J. Weber, P. Schuster, J. Grosjean: Mathematics for Physicists and Engineers, Fundamentals and Interactive Study guide, Springer, 2014. – K.F. Riley, M.P. Hobson, S.J. Bence: Mathematical Methods for Physics and Engineering: A Comprehensive Guide, 3rd ed., Cambridge Univ. Press, 2006. 

BMETE93AF00  Analysis for Physicists  4  2  0  E  6  6  Dr. Illés Tibor  
Rational and real numbers, sets, convergence of real series. Functions of one variable: continuity, properties of continuous functions, monotonicity, properties of monotonic functions, differentiability, significant limits, elemental functions and their inverse functions, intermediate value theorems, properties of differentiable functions, function analysis. Taylor polynomial, definite and indefinite integral, technique of integration, usage of integration, improper integral, simple differential equations. Infinite series. Convergence criteria.


BMETE93AF01  Multivariate Analysis for Physicists  4  2  0  E  6  6  Dr. Illés Tibor  
Function of 2 real variables. Continuity, level curves, differentiation. Young theorem, total differential. Local, global and conditional extremum. Implicit functions. Functions of several variables. Derivative vector, directional derivative. Geometric visualization, level surfaces, chain rule. Integration: double, triple integrals, integral transformations. Cylindrical, spherical coordinates. Spatial curves. Arc length, curvature, torsion. Surfaces. Tangent plane, normal vector, surface area. Scalar and vector fields. Line and surface integrals. Divergence and curl, theorems of Gauss and Stokes, Green formulae. Conservative vector fields, potentials. Some applications of vector analysis. Functional sequences, series. Power, Taylor, Laurent, Fourier series. Software applications for solving some elementary problems. 

BMETE95AF00  Probability Theory for Physicists  2  2  0  E  4  4  Dr. Bálint Péter  
Introduction: empirical background, sample space, events, probability as a set function. Enumeration problems, inclusionexclusion formula, urn models, problems of geometric origin. Conditional probability: basic concepts, multiplication rule, law of total probability, Bayes formula, applications. Independence. Discrete random variables: probability mass function, Bernoulli, geometric, binomial, hypergeometric and negative binomial distributions. Poisson approximation of the binomial distribution, Poisson distribution, Poisson process, applications. General theory of random variables: (cumulative) distribution function and its properties, singular continuous distributions, absolutely continuous distributions and probability density functions. Important continuous distributions: uniform, exponential, normal (Gauss), Cauchy. Distribution of a function of a random variable, transformation of probability densities. Quantities associated to distributions: expected value, moments, median, variance and their properties. Computation for the important distributions. Steiner formula. Applications. Joint distributions: joint distibution, mass and density functions, marginal and conditional distributions. Important joint distributions: polynomial, polyhypergoemetric, uniform and mutlidimensional normal distribution. Conditional distribution and density functions. Conditional expectation and prediction, conditional variance. Vector of expected values, Covariance matrix, CauchySchwartz inequality, correlation. Indicator random variables. Weak Law of Large Numbers: Bernoull Law of Large Numbers, Markov and Chebyshev inequality. Weak Law of Large numbers in full generality. Application: Weierstrass approximation theorem. Normal approximation of binomial distribution: Stirling formula, de MOivreLaplace theorem. Applications. Normal fluctuations. Central Limit Theorem. Literature: – Ross, Sheldon: A First Course in Probability, 8th Edition, Pearson Education International, 2010. 

Total  Mathematics (28 credits)  12  12  4 
Fundamental Physics
(24 credits)
Code  Subject  L  P  B  R  C  I  II  III  IV  V  VI  Responsible 

BMETE13AF02  Experimental Physics 1  4  0  0  E  4  4  Dr. Vankó Péter  
Basic concepts of kinematics, kinematics of points. Force, Newton’s laws, momentum. Gravity, the equivalence principle, SI units. Dynamics, frames of reference, the principle of relativity, inertial forces. Work, kinetic and potential energy. Pointmass systems, conservation laws in mechanics. Statics, kinematics and dynamics of rigid bodies. Elasticity. Fluids: statics, surface phenomena, frictionless and viscous flow, drag. Oscillations: free, damped and driven harmonic oscillators. Superposition of vibrations. Coupled vibrations. Waves, classical wave function. Harmonic waves, phase and group velocity. Wave equation in elastic rods, energy transport in waves. Polarization. Reflection and refraction. Interference, coherence, diffraction. Standing waves. Wave equation in gases and on strings. Standing wave equation, whistles, strings, the physics of music. Doppler effect. Ultrasonic medical diagnostic. Literature: – Alonso, Finn: Fundamental University Physics (https://www.scribd.com/doc/59147822/FundamentalUniversityPhysicsAlons...) 

BMETE11AF26  Practical Course in Experimental Physics 1  0  4  0  M  4  4  Dr. Vankó Péter  
Problem solving. Topics (Experimental physics 1): differentiation, integration, vectors, point kinematics, dynamics of points, work, energy, power, masspoint systems, rigid bodies, flexible bodies and liquids, oscillations, waves. Literature: – Alonso, Finn: Fundamental University (Physics https://www.scribd.com/doc/59147822/FundamentalUniversityPhysicsAlons...). 

BMETE13AF03  Experimental Physics 2  4  0  0  E  4  4  Dr. Koppa Pál  
The basic electric phenomena, electric charge, Coulomb's law. Electric field strength. Electric potential, the first law of electrostatics. Flux, the second law of electrostatics in vacuum. Calculation of the electric field of simple charge distributions. Conductors in electric field. Potential and capacity of charged conductors. Electric dipoles. Polarization, the first and second laws of electrostatics in insulators. The dielectric displacement vector, electric susceptibility and permittivity. The energy density of the electric field. Electric current, Ohm's law, resistance, conductivity and mobility. Kirchhoff's laws. Joule's Law. Mechanisms of conduction in different materials. Contact phenomena. Basic magnetic phenomena, magnetic induction. Forces in a magnetic field. Magnetic dipole moment. Magnetic field of currents, BiotSavart law and the first law of magnetostatics in vacuum. Calculation magnetic field of simple current arrangements. Induction flux, second law of magnetostatics in vacuum. Interaction of currents, unit of current in SI. The magnetization vector, the first and second laws of magnetostatics in materials. Magnetic susceptibility and permeability. The magnetic field strength vector. Electromagnetic induction. Lenz's law, eddy currents. Selfinduction, mutual induction. The energy of magnetic field. Displacement current, Maxwell's equations. The fundaments of special relativity. Electromagnetic oscillations. Electromagnetic waves. Refraction, reflection and interference of light waves. Diffraction, Fraunhofer diffraction on a slit and on a grating. Xray diffraction. Fresnel diffraction. Literature: – A. Hudson, R.R. Nelson: University physics, Saunders College Pub., 1990. – R. Feynman, R. Leighton, M. Sands: The Feynman Lectures on Physics . Desktop Edition Volume II, Addison–Wesley, 1964. 

BMETE12AF20  Practical Course in Experimental Physics 2  0  4  0  M  4  4  Dr. Beleznai Szabolcs  
Electric field, electric currents, magnetic field, electromagnetic induction, electromagnetic oscillations, electromagnetic waves, wave optics. The problemsolving is partially differentiated into groups based on the experience and knowledge of the students. Literature: – K.E. Lonngren, S.V. Savov, R.J. Jost: Fundamentals of Electromagnetics with MATLAB. 

BMETE15AF21  Experimental Physics 3  3  0  0  E  3  3  Dr. Újsághy Orsolya  
Thermodynamics: Temperature, Temperature scales. Equation of States of the ideal gas. Basics of the kinetic theory of gases, pressure, temperature, kinetic energy. Maxwellian velocity distribution. Real gases and the van der Waals equation. Transport properties of gases: mean free path, diffusion, heat conduction, viscosity. State of a system, equation of state. Quasistatic and reversible processes. Heat, internal energy, work, first law of Thermodynamics. Specific heat, enthalpy. Isothermal, isobaric, isochoric, adiabatic processes of ideal gases. Thermodynamic cycles. Second law of Thermodynamics. Efficiency of the Carnot Engine. Entrophy. Basics of statistical mechanics: microstate, macrostate, interpretation of the entropy. Conditions for thermodynamic equilibrium in homogeneous systems. Thermodynamical potentials. Fundamental equations, Maxwell relations, GibbsHelmholtz equations. Third law of Thermodynamics. Chemical potential, Euler equations, GibbsDuhem relation. Phase transitions, Clapeyron equation. Gibbs phase rule. Introduction to Quantum Mechanics: Blackbody radiation, Photoelectric effect, Comptoneffect. Atomic spectra, Thomson, Rutherford, and Bohr model of the atom. DeBroglie wavelength. Wave function, Schrödinger equation. Quantum tunneling. Quantum numbers, Pauli principle. Literature: – Alonso, Finn: Physics (Chapters 1518). – Alonso, Finn: Fundamental university physics, vol. 1 

BMETE15AF22  Practical Course in Experimental Physics 3  0  2  0  M  2  2  Dr. Újsághy Orsolya  
Problem solving class accompanying Experimental Physics 3. Literature: – Alonso, Finn: Physics (Chapters 1518). – Alonso, Finn: Fundamental university physics, vol. 1 

BMETE80AF18  Experimental Nuclear Physics  2  1  0  E  3  3  Dr. Dóczi Rita  
Composition of the atomic nucleus, nuclear force, mass defect and stability of the nucleus, binding energy. The liquid drop model and the semiempirical mass formula. Two ways to release nuclear energy. Types of radioactive decay, exponential decay law, radioactive decay chains; alpha, beta and gamma decay. Types of nuclear reactions, conservation of quantities with nuclear reactions, direct nuclear reactions and compound nucleus reactions. Microscopic and macroscopic cross sections. Types and properties of the neutron induced nuclear reactions. The energy dependence of the cross section of neutron induced nuclear reactions. Neutron slowingdown. Fast neutrons, epithermal neutrons, thermal neutrons. Interaction of radiation with matter: interaction of charged particles (alpha and beta radiation), neutron and gamma radiation with matter, the exponential attenuation of the radiation. Basic properties of the nuclear radiation detectors: gasfilled detectors, scintillation detectors, semiconductor detectors, thermoluminescent dosimeters, solidstate nuclear track detectors. Neutron detectors. Nuclear fission. Fission products, fission neutrons; the energy balance of the fission process. Chain reaction with neutrons, timebehavior of the chain reaction, effective neutron multiplication factor, the basic constituents of a thermalneutron reactor. Nuclear reactions capable to produce fissile material. Types of particle accelarators. Literature: – K.S. Krane: Introductory Nuclear Physics. John Wiley and Sons, Inc. 1988. 

BMETE13AF11  Comprehensive Examination in Experimental Physics  0  0  0  S  0  0  Dr. Vankó Péter  
Topics of Experimental physics 1, 2, 3 and Experimental nuclear physics. 

Total  Fundamental Physics (24 credits)  8  8  5  3 
Advanced Physics
(30 credits)
Code  Subject  L  P  B  R  C  I  II  III  IV  V  VI  Responsible 

BMETE15AF23  Mechanics 1  2  0  0  E  2  2  Dr. Zaránd Gergely Attila  
Hamiltonian mechanics: Newton axioms, motions of systems of point particles and conserved quantities (first integrals). Motion in one dimension. Forces of inertia. Rigid body and tensor of inertia, Euler equations. Constraints, generalized coordinates, D'Alambert's principle and Lagrangeformalism. Charged particle in an electromagnetic field. Lagrange theory of a symmetrical top. Hamilton's principle. Motion in central potential, twobody problem, stellar motion, scattering theory and crosssection. Theory of small oscillations. Hamilton formalism. Liouville's theorem. Continuum mechanics: Deformations, strain and Stress tensor. Continuity equation. Lagrange's equation of motion. Viscous fluids and NavierStokes equation, laminar flow. Sound waves in isotropic solids. Literature: – H. Goldstein: Classical Mechanics. 

BMETE15AF24  Practical Course in Mechanics 1  0  2  0  M  3  3  Dr. Zaránd Gergely Attila  
Tutorial supplementing the Mechanics 1 lecture series. General description of the motion of point particles (curvilinear coordinates). Onedimensional damped oscillations, motion in one dimension. Forces of inertia, motion on the rotating Earth. Rigid body and tensor of inertia, Euler angles. Principle of virtual work. Lagrange I and Lagrange II equations. Theory of small oscillations. Hamilton formalism. Deformations, strain and Stress tensor. Laminar flow. Literature: – H. Goldstein: Classical Mechanics. 

BMETE15AF27  Quantum Mechanics 1  2  0  0  E  2  2  Dr. Szunyogh László  
This course serves to ground the concepts and methods of Quantum Mechanics. Topics to be covered: Experimental backgrounds of Quantum Mechanics. Mathematical apparatus: Hilbert space, operators, properties of Hermitian operators, eigenvalue problem of operators. Schrödinger equation, wavefunctions, probability density. Eigenvalues and eigenfunctions of the coordinate and linear momentum operators. Quantum theory of measurements, Heisenberg's uncertainty principle. Tunneling effect. Harmonic oscillator, step operators. Eigenproblem of angular momentum operators, addition of angular momenta. Central potential, radial Schrödinger equation. Hydrogen atom. Approximation methods: variational principle, stationary and timedependent perturbation theory. The spin: experimental evidences, spin operators, Pauli equation. Identical particles, Pauli principle. Atoms and periodic table. Literature: – F. Schwabl: Quantum Mechanics, Springer, 1990. – A. Messiah: Quantum Mechanics, Vol. 12, North Holland, 1986. 

BMETE15AF28  Practical Course in Quantum Mechanics 1  0  2  0  M  3  3  Dr. Szunyogh László  
Problem solving course related to Quantum Mechanics 1. The topics cover: Mathematical basis, Hilbert space, operators and their eigenproblem. Solution of the Schrödinger equation for simple systems. Onedimensional potential barrier, tunneling effect. Sommerfeld polynomial method. Harmonic oscillator. Eigenvalue problem of the angular momentum operators. Energy eigenvalues and eigenstates of the Hydrogen atom. Rayleigh–Ritz variational approach. Rayleigh–Schrödinger stationary perturbation theory. Dirac's timedependent perturbation theory. Many particle systems: Helium atom, Hartree method. Literature: – S. Flügge: Practical Quantum Mechanics, Springer, 1994. 

BMETE15AF25  Electrodynamics 1  2  0  0  E  2  2  Dr. Takács Gábor  
Electrostatics: point charge and charge distribution, Gauss's Law, Maxwell's equations for electrostatics. Potential, Poisson and Laplace equations, boundary conditions. Green's function, capacitance, method of images. Electric dipole and quadrupole, multipole expansion. Dielectrics, polarization, electric displacement field, surface charge density. Energy of electrostatics field. Magnetostatics: current density, charge conservation. BiotSavart law. Maxwell's equations for magnetostatics. Vector potential. Magnetic dipole. Magnetostatics in the presence of matter. Boundary conditions, surface current density. Linear and nonlinear materials, hysteresis. Quasistatic fields: electromotive force, Faraday's law, Lenz's law. Inductance. Quasistatic magnetic field in conductors, skin effect. Energy of magnetic field. Dynamics: displacement current, full Maxwell's equations in vacuum and matter. Potentials, gauge fixing, d'Alambert equation. Energy and momentum of the elctromagnetic field. Plane waves, polarization, energy and momentum. Electromagnetic waves in matter, reflection and refraction. Retarded and advanced Green's functions. Dipole approximation, Larmor formula. Thomson and Rayleigh scattering. Literature: – D.J. Griffiths: Introduction to Electrodynamics , Pearson. – J.D. Jackson: Classical Electrodynamics, Wiley. 

BMETE15AF26  Practical Course in Electrodynamics 1  0  2  0  M  3  3  Dr. Takács Gábor  
Problem solving class accompanying Electrodynamics 1. Literature: – D.J. Griffiths: Introduction to Electrodynamics , Pearson. – J.D. Jackson: Classical Electrodynamics, Wiley. 

BMETE15AF29  Statistical Physics 1  2  0  0  E  2  2  Dr. Zaránd Gergely Attila  
Micro and macrostates, closed systems, equilibrium and ergodicity. Principle of equal probabilities, Boltzmann entropy, connection to thermodynamics. Statistical physical ensembles and their equivalence. Thermodynamic potentials and fluctuations. Ideal gases, FermiDirac, BoseEinstein and MaxwellBoltzmann statistics. Black body radiation. Interacting systems, pair correlation functions, screening. Virial expansion. Van der Waals equation, mean field theory and critical behavior. Literature: – D. Chandler: Introduction to Modern Statistical Physics. 

BMETE15AF30  Practical Course in Statistical Physics 1  0  2  0  M  3  3  Dr. Zaránd Gergely Attila  
Tutorials supplementing the Statistical Physics 1 lecture series. Micro and macrostates, closed systems, equilibrium and ergodicity. Principle of equal probabilities, Boltzmann entropy, connection to thermodynamics. Statistical physical ensembles and their equivalence. Thermodynamic potentials and fluctuations. Ideal gases, FermiDirac, BoseEinstein and MaxwellBoltzmann statistics. Black body radiation. Interacting systems, pair correlation functions, screening. Virial expansion. Van der Waals equation, mean field theory and critical behavior. Literature: – D. Chandler: Introduction to Modern Statistical Physics. 

BMETE11AF05  Introduction to Solid State Physics  2  0  0  E  2  2  Dr. Kézsmárki István  
Symmetries of crystals, crystal structures, Bravais lattices. Theory of diffraction, structural factor, atomic scattering factor. XRay, electron and neutron scattering experiments. Lattice vibrations in harmonic approximation, dynamical matrix, normal coordinates, dispersion relation, density of states. Quantum description of lattice vibrations, energy and momentum of phonons, experimental measurement of the dispersion relation. BoseEinstein statistics, heat capacity of solid bodies, Debye approximation. Drude model of electrons, transport and optical properties. FermiDriac statistics, Sommerfeld expansion, heat capacity, magnetic susceptibility of an electron gas. Electrons in the periodic potential of a crystal, Bloch electrons. Band structure in the nearly free and tight binding approximation, effective mass. Literature: – C. Kittel: Introduction to Solid State Physics, Wiley, New York, 1986. – N.W. Ashcroft, N.D. Mermin: Solid State Physics, Saunders, Philadelphia, 1976. 

BMETE11AF06  Practical Course in Solid State Physics  0  2  0  M  2  2  Dr. Kézsmárki István  
Crystal structures, Bravais lattices: basis, unit cell, reciprocal lattice, packing faction. Theory of diffraction: structural factor, atomic scattering factor. Noncrystalline solids, liquid crystals. Real crystals, classification of defects, thermodynamics of point defects. Lattice vibrations in harmonic approximation: dispersion relation, effects of lattice vibration in the scattering pattern. Quantum description of lattice vibrations, energy and momentum of phonons, density of state, melting point of the crystal (Lindemann criterion). BoseEinstein statistics, heat capacity of solid bodies, Debye approximation. Drude model of electrons, transport and optical properties. FermiDriac statistics, Sommerfeld expansion, heat capacity, magnetic susceptibility of an electron gas. Electrons in the periodic potential of a crystal, Bloch electrons. Band structure in the nearly free and tight binding approximation, effective mass. Literature: – C. Kittel: Introduction to Solid State Physics, Wiley, New York, 1986. – N.W. Ashcroft, N.D. Mermin: Solid State Physics, Saunders, Philadelphia, 1976. 

BMETE11AF11  Applied Solid State Physics  2  0  0  E  2  2  Dr. Csonka Szabolcs  
Band structure of metals and semiconductors, electron transport, electron scattering mechanisms, 2 dimensional electron gases, Si technology (FET, SSD memory), semiconductor heterostructure (semiconductor laser, MEMT), nanoelectronics, single electron transistor. Magnetic materials, origin of magnetic momentum and interaction between moments, magnetic structures. Magnetism of metals, spin polarized bands, spintronics devices (spin valve, MRAM). Spin transistor, magnetic semiconductors.


BMETE12AF35  Optics  2  2  0  E  4  4  Dr. Erdei Gábor  
Models of light, Fermatprinciple, Huygensprinciple. Reflection and transmission of light at planar surfaces. Total reflection, evanescent wave. Geometrical optics. Paraxial optics, matrix optics.Concept of principle planes. Interference; single beam , multiple beam (Michelson, MachSender). Resolution of optical gratings. Description ef systems of thin films by matrix formalism. Antireflection coating, interference mirror. FabryPerot interferometer. Diffraction, Fresnel Kirchoff and RaylighSommerfeld formulas. Fraunhofer and Fresnel diffraction. Square and round apertures. Fraunhofer diffraction image of a sinusoidal grating. Polarization. Polarization sensitive optical elements. Birefringence. Ordinary and extraordinary beams. Propagation of light in anisotropic media. Polarization prisms. polarizing and phasese shifting plates. Interaction of light and matter. Energy levels, inverse population. Spontaneous emission, induced emission and absorption. Lasers, resonators,amplification, pumping. Temporal and spatial coherence. Acoustooptics. Modes of planar waveguides.Ray optics description.Propagation constant. Graphic solution of the mode equation. Literature: – Klein, Furtak: Optics. 

Total  Advanced Physics (30 credits)  5  5  13  7 
Laboratory work, measurement techniques, electronics
(32 credits)
Code  Subject  L  P  B  R  C  I  II  III  IV  V  VI  Responsible 

BMETE11AF27  Introductory Laboratory Exercises  0  0  2  M  2  2  Bordács Sándor  
Basic error analysis. Evaluation and plotting of the experimental data, linear regression, nonlinear curve fitting. Simple experiments to practice data evaluation and error analysis. Basic functions of multimeters, oscilloscopes, function generators and data acquisition cards are introduced to the students. Students must attend to 6 laboratory practices each of them is 4 hour long.


BMETE11AF28  Laboratory Exercises in Physics 1  0  0  3  M  4  4  Dr. Vankó Péter  
Basic instruments, procedures and methods. Evaluation of measurements, error calculation, protocol writing. Measurement of basic electrical, mechanical, optical and thermal quantities. Data collection (manual and by computer interfaces). Basic use of power supplies, sound generators, multimeters, oscilloscopes, etc. Measurements related to Experimental physics 1 and 2. 

BMETE11AF29  Laboratory Exercises in Physics 2  0  0  4  M  5  5  Dr. Vankó Péter  
Basic instruments, procedures and methods. Evaluation of measurements, error calculation, protocol writing. Measurement of complex electrical, mechanical, optical and thermal quantities. Advanced data collection. Advanced use of power supplies, sound generators, multimeters, oscilloscopes, etc. Measurements related to Experimental physics 1, 2 and 3. 

BMETE12AF27  Electronics  2  0  0  M  2  2  Dr. Kiss Gábor  
The primary aim is to teach the operation and planning of the basic circuits used in the experimental and applied physics. This subject is based on the thematics of Experimental physics 2 and Practice in experimental physics 2, giving knowledge in the physical bases of linear electronics (Maxwellequations, Kirchofflaws, resistance, capacity, inductivity, complex impedance, transient phenomena, RLC circuits). The detailed physics of semiconductor devices is tought later (Theoretical solid state physics, Applied solid state physics). In Electronics only the phenomenological models of semiconductor devices are treated. Thematics: Brushup the physical bases of linear electronics. Linear electronic elements: ideal resistor, capacitor, inductor, distributed (parazite) parameters, volt and amper meters, voltage and current sources. Basic AC and DC circuits: bridges, voltage dividers, filter circuits, transformers. Introduction into the calculational methods of complex linear AC and DC circuits. Analysis methods of nonliner circuits. Smallsignal models, notion of distorsion. Characteristics of diodes, bipolar and fieleffect transistors, small and large signal models of the devices. Active analogue circuits, bipolar and field effect transistor amplifiers, rectifiers. Feedback and its application. Parameters of operation amplifiers and their applications. Inverting and noninverting amplifiers, summarizing, differentiating and integrating circuits, schmitttrigger circuit, oscillators. Special complex circuits (power supplies, regulators), protection of circuits.


BMETE80AF03  Laboratory of Electronics  0  0  2  M  2  2  Dr. Pór Gábor Géza  
This is a practical course, where students build basics electronics circuits like Smitt trigger, Miller effect and electronics of coincidence measurement. We pay attention mainly to electronics applied in nuclear measuring chains including signal formation differential and integral electronics, analog digital converters, transfer function signal/noise ratio, dead time, and jitter. Students get practice in electronics oscilloscopes, measuring automatically amplitude and spectrum. Using LABVIEW they learn how to build a spectrum analyzer in one day, measuring propagating perturbations to estimate velocity of natural convention in the water. All practice should be reported in form of well formatted measuring report including error estimation as well.


BMETE11AF30  Measurement Techniques  2  0  0  E  2  2  Dr. Halbritter András Ernő  
Voltage and current sources/meters. Measurement of resistance, four probe method. Voltage and current amplifier circuits. A/D and D/A converters, data acquisition cards. Analog and digital oscilloscopes, sampling modes, triggering, waveform measurements, aliasing. Suppression of disturbing signals: electrostatic and inductive coupling, grounding and guarding, twisted pairs, thermo electric power and offset compensation, stray capacitance. Wave propagation in coaxial lines, telegraph equations, reflections at the cable termination. Fourier analysis considering finite temporal window: spectral leakage, frequency resolution and amplitude accuracy of various window functions. The role of finite sampling, sampling theorem. Discrete Fourier transform, and its implementation by the fast Fourier transform algorithm. Spectrum analyzers: FFT, swept tuned and hybrid devices. Phase sensitive measurements: lockin amplifiers, phase locked loops. The application of PID control from temperature controllers to scanning probe microscopes. Electronic noise phenomena, thermal noise, noise limit of current amplifier circuits, cross correlation noise measurement. Fundamental measurement units (SI) and their definitions. Measurement standards: atomic clocks, conversion between voltage, current and frequency (josephson effect, Quantized Hall effect, elcetron pump), measurement of mass by Watt balance, measurement of temperature by the speed of sound and the thermal noise. Magnetic field sensors: inductive, magnetoresitive, spin valve, and Hall sensors, SQUID magnetometers. Distance and position sensors: linear differential transformers, capacitive tansducers, LASER and ultrasound based measurement of distance, LIDAR systems. Temperature sensors: thermocouples, resistance thermometers, thermistors. Light sensors: photo diodes, CCD sensors, CMOS active pixel sensors, bolometers. Measurement of acceleration: MEMS accelerometers and gyroscopes, piezoelectric accelerometers. Literature: – Keithley: Low Level Measurements handbook. – J.A. Blackburn: Modern Instrumentation for Scientists and Engineers. 

BMETE11AF32  Advanced Laboratory Exercises in Physics 1  0  0  4  M  5  5  Dr. Fülöp Ferenc  
Advanced level experiments related to various topics of the modern physics and the current research activities in the BME TTK: experiments in basic quantum physics; measuring basic physical constants; optical measurements, experiments in wave optics; mastering of modern measurement techniques. 

BMETE11AF33  Advanced Laboratory Exercises in Physics 2  0  0  4  M  5  5  Dr. Fülöp Ferenc  
Advanced level experiments related to various topics of the modern physics and the current research activities in the BME TTK: experiments in solid state physics, material sciences, optical phenomena and nuclear physics; investigation of ionising radiations and radiation detectors; acquirement of modern measurement techniques. 

BMETE12AF21  Advanced Laboratory Exercises in Physics 3  0  0  4  M  5  5  Dr. Ujhelyi Ferenc  
Advanced laboratory experiments related to the modern physics and the research fields of BME TTK mainly in the following fields: Semiconductor physics, material science, surface physics, vacuum techniques. Advanced optical measurements. Nuclear measurements. Modern measurement methods.


Total  Laboratory work, measurement techniques, electronics (32 credits)  2  4  9  7  5  5 
Computer programming, numerical methods
(10 credits)
Code  Subject  L  P  B  R  C  I  II  III  IV  V  VI  Responsible 

BMEVIEEA024  Programming  2  0  2  M  4  4  Dr. Pohl László  
Synopsys of the subject, requirements, algorithm, data, language, programming languages, why the C? specification, design, coding, testing, documenting, algorithm choice questions in connection with GCD (trial and error, prime factors, Euclidean formula), elements of algorithms: sequence, branching, cycles, n! calculation: algorithm selection, parts, data structure, narrative description of the algorithm, algorithm by block diagram, encoding; a small analysis: mandatory elements of a C program, the frame, the main function, return 0; the purpose and significance of indenting, scanf for reading integer values, printf for writing integer values. Storage units: variables, constants, functions; mandatory declaration / definition, syntax / semantics: Syntax diagram, syntax of an integer value, Basic syntax rules: free writing mode (white spaces), a != A, #preproc, / * comment * /, regular identifiers; predefined types, why we use int and double, constant int definition in dec, oct, hex forms, lack of the logic type, logic value of numbers. Instructions: ;, declaration/definition, expression instruction, conditional instruction, cycle (now just the while), control statements (switch/case just mentioned), {}, block diagram of if..else and while. Conditions: relational operators ('==' != '=', the dangers), logic operators !, &&, . Supplement and deepen the knowledge of the past week. control structures, instructions, builtin types, number representation. Use of library functions. Basic operators: arithmetic, integer, real, type cast, assignment, sizeof, relational, logic, bitwise, shortcut, ?:. Iterative solutions, =, pre/post ++ , dangers of post, arrays, 1D, 2D, strings, pointers. 1D dynamic array (example of use), (only breefly, at the level of usage: getchar, putchar, EOF, ctrl+z/ctrl+d) filter program template, enum type, finite automaton example: writing out the comments from a C code, ly counter. Functions, memory areas allocated in the program, what is/will be where, the heap, behavior of the stack, the consequences of the differences. Storage classes (for local variables), the function call mechanism, multiple return values: void descart2polar(double, double, double*, double *), why forbidden to return local variable address. Struct, ., >, typedef, direct selection sort, bubble sort, for structure array also, comparing functions, strcmp, sorting by text. Function pointers, useage of qsort. Making of string, int and double comparing funtion (by a structure array sorting example), introducing recursive structure, ONLY drawn. Unidirectional, bidirectional, "arranged according to several criteria" list, binary tree, coded only the search in the list by cycle. Managing lists, insertion, search, deleting functions, the two possible head handling: head=insert(head..., and insert (&head,..., interpretation of recursion by n!, binary tree management, inorder traversal only in code level. I/O, FILE fopen, fclose, feof, f/sprintf, f/sscanf, getc/s, Putc/s, parameters of main. In short, what is missed: the comma op, (union, bitfield vararg), the C preprocessor. Backup (if there is no need to make up missed lectures then: making programs from multiple source file). Literature: – B.W. Kernighan, D.M. Ritchie: The C Programming Language. Prentice Hall, 2nd edition, 1988. – D. Gookin: C AllinOne Desk Reference For Dummies, Wiley Publishing, Inc, 2004. 

BMEVIEEA026  Programming 2  2  0  2  M  4  4  Dr. Pohl László  
Overview, C repeat, process of function call, const, make, purpose and possibilities of profiling. Number representation questions in simulations, inaccuracy, instability, InF, NaN, different real types, fitting function versions. Function overload, default arguments, inline function to replace macros, the reference type, dynamic memory management: new, new [], delete, delete []. Objectoriented programming concepts, principles, objects, classes, member variables and member functions, the this pointer, encapsulation, visibility and data hiding (complex number class). Constructors and destructors, exception handling, operator overload by member function and by global function (rational number class). Dynamic classes with members, copy constructor, assignment operator, the destructor. (Vector and matrix classes). Member variable initialization, constants and static members, namespaces, C++ I/O, overload of >> and
Literature: – B. Stroustrup: The C++ programming language AddisonWesley, 3rd Edition, 2000. – A. Alexandrescu, H. Sutter: C++ Coding Standards: 101 Rules, Guidelines, and Best Practices, AddisonWesley Professional; 1st ed. 2004. – Scott Meyers: Effective C++: 55 Specific Ways to Improve Your Programs and Designs, AddisonWesley Professional; 3rd ed. 2005. 

BMETE92AF01  Numerical Computations for Physicists  0  0  2  M  2  2  Dr. Szabó Sándor  
In this course we use the Matlab and Maple softwares to solve linear algebraic, one and multivariable analysis problems. We consider the following topics. Linear Algebra: Solution of linear systems, Eigenvalues, eigenvectors, Column space, row space, rank, GramSchmidt orthogonalisation process, Inverse, determinant. Analysis: Solution of nonlinear systems by numerical methods, calculating integrals by quadratures, multiple integrals. Interpolation, limit, differentiation, determining potential function. Differential equations: Numerical (Euler, RungeKutta methods) and symbolical methods. Matlab: Programming in Matlab, Vectors, matrices, functions, graphics. Maple: Basic commands, LinearAlgebra, DEtools, VectorCalculus and plots packages. Literature: – www.maplesoft.com/support/ – www.mathworks.com/support/?s_tid=gn_supp 

Total  Computer programming, numerical methods (10 credits)  4  6 
Others
(10 credits)
Code  Subject  L  P  B  R  C  I  II  III  IV  V  VI  Responsible 

BMEVEFKA144  Chemistry  4  0  0  E  4  4  Dr. Kállay Mihály Balázs  
General chemistry (introduction, basic chemical terms, notion of mole, reaction equations, stoichiometry, basics of chemical calculations, types of concentration). Basics of inorganic chemistry (constitution of atoms and molecules, types of chemical bonds, types of chemical formulae, the periodic table, states of matter, properties of the elements, most important inorganic compounds). Basics of chemical thermodynamics (basic terms, internal energy, work, heath, the first law of thermodynamics, enthalpy, heat of reaction, standard enthalpies, Hess's law, second law of thermodynamics, entropy, free energy, free enthalpy, standard free enthalpies, free enthalpy of the ideal gas, chemical potential, mixtures, activities, equilibria, thermodynamic equilibrium constant). Chemical kinetics (notion of reaction rate, molecularity of reactions and reaction order, first and second order reactions, stepwise reactions, the effect of temperature on the reaction rate). Electrochemistry (properties of electrolytes, electrolytic dissociation of water and the concept of pH, galvanic cells, Nernst equation, types of electrodes, electrochemical power sources, zinc coal cells, batteries, fuel cells, electrolysis). Organic chemistry (hydrocarbons, aromatic compounds, halogen derivatives, alcohols, amines, ethers, aldehydes, ketones, carbonic acids, anhydrides, esters, carbohydrates, proteins, nucleic acids – definition, nomenclature, structure, most important reactions). Colloid chemistry (basics of colloid chemistry, dispersions, macromolecular and micellar solutions, gels, stability of colloids, preparation of colloids, examination methods of colloid systems). Materials science (basics of polymer chemistry, types of polymers, structure of polymers, polymerisation reactions, most important plastics, composites, ceramics, liquid crystals). Chemical examination and analytic methods (spectroscopic methods, classical analytic procedures, chromatography, electroanalysis). Literature: – P. W. Atkins: Physical Chemistry IIII. 

BMETE80AF24  Radiation Protection and its Regulatory Issues  2  0  0  M  2  2  Dr. Pesznyák Csilla  
Basic knowledge of radioactivity. Interactions between ionizing radiation with matter. The physical, chemical, biochemical and biological effect of radiation energy. The effects of radiation on whole organisms, tissues and cells, as well as on cellular causes leading to the death of normal and malignant cells. Dose definitions. External and internal exposure. The appearance of different radionuclides in organisms. The basic principles of radiation protection. The radiation dose limit system. Regulations of radiation protection. Dose and dose rate measurement and their calculation in direct and indirect ways. The relationship between emissions and immissions. Technical radiation protection. Handling accidental situations. The occurrence of natural radioactivity in inorganic and living environment. The ingredients of public exposure from natural sources. The applications of artificial radioisotopes and how they get into the environment. The forms of nonionizing radiation and their possible physiological effects. The applications of nonionizing radiations and their system of limitations. Safety issues in the application of lasers. Radiation protection of particle accelerators. 

BMEGT20A003  Management and Business Economics  2  0  0  M  2  2  Dr. Kövesi János  
The course introduces the essentials of management as they apply within the contemporary work environment and gives a conceptual understanding of the role of management in the decision making process. Particular attention is paid to management theories, corporate finance, leadership, teamwork, quality management, management of technology, economics calculation and operations management. For problem formulation, both the managerial interpretation and the mathematical techniques are applied. Principles of management: Organizational resources. The enterprise as an organization. Functions of managerial processes. Managerial roles. Role of an engineer. Team work, communication in an organization. Lifecycle management and its managerial aspects. Costing: costing, cost effectiveness, traditional costing systems. Break even analyses, standard costing, activity based costing. Quality management: Principles of quality management, the brief history of quality management systems. Overview of quality assurance systems based on ISO 9001:2000. Overview of quality assurance systems based on Total Quality Management System. Literature: – R.W. Griffin: Management (2nd ed.). Houghton Mifflin Company, Boston, MA, 1987 (vonatkozó fejezetei). – L.W. Rue, L.L. Byars: Management. Theory and Application. (4th ed.) Richard D. Irwin, INC. Homewood, Illinois, 1986 (vonatkozó fejezetei). – A. Tenner, I. DeToro: Total Quality Management: Three Steps to Continuous Improvement, AddisonWesley, Reading, MA, 1992. – D. Waters: Operations Management. Producing goods & Services. AddisonWesley, 1996. 

Total  Others (10 credits)  4  4 
Specialized courses chosen from the list below
(27 credits must be completed)
Code  Subject  L  P  B  R  C  I  II  III  IV  V  VI  Responsible 

Total  Specialized courses chosen from the list below (27 credits must be completed)  9  10  8 
Advanced Mathematics
Code  Subject  L  P  B  R  C  I  II  III  IV  V  VI  Responsible 

BMETE15AF31  Modern Mathematical Methods in Physics  2  2  0  E  4  4  Dr. Lévay Péter Pál  
Definition of generalized functions (distributions). Dspace, convergence properties. Regular and singular distributions. Manipulating distributions. Convergence in D' space. Multiplying functions and distributions. The derivative and integral of distributions. The derivative and integral of distributions with respect to parameters. Regularization of distributions. Diracdelta series. Convolution of distributions, properties of convolutions. Multivariable distributions. Fourier transform of distributions. Properties of Fourier transform. Fourier transform of shifted, rescaled and derived distributions. Fourier transform of convolution. Solving initial value problems via Fourier transform. Greenfunction of linear differential operators. Titchmarshtheorem, dispersion relations. Basic solutions and Green functions of famous partial differential equations of mathematical physics. (Poisson equation, wave equation, Schrödinger equation etc.). Applications. The lectures are connected to a practice with an aim to apply the material of the lectures for problem solving. Literature: – R. Strichartz: A guide to distribution theory and Fourier transforms, CRC Press London, 1994. 

BMETE80AF25  Introduction to Experimental Data Handling  2  0  0  E  2  2  Dr. Kis Dániel Péter  
Basic concepts of probability theory. Measurement results, distribution function, mean value, standard deviation, and covariance. Poisson distribution, Gauss distribution, Student distribution, chi square distribution, confidence intervals. Parameter estimation. Concept of statistics, estimated parameters. Properties of estimates: unbiasedness, efficiency, consistence. Method of least squares. Maximum likelihood method. Normal equations and their solution. Estimating the deviation of estimated parameters. Examples of evaluation of measurements. Linear regression. Curve smoothing. Handling nonlinear fittings, iteration. Corrections, e.g. dead time correction. Basic concepts of metrology. Systematic and statistical error. Consideration of corrections. Concept of measurement uncertainty, methods of estimation. Examples of forms of presenting measurement results. Preparing diagrams. Erroneous measurements. Detection and handling outliers. Literature: – J. Mandel: The statistical analysis of experimental O52 Dover Publications, New York,1964. 

BMETE92AF02  Functional Analysis for Physicists  4  2  0  E  6  6  Dr. Petz Dénes  
Vector spaces (linear maps, algebraic dual of a vector space, matrix of linear maps). Tensor product of vector spaces (symmetric and anti symmetric tensor products, bases, determinant). Normed spaces (examples, Hölder's and Minkowski's inequalities, continuity and boundedness of linear maps, norm of operators). Banach spaces (convergence, rearrangement and unconditional convergence of absolute convergent sequences; the exponential function, Neumann series). Main theorems in Banach spaces (open mapping theorem, uniform boundedness theorem, application to Fourier series). Dual space (dual of l^p spaces, HahnBanach theorem, dual of the space of continuous functions). Hilbert space (expansion in a basis, Riesz lemma, projection theorem, Riesz representation theorem). Special functions (Hermite, Legendre polynomials, expansions). Tensor product of Hilbert spaces and their operators (difference between algebraic and Hilbert tensor product, tensor product of L^2 spaces, norm of elementary tensor). Adjoint (adjoint of bounded linear operator, selfadjoint operators, unitary operators, projections, examples). Topologies (Weak topology on Hilbert space, pointwise and pointwise weak convergence of operators, monotonic sequence of selfadjoint operators, topological group of unitaries). Spectrum of bounded operator (parts of the spectrum, spectral radius, resolvent set, properties of the spectrum (nonempty, closed)). Compact operators (ideal of compact operators, HilbertSchmidt integral operator, Green's function, RieszSchauder theorem). Fourier transformation (definition on L^1, its extension to a unitary transformation of L^2, its spectrum, differentiability of the image, Schwartz space and its topology, dual space of the Schwartz space, distributions). Unbounded operators (adjoint, symmetric operators, Laplace operator, examples). Spectral theorem. One parameter unitary groups.


BMETE11AF35  Group Theory for Physicists  2  2  0  E  4  4  Dr. Fehér Titusz  
The aim of the course is to introduce the principles of group theory to physics students: we learn how the symmetries of a system can be used to describe it, and how the symmetries of nature manifest themselves in laws of physics. We apply the concepts of group and representation theory to practical problems. Theory: Symmetries in nature and physics. Definition and basic properties of groups. Some special groups. Homomorphism, isomorphism. Subgroups, cosets, Lagrange's theorem. Normal subgroup, quotient group, first isomorphism theorem. Conjugate, conjugacy classes, centralizer. Group action, orbit, stabilizer. Representations and their properties, equivalent representations, irreducible representations. Schur's lemma. Character of representations, propertires of characters, character tables. Direct sum of representations and their reduction. Product representations. Lie groups, infinitesimal generators, Lie algebras. Topological properties, universal covering group. Rotation group and its representations. Lorentz group and other matrix groups. Calculation: Description of normal modes, crystals, and quantum mechanical wave functions using group theory. Selection rules. Literature: – H.F. Jones: Groups, Representations and Physics (IOP Publishing, 1998) – R.L. Liboff: Primer for Point and Space Groups (Springer, 2003). – M.S. Dresselhaus, G. Dresselhaus, A. Jorio: Group Theory  Application to the Physics of Condensed Matter (Springer, 2008). 
Advanced Physics
Code  Subject  L  P  B  R  C  I  II  III  IV  V  VI  Responsible 

BMETE15AF32  Mechanics 2  2  0  0  E  2  2  Dr. Zaránd Gergely Attila  
Relativistic mechanics: Lorentztransformations, fourvectors and Minkowski space, relativistic collisions, relativistic action and equations of motion. Relativistic particle in an electromagnetic field. Lagrangetheory of continuum mechanics: Lagrange density of a string, EulerLagrange equations, energy density. Application to quantum mechanics and to harmonic media, KleinGordon equations. Hamiltonian formulation of continuum mechanics. Symmetries: Noether's theorem, symplectic formulation of Hamiltonian mechanics. Poisson's brackets, integrability. Canonical transformations, HamiltonJacobi equations, actionangle variables. Nonlinearity, second harmonic generation, parametric resonance. Basics of dynamical systems and chaos. Literature: – H. Goldstein: Classical Mechanics, AddisonWesley. – J.R. Taylor, Classical Mechanics, University Science Books. 

BMETE15AF44  Practical Course in Mechanics 2  0  2  0  M  3  3  Dr. Zaránd Gergely Attila  
Problem solving class accompanying Mechanics 2. Literature: – H. Goldstein: Classical Mechanics, AddisonWesley. – J.R. Taylor, Classical Mechanics, University Science Books. 

BMETE15AF36  Quantum Mechanics 2  2  0  0  E  2  2  Dr. Szunyogh László  
This course conveys advanced knowledge on Quantum Mechanics according to the following topics: The WKB approach, quasiclassical quantization. Scattering theory, scattering amplitude and cross section, Green functions, LippmannSchwinger equation, Born series, method of partial waves. Motion in electromagnetic field, AharonovBohm effect, Landau levels. Time evolution and pictures in Quantum Mechanics (Schrödinger, Heisenberg and Dirac pictures). Adiabatic motion and Berry phase. Relativistic Quantum Mechanics, KleinGordon equation, Dirac equation, continuity equation, Lorentz invariance, spin and total angular momentum. Free electron and positron. Nonrelativistic limit, spinorbit interaction. Literature: – Franz Schwabl: Quantummechanics, Springer 1990 – Albert Messiah: Quantummechanics, Vol. 12, North Holland, 1986 

BMETE15AF43  Practical Course in Quantum Mechanics 2  0  2  0  M  3  3  Dr. Szunyogh László  
Problem solving course related to the topics of the course Quantum Mechanics 2. Literature: – Siegfried Flügge: Practical Quantum Mechanics, Springer, 1994 

BMETE15AF34  Electrodynamics 2  2  0  0  E  2  2  Dr. Takács Gábor  
Electrostatics: Solving Laplace's equation in spherical and cylindrical coordinates. Grounded sphere in external field, electric field near a sharp cone. Multipole expansion in spherical harmonics. – Magnetic and quasistatic fields: magnetic scalar potential, solution methods in nonlinear materials. – Electromagnetic waves in vacuum and matter. Microscopic model for polarizability. Dispersion, plasma frequency, KramersKroing relations. – Wave guides, resonant cavity. Losses, quality factor. – Radiation field of oscillating charges. Electric dipole and quadrupole, magnetic dipole radiations. – Scattering of electromagnetic waves, cross section. Scattering on solids and gases. – LienardWiechert potential of moving charge, field strength, radiated power, angular distribution, spectrum. Synchrotron radiation. Cherenkov and transitional radiations. – Elements of relativistic electrodynamics. Literature: – David J. Griffiths: Introduction to Electrodynamics (Pearson) – John D. Jackson: Classical Electrodynamics (Wiley) 

BMETE15AF42  Practical Course in Electrodynamics 2  0  2  0  M  3  3  Dr. Takács Gábor  
Problem solving class accompanying Electrodynamics 2. Literature: – David J. Griffiths: Introduction to Electrodynamics (Pearson) – John D. Jackson: Classical Electrodynamics (Wiley) 

BMEGEÁTAMF4  Fluid Mechanics  2  0  0  M  2  2  Dr. Kristóf Gergely  
Properties of Fluids, Newton’s law of viscosity. Cavitation, description of fluid flow, force fields. Characterisation and visualisation of flows, free (irrotational) vortex, continuity theorem, hydrostatics. Fluid acceleration, Eulerequation, Bernoulliequation, total, static, and dynamic pressure. Basic examples for the Bernoulliequation: flow rate measurement using a Venturitube, measurement of pressure, velocity, and volume flow rate. Syphon, rotating pipe pump, unsteady discharge from a vessel. Euler equation in the streamline coordinate system, vortex theorem, floating bodies. Momentum theorem and its applications, jet contraction, BordaCarnot expansion, Pelton turbine. KuttaJoukowsky theorem, Allievi theorem, Euler turbine equation, propeller, wind turbine. Nonnewtonian fluids, momentum equation, NavierStokes equation, laminar flow in a pipe, laminar / turbulent flow. Hydraulics, dimension analysis, Bernoulliequation with losses, friction factor, losses in pipe components. Bernoulli equation for compressible fluids, similarity of flows, boundary layer, mixing length model of turbulence, flat plate boundary layer. Energy equation, speed of sound, wave propagation in gases. Discharge from a vessel, use of a Laval nozzle and its simplified calculation. Force acting on solid bodies. Literature: – Frank M. White: Fluid Mechanics, Mc Graw Hill, 2011, ISBN 9780073529349 – https://docs.google.com/file/d/0B9JtpWUzKcwkNC11OC1mQ0NzZjg/edit?pli=1 

BMETE15AF39  Classical and Quantum Chaos  2  0  0  E  2  2  Dr. Varga Imre  
Hamiltonian formalism, integrability in general, examples in physics for chaotic behavior in case of continuous and discrete dynamics; Continuous, nonautonomous differential equations; Anharmonic, dissipative oscillator; Mappings, Poincaremapping; Periodically excited systems; Billiards. For some of these cases: application of techniques introduced for the analysis of chaos: Lyapunov exponent, invariant measures; FrobeniusPerron equation. Stability analysis; Bifurcations, attractors, strange attractors; Kolmogoroventropy; KAMtheorem; Chaotic dynamics and its traces in quantum mechanics. Semiclassical quantization, WKB method; Gutzwillertrace formula; Spectral statistics, Loschmidtecho. Literature: – E. Ott: Chaos in Dynamical Systems, Cambridge Univ. Press. – A.M. Ozorio de Almeida: Hamiltonian Systems: Chaos and Quantisation, Cambridge Univ. Press. – H.J. Stöckmann: Quantum Chaos, An Introduction, Cambridge Univ. Press. 

BMETE15AF38  Theory of Relativity  2  0  0  E  2  2  Dr. Lévay Péter Pál  
Minkowski spacetime, four vectors. Lorentz and Poincaré groups. Time dilation, Lorentz contraction, relativity of simultaneity. Addition of velocity, rapidity. Causality, Zeeman's theorem. Proper time, four velocity, four acceleration. Relativistic dynamics. Hyperbolic motion. Principle of Equivalence. Geodesic hypothesis. Principle of covariance. Local systems of inertia. Riemann and pseudo Riemann geometry, Christoffel symbols. Geodesics. Covariant derivative, parallel transport. The Newtonian limit. Connection between the metric tensor and the gravitational potential. Geodesics from a variational principle. Riemann tensor and its properties. Riemann tensor and its connection with parallel transport. Geodesic deviation. Ricci tensor, scalar curvature. Bianchi identity, Einstein tensor. Energymomentum tensor. Conservation of energy and momentum. Einstein's equation, EinsteinHilbert action, cosmological term. Schwarzschild's solution. The perihelium precession of Mercury. Literature: – G. Naber: The Geometry of Minkowski SpaceTime, Springer 1992. – S.M. Carroll: An Introduction to General relativity Spacetime and Geometry, Addison Wesley 2004. 
Computer programming
Code  Subject  L  P  B  R  C  I  II  III  IV  V  VI  Responsible 

BMETE11AF37  Computer Controlled Measurements  0  0  2  M  2  2  Dr. Halbritter András Ernő  
The participants gain experience in computer controlled measurements and in the programming of scientific instruments and data acquisition system. To this end the following topics are covered: communication with the instruments via serial, GPIB, and USB ports. Programming of data acqusitin cards. Programming of complex measurement control platforms, plotting and saving the data, programming of timelines, in situ data analysis. The course consists of 4 hour long computer laboratory exercises every second week. In the first part of the semester fundamental programming skills are obtained through simple example programs. In the second part the participants individually program complex measurement control and data analysis platforms, like nonlinear curve fitting by Monte Carlo method, full computer control of a digital multimeter, digital oscilloscope program using a data acquisition card. 

BMETE12AF24  The Fundaments and Applications of Finite Element Modeling  0  0  2  M  2  2  Dr. Beleznai Szabolcs  
Summary of theoretical and practical aspects of the finite element method to solve practical physical problems. The most important subjects are: numerical solution of the most common physical problems described by ordinary and partial differential equations: PoissonLaplace equation, Heat transfer, Particle convection, Diffusion, Helmholtz equation, Wave equation, Eigenvalue problems, Complex problems.


BMETE11AF36  Computer Solution of Technical and Physical Problems  0  0  2  M  2  2  Dr. Varga Gábor  
In the frame of this course several areas of technical and physical problems (one and many particle problems, Poisson equation, fluid flow, sheet deformation, heat transport, wave equation, Schrödinger equation) are investigated. Investigated problems can be described by ordinary or partial differential equations. For every problem computer program is written. During the computer implementation not only the physical models but the needed numerical methods are analyzed. MATLAB program language is applied as a programming tool. The course is complemented at beginning of the semester with optional MATLAB training. Literature: – G.D. Smith: Numerical Solution of Partial Differential Equations, 1979. – MATLAB documentation (http://www.mathworks.com/help/matlab/). 

BMETE80AF26  Monte Carlo Methods  2  1  0  M  3  3  Dr. Fehér Sándor  
Random number generation. Experimental and algorithmic methods. Generation of uniformly distributed pseudorandom numbers on computers. Multiplicative, congruential and other algorithms. Statistical tests of random number series. Randomness, independency. Chisquare test. One and twodimensional frequency tests, digit test, gap test, poker test, run test, test of subseries. Sampling discrete random variables by Monte Carlo method. Techniques for acceleration of sampling. Sampling continuous random variables. Methods for sampling onedimensional density functions. Inverse cumulative function method, acceptancerejection algorithm, composition method, table lookup techniques. Application of Monte Carlo methods for particle transport simulation. Methods for choosing uniformly a random point from the surface of a sphere. Sphere slicing, cube rejection and Marsaglia’s algorithm. Free flight sampling in homogeneous, regionally homogeneous and inhomogeneous media. Woodcock’s method. Analog and nonanalog simulation of particle transport. Variance reduction techniques. Statistical weight, implicit capture, spatial importance, biasing, splitting, Russian roulette. Monte Carlo integration. Interpolation of multivariable functions using Monte Carlo method.

Optics
Code  Subject  L  P  B  R  C  I  II  III  IV  V  VI  Responsible 

BMETE12AF28  Spectroscopy  2  0  0  E  2  2  Dr. Lenk Sándor  
Classification of spectroscopic techniques: gamma, Xray, UVVISNIRFIR, radiofrequency, NMR, particle and mass spectroscopy. Optical spectroscopy: emission, absorption, fluorescence, Raman, multiphoton, laser. Optical spectrometers: prism, grating, Fourier, FabryPerot, acoustooptic, photoacoustic. Non optical spectrometers: gamma spectrometer, Xray spectrometer, nuclear magnetic resonance, mass spectrometers. Application of spectrometers in metrology. Literature: – W. Demtröder: Laser spectroscopy, Springer. – N.V. Tkachenko: Optical Spectroscopy: Methods and Instrumentations, Elsevier. – M.A. Linne: Spectroscopic Measurement: An Introduction to the Fundamentals, Elsevier. 

BMETE12AF07  Laser Technique  2  0  0  M  2  2  Dr. Ujhelyi Ferenc  
Light and material interaction, spontaneous emission, absorption, stimulated emission. Coherent optical amplifier. Pumping methods in practice. Saturation of gain. Properties of materials with homogeneous and inhomogeneous gain. Continous and pulsed laser operation, gain and phase condition, Feedback system, properties of the optical resonator, definition of the modes. Gain and Qswithing, mode locking. Properties of the laser light, bandwidth, coherence, propagation, brightness. Types of laser: solid state, semiconductor, gas, fluid, and others. Laser applications: industrial, medical, data communication, and metrology. Literature: – Svelto: Principles of lasers. 

BMETE12AF09  Microscopy  2  0  0  M  2  2  Dr. Maák Pál  
The scope of the course is to make the microscopic techniques and approaches familiar to the students as well as to get insight into the development of microscopy from classical to the newest technical achievements. Detailed topics: History of the microscope, development of the combined microscope. Classification of the old and new microscopy techniques. Geometric optical basis of the optical microscope. Abbe theory of image formation. Estimation of the lateral resolution based on diffraction theory. Buildup of the compound microscope, roles of the imaging and illuminating systems. Specific properties of the objective and ocular. Role of the immersion fluid. Errors and aberrations in imaging, depth of field, brightness. Methods of optical design to eliminate aberrations. Illumination techniques: bright field, oblique, dark field illuminations. Role of diaphragms. Special condensers. Role of sample preparation. Phase contrast and polarization microscopy: physical optical background, diffraction theory and practical realization. Use of the microscope in the practice – laboratory demonstration. Theoretical and practical limitations of the increase of the lateral resolution: techniques to overcome the fundamental diffraction limit. Techniques for image registration. Analysis of the registered images, image processing based on optical and electronic methods. Fluorescence microscopy. Overview of new research directions in microscopy: confocal, Xray, UV, differential interference contrast, electron, atomic force, tunneling. Confocal and multiphoton microscopes: operation principles in detail, parameters, experimental results. Scanning and transmission electron microscopes: theory, parameters, applications. Practical laboratory work on scanning electron microscope, sample preparation, limitations. Discussion of tunneling and atomic force microscopes, parameters, practical tutorial.

Materials science
Code  Subject  L  P  B  R  C  I  II  III  IV  V  VI  Responsible 

BMETE12AF10  Foundations of Biophysics  2  0  0  M  2  2  Dr. Barócsi Attila  
The aim of the course is to familiarize students with the fundamental physical properties that govern biological (living) systems having higher complexity to inert physical systems and illustrate the physical modeling of such biological systems. Unlike medical courses, the present one aims at providing extensive biological information to the topics of physics with the prerequisite that students are familiar with the basics of classical and modern physics. Detailed topics: Biological basics of biophysics (criteria of life, the cell, descriptive genetics). Material structure and its relation to function (bond types, the water, biological macromolecules, molecular basics of the genetic code). Interaction of biophysical systems with radiation (light absorption in macromolecules, biological impact of optical and Xray radiations, radiobiology). Thermodynamics of biological processes (thermal homeostasis, irreversible thermodynamics, cellular respiration and photosynthesis). Metabolism and transport (transport phenomena, drift, diffusion and osmosis). Biological membranes (ion transport, electric phenomena, stimulated processes, propagation of stimulus, the patchclamp measuring technique). Biophysics of sensory organs (receptors): vision and hearing. Collective phenomena (trafficlike motion, ASEP models, fundamental mechanisms of molecular motors).


BMETE12AF25  Fundaments and Applications of Materials Science  2  0  0  E  2  2  Dr. Réti Ferenc  
The aim of the subject is to give a basic knowledge in the modern materials science and its use in different areas of physics and engineering. Topics: Materials science and engineering. Modern materials, requirements in their use. Role of primary and secondary bonding in properties of materials. Importance of thermal processes, thermodynamics, thermochemistry, Hess principle, BornHaber cycle. Chemical potential, equilibrium constant. Reaction rate equations. Arrhenius and Eyring equation. Importance of crystal imperfections e.g. in electrical and mechanical properties. Equilibrium concentration of crystal imperfections. Sensors in engineering. Principles, physical and chemical sensors. Pressure sensors, thermometers, strain gauges, magnetic sensors. Nondestructive testing. Flaw detection by ultrasound, Xray. Magnetic tests. Practical examples. Alternative energy sources and energy carriers; contradictions of the field. Hydrogen economy, bioethanol. Fuel cells as continuous power sources.


BMETE12AF08  Microtechnology and Nanotechnology  2  0  0  M  2  2  Dr. Kiss Gábor  
Definition of microtechnology, nanotechnology and molecular nanotechnology, their comparison and interrelation. Conditions of the technology. Micro and nanophysics. Thin layer deposition methods: physical (vacuum evaporation, laser ablation evaporation, molecular beam epitaxy, sputtering). Doping (diffusion, ion implantation). Litography (photo, Xray, electron beam, ion beam). Layer removing technologies: wet „chemical” etching, dry etching (plasma, ion beam). Layer characterisation methods: Xray diffraction, transmission electron microscopy, scanning electron microscopy, secondary ion mass spectrometry, Xray photoelectron spectroscopy, Auger electron microscopy, scanning tunneling microscopy, atomic force microscopy. Conventional electronic devices: bipolar transistor, field effect transistor. Thick layer technology: screen printing, burning, thick layer pastes. Nanometer devices: single electron devices, resonant tunnel effect devices, microelectromechanical systems, sensors, image detectors, displays.

Nuclear technology
Code  Subject  L  P  B  R  C  I  II  III  IV  V  VI  Responsible 

BMETE80MD00  Nuclear Physics  3  1  0  E  5  5  Dr. Sükösd Csaba  
Stability of the nucleus, mass defect. Semiempirical binding energy formula. Types and basic theory of radioactive decays. Nuclear models: Fermigas, Shellmodel, Basics of collective model. Nuclear forces. Nuclear reactions. Cross sections and their two additivities. Mechanism of fission and fusion. Main types and working principles of accelerators. 

BMETE80MD01  Nuclear Measurement Techniques  1  1  0  E  3  3  Dr. Szalóki Imre  
Electromagnetic and particle radiations, basic interactions between radiations and matter. General measuring properties of radiation detectors. Detectors: ionization chambers, proportional counters, GM counters, scintillation detectors, semiconductor and solid state detectors. Special detectors: detection of neutrons, detectors for dosimetry, TLD, particle detectors. Detection of gamma, alpha, beta and Xrays, nuclear spectrometers. Counting statistics and error prediction. Evaluation of gamma and Xray spectra. Electronics of nuclear spectrometers. Nuclear accelerators. 

BMETE80MD05  Nuclear Safety  2  0  0  E  2  2  Dr. Czifrus Szabolcs  
Introduction into nuclear safety – basic terms, safety functions, physical barriers, defence in depth. Plant states, design basis of a nuclear plant. Safety of nuclear plants – safety systems, comparison of different reactor types. Deterministic analysis – methods, postulated initiating events. Probabilistic analysis – methods. Level 1, 2, and 3 PSA. Application of PSA in nuclear design. Design basis accidents – course of an LB LOCA accident in PWR reactors. Severe Accidents – typical phenomena during SA. International Nuclear Event Scale (INES) – classification of events. Exercise: group work for classification. Lessons learned from incidents, accidents. The Fukushima accident. National and international regulation of nuclear safety. Standards, limits. 

BMETE80MD07  Radioactive Waste Management  2  0  0  E  2  2  Dr. Zagyvai Péter  
Overview of dose concept, hazardous effects of ionizing radiations and elements of health physics regulations. Definitions of radioactive wastes. International guidance and national regulations on radioactive waste management. Classifications of radioactive wastes, role and significance of radioactive wastes in the system of radiation protection. Classification and radioactive waste according to their generation. Characteristic components of waste streams, radiation protection and technological properties of representative waste components. Nuclear analytical procedures applied for waste qualification and quantitation. Operations of radioactive waste processing. Collection, classification, storage, volume reduction, conditioning, transport. Methods for qualification of processed wastes. Examples of compound procedures for waste processing and management. Longterm interim storage and final disposal of radioactive wastes. Qualification of disposal, radiotoxicity. Special waste processing methods of closed fuel cycle systems: reprocessing, transmutation. 

BMETE80MD02  Plasma Physics  3  1  0  E  4  4  Dr. Pokol Gergő  
General introduction to plasma physics. Energy generation with fusion reactors, Lawson criterion, parameters of fusion plasmas. Inertial fusion. Collisionless motion of charged particles in magnetic field. Thermodynamic equilibrium, ionization and radiative processes in the plasma. Magnetic confinement: configurations. Particle collisions in plasma, transport processes. Plasma theory: kinetic description, fluid description, MHD. Equilibrium and instabilities in magnetically confined plasma, plasma waves. Laboratory plasmas: breakdown, plasma heating, plasmawall interaction. Plasma diagnostics, measurement methods. Recent results, achievements in fusion plasma confinement. 
Medical physics
Code  Subject  L  P  B  R  C  I  II  III  IV  V  VI  Responsible 

BMETE80AF17  Medical Imaging Systems  2  0  0  M  2  2  Dr. Légrády Dávid  
A kép fogalma, matematikai leírása. Képjellemzők matematikai és fizikai tárgyalása: kontraszt, geometriai felbontás, zaj, detektálási kvantum hatásfok, modulációs transzfer függvény, jelzaj viszony. Képalkotási módszerek: transzmissziós, emissziós és gerjesztett technikák. Képalkotás gamma fotonokkal: gamma sugárforrások. Projekciós radiográfia: a képalkotás szakaszai és matematikai modellezése. Hagyományos filmbázisú és elektronikus rendszerek paraméterei. Transzmissziós tomográfia: vetületek mérése és rekonstrukciós (analitikusalgebrai és modell bázisú) algoritmusok áttekintése. A képjellemző paramétereket befolyásoló tényezők. Emissziós tomográfia: SPECT és PET. Pásztázás és emissziós képrekonstrukciós algoritmusok áttekintése. A képjellemző paramétereket befolyásoló tényezők. Nukleáris medicina. A Mágneses Rezonancia (MRI) képalkotás fizikája és technikai eszközei. Ultrahang képalkotó eljárási módszerek. A képalkotó eljárások összehasonlító komplex értékelése.

Elective courses
(any course of the university, 9 credits)
Code  Subject  L  P  B  R  C  I  II  III  IV  V  VI  Responsible 

Total  Elective courses (any course of the university, 9 credits)  6  3 
Supervised research
(10 credits)
Code  Subject  L  P  B  R  C  I  II  III  IV  V  VI  Responsible 

BMETE15AF11  BSc Thesis Project  0  0  10  M  10  10  Dr. Szunyogh László  
This course serves for the evalutation of the diploma work and preparing the diploma theses. 