Theoretical and mathematical physics students – bachelor studies

##http://hdl.handle.net/20.500.11956/197397

##http://hdl.handle.net/20.500.11956/202153

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Work tasks: 1) become familiar with hydrodynamic code Athena++ 2) based on simulation setup of Pejcha & Gagnier (2023), try to reproduce simulations of Morris & Podsiadlowski (2009) 3) evaluate similarities and differences with respect to Morris & Podsiadlowski (2009) 4) evaluate the presence and importance of hydrodynamical instabilities The Bachelor thesis will have these components: 1) literature review of SN1987A, its progenitor, and stellar mergers 2) overview of numerical method used in Athena++ 3) results from simulations with Athena++ ##http://hdl.handle.net/20.500.11956/188514

##http://hdl.handle.net/20.500.11956/184011

The aim of this work will be a systematic investigation of the complete class of vacuum and pure radiation solutions to the Einstein-Gauss-Bonnet gravity theory with a cosmological constant, which belong to the Kundt family of geometries with vanishing optical scalars. Some particular exact spacetimes of this type have been previously found, but only without the off-diagonal “gyratonic” terms. We will attempt to derive the explicit form of the field equations, and then to proceed with their integration, paving the way for their subsequent analysis. ##http://hdl.handle.net/20.500.11956/183966

Nonlinear Electrodynamics (NE) was founded almost a century ago and used mainly as a solution to the problem of divergent field of a point charge in the vicinity of its position (see e.g. [1]) also giving satisfactory self-energy of charged particle. The best-known and frequently used form of the theory was introduced already in 1934 by Born and Infeld [2]. Nice overview with a lot of useful information was given in a book by Plebański [3]. There are different models of Nonlinear Electrodynamics • Born-Infeld (BI) theory • Hoffmann-Born-Infeld (HBI) theory • Logarithmic Lagrangian • Power Maxwell • many other models! ##http://hdl.handle.net/20.500.11956/182548

##http://hdl.handle.net/20.500.11956/182524

##http://hdl.handle.net/20.500.11956/184031

##http://hdl.handle.net/20.500.11956/184006

Can we combine Rainich conditions and conditions on a perfect fluid? The aim is to see whether it is possible to determine whether a given metric can be interpreted as due to an electromagnetic field plus a perfect fluid. ##http://hdl.handle.net/20.500.11956/182577

##http://hdl.handle.net/20.500.11956/184012

##http://hdl.handle.net/20.500.11956/175583

Resonances are quasi-bound states of quantum systems (e.g. electron+atom) which are known to enhance or suppress scattering and other transitions in the system. How does the resonant scattering change in the presence of a laser field which allows for absorption of one or more photons? The goal of this work is to understand on simple 1D models the properties of laser-assisted electron scattering on potentials supporting resonances. The student will implement and test a program for numerical solut...

##http://hdl.handle.net/20.500.11956/173939

##http://hdl.handle.net/20.500.11956/175581

##http://hdl.handle.net/20.500.11956/174233

Interatomic Coulombic decay (ICD) [1] is auto-ionization relaxation mechanism, bych which an excited atom can efficiently transfer its excess energy to a neighboring atom or molecule. ICD belongs to the class of Feshbach resonances, which are inherently of many-electron character. Theoretical study of such processes requires the use of advanced methods such as many-particle Green's functions. For quantitative computations such methods can be efficient and successful, however, their complexity of...

In quantum mechanics of collision processes the basic quantity that is obtained in experiments is the scattering cross section which is a measure of the strength of interaction between the incoming particle and the target. The scattering cross section depends on the energy of the incoming particle and can have a complicated shape depending on the processes that take place during the collision. For example the process of electron-atom collision can be strongly influenced by formation of so calle...

##http://hdl.handle.net/20.500.11956/128234

Not only due to non-vanishing cosmological constant it is natural to ask why the fundamental group of transformations in special relativity should be the Poincare group, which is a semidirect product of the Lorentz group and spacetime translations. De Sitter group SO(1,4) has equal dimension and contains standard boosts and rotations, but it has no commuting translational part. This has many implications, that might be studied. ##http://hdl.handle.net/20.500.11956/149148

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##http://hdl.handle.net/20.500.11956/149190

##http://hdl.handle.net/20.500.11956/109996

Prostudovat literaturu o teoriích gravitace, zvláště o teoriích Horndeskiho typu, zaměřit se na prostoročasy representující černé díry nebo jednoduché kosmologické modely. Provést přehled metod umožňujících formulaci zákonů zachování pro perturbace přesných řešení rovnic pole v těchto teoriích. Soustředit se na metody vycházející z přístupů Katze, Bičáka, Lynden-Bella, Petrova a dalších a s jejich pomocí se pokusit vyjádřit například hmotnost černých děr v asymptoticky anti-de Sitterových případně dalších "kosmologických" prostoročasech.

##http://hdl.handle.net/20.500.11956/149260

LISA is an approved space-based mission for detection of mHz gravitational waves to be launched in 2030s. It will detect a number of compact binary stars, which emit gravitational waves on essentially a single frequency. This frequency can be smeared out by finer effects, for example by frequency shifts and light travel time effects for a binary orbiting a distant body (star or a planet) around a common barycenter. These perturbations, if detected, would provide a new probe of certain phenomena ...

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Molecular vibrations are usually described with Born-Oppenheimer approximation. This leads to motion of nuclei in the mean-field of electrons which is called potential energy surface. Dymensionality of such dynamics depends on the number of degrees of freedom of vibrational motion i.e. on the size of the molecule, which is in principle unlimited. Dependence of numerical demands for solution of such system on dimensionality is exponential. There is only small number of analytically solvable cases...

##http://hdl.handle.net/20.500.11956/109937

##http://hdl.handle.net/20.500.11956/99995

Rattleback is a small boat-shaped toy, that is placed on table and spun. If the direction of the spin is "incorrect" the boat starts to vibrate and the direction of the spinning is gradually inverted (see "rattleback" on wikipedia and the videos shown there). Apparent violation of angular momentum conservation is consequence of interaction with the table. The goal of this thesis is to design a molecular system attached to surface with similar behavior ##http://hdl.handle.net/20.500.11956/101909

##http://hdl.handle.net/20.500.11956/99994

##http://hdl.handle.net/20.500.11956/101913

##http://hdl.handle.net/20.500.11956/101891

Molecular vibrations are usually described with Born-Oppenheimer approximation, which leads to motion of nuclei in the mean-field of electrons. This approximation must be generalized in the presence of more electronic states or electron scattering states. In the presence of more vibrational modes the effects associated with nontrivial topology of potential energy surface are possible. In this thesis the student will familiarize him(her)self with description of electron scattering from molecules ...

Molecular clusters are a type of material with different properties from those of an isolated molecule and a bulk solid. Contemporary experimental techniques (supersonic jet expansion) are capable of preparing clusters of water molecules of various sizes which can serve for example as a model for the immediate environment surrounding the subunits of DNA. However, the experiment is not able to control very well the geometries of the clusters that are prepared. The goal of this project would be to...

Gravitational lenses are typically modeled either by a continuous mass distribution (in the case of strong and weak lensing by galaxies or galaxy clusters), or by a set of discrete point masses (in the case of microlensing by stars or stellar systems). However, continuously modeled astrophysical systems do have their discrete component: lensing galaxy clusters contain individual galaxies; lensing galaxies contain dwarf galaxies, globular clusters, and at a finer scale individual stars. The objec...

Numerical determination of the scattering length for low-energy collisions of atoms, which is needed e.g. when modelling the Bose-Einstein condensate. ##http://hdl.handle.net/20.500.11956/90982

Variety of physical processes in quantum mechanics can be described as interaction of the so called discrete state with continuum. Resonant scattering can serve as a most typical example. Here, the metastable state, which is described by a square integrable wave function, corresponding to capture of the particle near the target, decays into the continuum of scattering states. When dealing with such a problem exponential decay of the discrete state is commonly assumed with the decay rate determin...

Numerical evaluation of phase shifts in the quantum scattering theory relies on the good representation of continuum. Recently, novel approach to its discretization has been proposed, which is based on the so-called wave-packet basis. One of the advantages of this basis is perfect control over the density of the resulting discrete levels in various energy intervals. Student should study the usefulness of the basis for different numerical approaches to scattering problem. ##http://hdl.handle.net/20.500.11956/86249

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The aim of this work is to analyze new vacuum solutions of Einstein’s gravitational field equations that allegedly represent accelerating (non-rotating) black holes with a NUT parameter. Although such solutions were mathematically found in 2006, their clear physical interpretation is still missing. The geometric and physical analysis will be based on explicit evaluation of the curvature scalars determining the algebraic type, finding suitable coordinates in which the acceleration can be set to zero, and elucidating the parameters related to the non-trivial NUT charge and acceleration. ##http://hdl.handle.net/20.500.11956/84463

Synthesis of molecules in the cosmic space is still not well understood. Conditions inside interstellar molecular clouds are so dramatically different from common laboratory conditions, that most of the processes that can take place there are not much explored. Quantum scattering theory describes such processes in principle exactly, but in practice the calculation is beyond computational power available even in the case of small molecules and we have to use clever approximations. A combination o...

Techniques of manipulation, trapping and cooling of atoms in optical laser field considerably developed during nineties. One of interesting applications is trapping of atoms in optical lattice, which is periodic potential formed by electromagnetic standing wave. Atoms in such lattice behave similarly like electrons or quasiparticles in solid state, but their properties could additionally be tuned with help of auxiliary electric or magnetic fields. Such configuration is a model system convenient ...

During electron collisions with diatomic molecules, the incoming electron is often captured into a resonant state producing a temporary molecular anion whose nuclear dynamics is different from that of a neutral molecule. As a result the molecule can be vibrationally excited, if the electron is again released, or it can dissociate, if the electron stays at one of the atoms. To construct a model describing the nuclear dynamics of these collisions, it is first necessary to calculate the potential-e...

Causal sets represent one of the approaches to quantum gravity which from the beginning takes discrete structure of spacetime as given. To the set of discrete events are then added their causal relations. Dynamics of such a model might be given e.g. by sequential growth, where we add subsequent events and with certain probability generate their causal relations with original set. ##http://hdl.handle.net/20.500.11956/61774

Prediction of the relative abundance of the elements (H,D,He,Li) in the early Universe was one of the important arguments for the validity of the big bang theory. This prediction was based on models of nuclear reactions in hot dense plasma. Similar calculations are done also for later stages of universe, when the mater cooled significantly, so that atoms and molecules can form. Cross section for tens of reactions [2] are needed for such models. Goal of this work is critical revision of quality of these calculations (some of them are quite old). Calculation of selected reactions can be repeated or improved. Search of overlooked missing reactions can be attempted. ##http://hdl.handle.net/20.500.11956/81917

##http://hdl.handle.net/20.500.11956/81850

The aim of the bachelor thesis is to attempt to replace the singularity of a particular cylindrically symmetric solution of Einstein-Maxwell equations with a surface source and based on its properties, give a physical interpretation of the spacetime. ##http://hdl.handle.net/20.500.11956/81841

Theoretical description of electron-molecule collisions still represents a great challenge. In addition to involving many particles, this theory is complicated by contribution of electron continuum and breakage of the Born-Oppenheimer approximation. All this problems are properly solved in the nonlocal resonance model theory [1,2]. This theory is used only for diatomic molecules. On the other hand, the equations of the nonlocal resonance model theory can easily be formulated for harmonic vibrati...

Numerically exactly solvable problem of particle diffraction from structured barrier and its interpretation using approximate methods of solution. Student learns the key concepts of multichannel quantum scattering theory and resonances. This experience may serve as a basis for further work in atomic and molecular physics, particle physics or theory of electron transport in solid state. ##http://hdl.handle.net/20.500.11956/69195

##http://hdl.handle.net/20.500.11956/69214

Integral equations of scattering theory (Fredholm integral equations of the second kind) can be solved analytically for so-called separable interactions. It is the goal of this work to find such solutions for systems electron-atom or electron-molecule and to carry out their analytical continuation into the complex energy plane. ##http://hdl.handle.net/20.500.11956/72563

##http://hdl.handle.net/20.500.11956/72566

##http://hdl.handle.net/20.500.11956/63958

Calculation of resonance widths or decay rates of metastable states of atoms, molecules, and their ions represents very difficult problem for many-body quantum physics. The difficulty stems from the necessity of description of the electrons released to continuum. In practice, the continuum wave functions are often approximated using square-integrable functions that are widely used to describe bound states but cannot represent the scattered electron. However, the resulting pseudo-spectrum can sti...

Analytically solvable model for two-channel scattering and its discussion in connection to theory of inelastic collisions in atomic and molecular physics. Student learns the key concepts of multichannel quantum scattering theory and resonances. This experience may serve as a basis for further work in atomic and molecular physics, particle physics or theory of electron transport in solid state. ##http://hdl.handle.net/20.500.11956/2200

##http://hdl.handle.net/20.500.11956/65828

##http://hdl.handle.net/20.500.11956/60254

The aim of this work is to demonstrate the best way in which dynamical electromagnetic and gravitational fields are represented in the Newtonian physics formalism, i.e., by using of the concept of force acting instantaneously at a distance. In particular, absence of the so called aberration will be explained. As a suitable explicit model, the Kinnersley class of exact solutions of Einstein’s equations which describe the field of arbitrarily accelerated sources will be used, including a possible cosmological constant. ##http://hdl.handle.net/20.500.11956/55415

Review work on geometry of anti-de Sitter spacetime and its isometric identifications. ##http://hdl.handle.net/20.500.11956/56477

##http://hdl.handle.net/20.500.11956/60252

##http://hdl.handle.net/20.500.11956/40308

The aim of the work is to summarize, compare, and further investigate exact spacetimes of Einstein’s equations of gravitational field with a null matter field that model an accelerated motion of localized objects – photon rockets. In particular, geometric and physical analysis of the Kinnersley and Bonnor classes of such spacetimes will be presented. ##http://hdl.handle.net/20.500.11956/43461

##http://hdl.handle.net/20.500.11956/42036

##http://hdl.handle.net/20.500.11956/42064

When a distant quasar is gravitationally lensed by an intervening galaxy, several observable images of the quasar are formed by the gravitational field of the galaxy. In case the light of any of the images passes directly through the stellar population of the lensing galaxy, it is additionally influenced by all stars lying along its path. Taking into account the non-negligible angular motion of the quasar with respect to the caustic structure formed by the gravitational field of these stars, it ...

##http://hdl.handle.net/20.500.11956/37134

The motion of mass points in a gravitational field can be described and calculated rather easily. However, in the case of bodies with internal degrees of freedom, the situation is substantially more involved. Two mass points connected by a massless strut the length of which changes in time can modify the parameters of their orbit even in the Newtonian gravitational field. There are also examples of simple, extended objects that modify their orbit as they move on curved surfaces. This suggests a ...

##http://hdl.handle.net/20.500.11956/50292

Study of extremely narrow resonances in potential scattering. Development of numerical methods for very narrow resonances, or resonances with very different time scales for decay in two different channels. ##http://hdl.handle.net/20.500.11956/38433

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##http://hdl.handle.net/20.500.11956/50228

##http://hdl.handle.net/20.500.11956/37104

The goal of this project is to study analytic properties of bound state energies and resonance parameters by means of simple potentials (sum of Dirac delta functions). It is in principle possible to determine resonance parameters from the knowledge of bound state energies by analytic continuation in the coupling constants. It is assumed that the analytic continuation will be carried out by the statistical Padé approximation (type III Padé). ##http://hdl.handle.net/20.500.11956/37088

Summery of numerical methods for solving the time-dependent Schrödinger equation and comparison of their efficiency on simple one- and two-dimensional problems. ##http://hdl.handle.net/20.500.11956/37086

When attempting to join quantum theory and general theory of relativity (while retaining covariance) it was realized that differing nature of time in these theories presents a nontrivial obstacle to their fusion. In general relativity, hamiltonian is a constraint and thus vanishes on physical solutions. Since hamiltonian generates the time evolution (both classical and quantum), it raised a question whether time is a truly fundamental and necessary quantity. ##http://hdl.handle.net/20.500.11956/37095

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##http://hdl.handle.net/20.500.11956/27875

The work will summarize the basic models of the universe based both on the Newtonian theory of gravity and, above all, on the Einstein equations of general relativity. Linear perturbations of these models will be analyzed and the role of various gauges (i.e., of suitable coordinates) for solving specific physical situations like a condensation of a spherical mass or rotational effects, possibly also gravitational waves, will be investigated. At present nonlinear perturbations of the second order are started to be studied; in future, it would be interesting to touch upon this mathematically very complicated topic as well.

Survey of classical systems showing non-collision singularities in the solution of equations of motion. ##http://hdl.handle.net/20.500.11956/27947

##http://hdl.handle.net/20.500.11956/27961

##http://hdl.handle.net/20.500.11956/27876

Theoretical description of the current conduction through metal-molecule-metal junction in the tight binding model and the formulation of the problem within quantum scattering theory. ##http://hdl.handle.net/20.500.11956/37090

##http://hdl.handle.net/20.500.11956/17763

##http://hdl.handle.net/20.500.11956/17762

In this work we study dynamics of systems in which their mass can change. We start from general Newton's law of motion, and in equations of motion we consider momentum change connected with the change of mass. In the first part of work we formulate and solve equations of motion for different examples of such systems. Examples are divided into three groups. The first group include Buquoys' problems where the mass is a function of coordinates. The second group includes conveyor-belt problems where...

##http://hdl.handle.net/20.500.11956/16531

##http://hdl.handle.net/20.500.11956/16458

##http://hdl.handle.net/20.500.11956/16454

The aim of this work is to study relations between some families of exact solutions of Einstein's gravitational field equations, in particular the conformally flat spacetimes with pure radiation (or analogous solutions of the algebraic type N) and a cosmological constant. Specifically, we present explicit transformations between the metric forms which have been recently found by Edgar and Ramos and solutions of this type found by Ozsváth, Robinson and Rózga that are known since 1980's. ##http://hdl.handle.net/20.500.11956/16464

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##http://hdl.handle.net/20.500.11956/7539

##http://hdl.handle.net/20.500.11956/55415

##http://hdl.handle.net/20.500.11956/6485

In this work we study geometric formulation of Hamiltonian mechanics. At the beginning we look at situation, when the Hamiltonian function is time-independent. In this case our geometric arena is the n-dimensional symplectic manifold. After that we suppose time-dependent Hamiltonian function and we use 2n+1 dimensional extended phase space. Evolution of the mechanical system is given by integral curves of the vector field, which is determined by Hamilton's canonical equations. We also construct canonical transformations in both cases. In the end of this work we show geometric interpretation of the Hamiton-Jacobi equation. ##http://hdl.handle.net/20.500.11956/6280

In the present work we study application of differential geometry to the Lagrangian formalism. In the first chapter we summarize the foundations of geometric formulation of Lagrangian mechanics, in particular we show the principal meaning of the tagent bundle of the configuration manifold and dynamical vector field which solves the Lagrange equations in their geometrical form. The Noether's theorem is also formulated and proved. The second chapter introduces other geometrical definitions related...

##http://hdl.handle.net/20.500.11956/7602

##http://hdl.handle.net/20.500.11956/6366