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A new method for the solution of the Schrödinger equation
Kocák, Jakub ; Uhlík, Filip (advisor) ; Demel, Ondřej (referee)
Title: A new method for the solution of the Schrödinger equation Author: Jakub Kocák Department: Department of Physical and Macromolecular Chemistry Supervisor: doc. RNDr. Filip Uhlík, Ph.D. Abstract: In this thesis we study method for the solution of time-independent Schrö- dinger equation for ground state. The wave function, interpreted as probability density, is represented by samples. In each iteration we applied approximant of imaginary time propagator. Acting of the operator is implemented by Monte Carlo simulation. Part of the thesis is dedicated to methods of energy calculation from samples of wave function: method based on estimation of value of wave function, method of convolution with heat kernel, method of averaged energy weighed by wave function and exponential de- cay method. The method for the solution was used to find ground state and energy for 6-dimensional harmonic oscillator, anharmonic 3-dimensional octic oscillator and hydrogen atom. Keywords: imaginary time propagation, Monte Carlo method, variational principle, ground state 1
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Quantum variational Monte Carlo method
Kocák, Jakub ; Uhlík, Filip (advisor) ; Srnec, Martin (referee)
Title: Quantum Variational Monte Carlo method Author: Jakub Kocák Department: Department of Physical and Macromolecular Chemistry Supervisor: RNDr. Filip Uhlík, Ph.D. Abstract: In this thesis, we study variational Monte Carlo method in quantum- mechanical systems. We analysed choice of trial wave function and afterwards we optimized selected function for helium singlet and triplet state. In the first chapter, we discuss basic notions of quantum mechanics and general properties of wave function and properties of ground state of the system. In the second chapter, we consider computational algorithms used to calculate integrals, to estimate errors and for optimization. In the third chapter, we present results of optimization and properties of optimized trial wave function. Keywords: variational principle, Monte Carlo method, ground state, helium 1
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Study of resonance and threshold effects on simple two-channel model
Kocák, Jakub ; Čížek, Martin (advisor) ; Kolorenč, Přemysl (referee)
In this thesis we study simple one-dimensional two-channel scattering model where pointlike coupling between channels is provided by the delta potential. The scattering task can be completely solved analytically. The solution of the Lippmann-Schwinger equation leads to improper scattering eigenvectors, consequently to scattering S matrix elements and eigenphases. We study how the setting of parameters affects threshold and resonant behaviour (presence, position, width) and the mutual relationship between resonances and poles of the S matrix in complex k-plane. Then we apply projection- operator formalism to model with resonance and the on-shell T matrix is separated into the orthogonal, direct and resonant term. We discuss how choice of subspace of quasi-bound states effects the separation. Powered by TCPDF (www.tcpdf.org)
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