We apply geometrical methods to the quantum brachistochrone problem, whose equation of motion was recently written as a limit case of a well known Lax pair [MC24]. First we discuss the theories of...Show moreWe apply geometrical methods to the quantum brachistochrone problem, whose equation of motion was recently written as a limit case of a well known Lax pair [MC24]. First we discuss the theories of Lie groups, Poisson geometry and Riemann surfaces in that order. This knowledge is applied to the study of Lax equations and their integrability. The final chapter applies these methods to the quantum brachistochrone problem for the smallest case su(2).Show less
β−decay is associated with emitting or absorbing an electron (anti)neutrino. By measuring the energies of the other particles involved, the energy of the neutrino can be deduced and with that its...Show moreβ−decay is associated with emitting or absorbing an electron (anti)neutrino. By measuring the energies of the other particles involved, the energy of the neutrino can be deduced and with that its mass. It is proposed to isolate an isotope in a solid-state structure, e.g. graphene or fullerene and to study the detection spectrum from β−decay inside the isotope’s nucleus [E. Baracchini et al., arXiv:1808.01892]. The energy uncertainty due to the zero-point motion of the most popular isotope candidate, Tritium, has been proved to be too great for a feasible experiment [Y. Cheipesh, V. Cheianov, A. Boyarsky, Phys. Rev. D 104, 116004, (2021)]. There is, however, no theoretical objection to heavier isotopes as β−emitters in the experiment. In this thesis, the energy uncertainty (or equivalently the lifetime) of heavier isotopes due to the coupling to graphene is estimated. Also, the influence of the lifetime on the spectrum is studied.Show less
This paper delves in a dilute four boson collision system in one dimension in the zero temperature limit. These bosons experience attraction in the F = 0 channel and repulsion in the F = 2 channel...Show moreThis paper delves in a dilute four boson collision system in one dimension in the zero temperature limit. These bosons experience attraction in the F = 0 channel and repulsion in the F = 2 channel and will thus form dimers, pairs of bosons. To know if such a system is stable one must compute if the dimers repel or attract each other. This can be found by solving a four body collision. Such a system is then solved using the Skorniakov and Ter-Martirosian (STM) method to resolve the system into an integral equation. The goal of this study is to derive this equation. Further study can numerically solve this integral equation to find the scattering amplitude of the collision. From this one can then find the stability conditions of the Boson condensate.Show less
An adiabatic cycle around a degeneracy point in the parameter space of a one-dimensional band insulator is known to result in an integer valued particle transport in the thermodynamic limit. In...Show moreAn adiabatic cycle around a degeneracy point in the parameter space of a one-dimensional band insulator is known to result in an integer valued particle transport in the thermodynamic limit. In this thesis, we will focus on the Rice-Mele insulator and discuss a protocol which creates such an integer valued particle transport outside of the adiabatic limit. It is shown that in this protocol, there are certain conditions where the corrections due to a finite-sized system become negligible. Furthermore, we investigate the effects of the addition of a harmonic potential field and show that the corrections to a quantized particle transport scale with the square of the potential curvature. We show using numerical calculations that the parameters of the protocol can be chosen in such a way that these corrections vanish completely. Lastly, it is shown that the change of center of mass of the density distribution is not a precise method for the measurements of the particle transport.Show less
