In this report, we have taken a look at three different types of trebuchets: the seesaw, the traction trebuchet or mangonel, and the counterweight trebuchet. We have created models to describe their motion by making use of Lagrangian mechanics and have compared them based on their maximum ranges. Moreover, we have also calculated a lower bound for the dimensions of themain beam needed to prevent it from breaking based on a static state. We have seen that the mangonel and trebuchet improved on the seesaw by increasing the range roughly sixfold which was expected. The maximum range of the trebuchet, however, slightly decreased compared to the mangonel. This was something we did not expect but think is due to the fact that the dimensions of the catapults were not optimised. Lastly, we found that the beam needed to be at least 18.7 cm to prevent it from breaking. However, for the real catapults, the beam would most likely need to be thicker to prevent it from breaking.

The aim of this thesis is to simulate quantum spin chains at a finite temperature. This has been achieved by using purification to write the incoherent thermal states of the spin chain as pure states, after which the matrix product state (MPS) formalism to simulate the spin chain. The influence of temperature on the spin chain has been tested by comparing chain magnetization to the strength of the external magnetic field at different temperatures. Real time evolution has also been applied under the assumption that the time evolution is adiabatic. The results have been compared to the analytical solution found by directly calculating the density matrix. The model is very accurate for imaginary time evolution, which is required to evolve the system to the desired temperature. During real time evolution the results oscillate slightly around the analytical solution, which remains constant. This oscillation is a consequence of model truncation and the Suzuki-Trotter approximation. The effect of allocating more numerical resources to limit this phenomenon is explored, and a method to increase reachable timescales by decreasing entanglement growth in the chain over time is tested and verified.

In this work, Clebsch-Gordan coefficients are studied from both a quantum mechanical and a mathematical perspective. In quantum mechanics, Clebsch-Gordan coefficients arise when two quantum systems with a certain angular momentum are combined and the total angular momentum is to be found. We start by discussing the relevant postulates of quantum mechanics. From there we move on to a discussion of quantum angular momentum, the combining of quantum systems, and Clebsch-Gordan coefficients. Finally, an algorithm for calculating the Clebsch-Gordan coefficients is developed. A firm grasp of linear algebra is required to understand this quantum mechanical perspective. The mathematical perspective is rooted in representation theory. The Clebsch-Gordan coefficients show the decomposition of the tensor product of irreducible representations of the three dimensional rotation group into the direct sum of irreducible representations. We explain the necessary theory to understand how this works. This includes a discussion of irreducible unitary representations, spherical harmonics, direct sum representations, and tensor product representations. Some experience with group theory is required to understand the mathematical perspective.

This thesis investigates intrinsic frustrated magnetic systems on an insulating square lattice. The frustrated systems were successfully engineered from magnetic iron atoms on top of an insulating square c(2x2) reconstruction of nitrogen on Cu3Au(100) (copper-gold) surface. Vertical atom manipulation was used to position the magnetic atoms atomically precise on the surface and methods were found to increase the success rate of this. For the investigation of frustration, a numerical simulator was made in Python for modelling frustrated spin structures based on already existing spin models and the following experimental measurements were taken: topography, spectroscopy and current-time traces. The simulations of a frustrated spin system show low energy excited states and the measurements show switching of the system between states.