This project looks into the time evolution of a wave function within a two-dimensional quantum well. We start by solving the time-dependent Schrödinger equation for stationary states in a quantum well. Next, we express any wave function as a linear combination of stationary states, allowing us to understand their time evolution. Two methods are presented: one relies on decomposing the wave function into a basis of stationary states and the other on discretisation of the time-dependent Schrödinger equation, incorporating three-point formulas for derivatives. These approaches necessitate confronting intricate boundary conditions and require maintaining energy conservation for numerical accuracy. We further demonstrate the methods using a wave packet, revealing fundamental phenomena in quantum physics. Our results demonstrate the utility of these methods in understanding quantum systems, despite the challenges in determining stationary states for a given potential. This study enhances our comprehension of the dynamics of quantum states in constrained systems, essential for fields like quantum computing and nanotechnology.

This chapter focuses on the project of finding the potential for a given distribution of charges in a two-dimensional system, which does not possess any symmetrical properties, an extension of the cylindrical potential problem discussed in the previous chapter. Using a method of minimising a functional, specifically the Gauss–Seidel method of iterative minimisation, the Poisson’s equation is adjusted to a 2D case, neglecting one partial derivative in Cartesian coordinates. We subsequently derive a discretised form of the functional, leading to a multi-variable function, following which the problem can be solved using the Gauss–Seidel iterative method. The numerical method discussed here is the finite elements method (FEM), with an emphasis on the need for a specific sequence for updating values to optimise computation efficiency. The discussion sheds light on the importance of the uniqueness of solutions in electrostatic systems, thereby exploring a fundamental question in electrostatics. The concluding part of the chapter provides an outline of a numerical algorithm for the problem, suggesting potential modifications and points for further exploration.

Modern computational fluid dynamics (CFD) methods are described. CFD simulations of whole compressor stages using the Reynolds-averaged Navier–Stokes equations (RANS), with a steady-state rotor–stator interaction model between the blade rows and a two-equation turbulence model, are the basis of aerodynamic calculations for the design of radial compressors. More detailed simulations using unsteady RANS (URANS) equations with unsteady rotor–stator interaction models are now used in a research environment. The source of errors and the importance of validating the CFD methods are described. The prediction of the turbulent flow is one of the main error sources in CFD simulations, and an overview is given of turbulence models. Up-to-date guidelines are provided on the use of CFD for the design and development of components in radial turbocompressors. A practical description of the CFD process is given, including the technologies used to support the calculations, such as geometry generation, mesh generation and postprocessing.