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 numerical simulation of light diffraction by single or multiple slits, which serves to illustrate key principles of wave physics and interference. Students will become acquainted with numerical differentiation and quadrature procedures, particularly in relation to grid parameter convergence. The physics background emphasises wave physics elements, such as the superposition principle and phase difference, as well as their practical applications in real systems. Concepts such as optical paths and coherence are addressed. To understand diffraction phenomena, the Huygens principle is introduced, leading to the diffraction integral formulation for infinite slits. The chapter then explores numerical methods based on local approximations of functions, such as the two-, three-, and five-point schemes for derivatives. This study culminates in the presentation of quadrature schemes, the application of power series expansions for numerical differentiation, and the Simpson algorithm for accurate numerical integration.

Unlike in Chapter 5, this project aims at finding a real mass density distribution of a hydrogen star of given mass. For that purpose an equilibrium condition for the gravitational and pressure-induced forces acting on a mass element is utilised. Using the integral form of Gauss’s law and the equation of state, we establish an integro-differential equation describing the mass density distribution. To numerically solve the integro-differential equation, we adapt the Adams–Bashforth method and implement a linear extrapolation based on known data points. This approach involves modelling the star as a gas under pressure using an exponential form for the equation of state, which helps in avoiding gravitational collapse. The equation of state is derived based on density functional theory data. We also discuss the constraints of this model and the significance of the parameters within it. The chapter concludes by suggesting potential numerical experiments to examine the influence of these parameters and their physical interpretation. This analysis aims to provide a more comprehensive understanding of stellar structure and the behaviour of mass density distribution within stars.

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.