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.

Welearn the key aspect of quantum mechanics – how to predict the future with Schrödinger’s equation. We learn the general recipe for solving time-dependent problems by diagonalizing the Hamiltonian to find the energy eigenvalues and eigenvectors.

An overview of the foundational aspects of complex dynamic systems is provided. This chapter serves as a reference for applications of complex dynamic systems concepts and methods to concrete topics. This chapter argues that complex dynamic systems is an ideal candidate for realizing a process approach in psychology, without it necessarily being a monolithic framework.