Time-Dependent Quantum Wave Packet Evolution in Two-Dimensional Quantum Dots and Wells

This work presents a comprehensive study of the time-dependent evolution of quantum wave packets confined in two dimensional potential landscapes, specifically circular quantum dots, square quantum wells, and harmonic oscillator potentials. Employing the time-dependent Schr¨odinger equation (TDSE), we implement a Crank–Nicolson finite difference scheme to simulate the temporal dynamics of initially localized Gaussian wave packets. We analyze the effects of confinement geometry on quantum oscillations, tunneling, and the evolution of expectation values of position and momentum. Additionally, stationary eigenstates are extracted numerically to correlate with the observed time-dependent behavior. The inclusion of harmonic confinement offers insights into quantized orbits and coherent oscillations, closely related to quantum optics and quantum information systems. This study emphasizes the role of geometric and potential symmetry in shaping quantum evolution in nanoscale systems.