The Schrödinger equation, introduced in 1925, is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is a key result of wave mechanics, one of the two main formulations of quantum mechanics (the other being matrix mechanics). The equation is named after Erwin Schrödinger, who developed it and won the Nobel Prize in Physics in 1933 for his work.

The time-dependent Schrödinger equation is given by:
\begin{equation}
    i\hbar \frac{\partial}{\partial t} \psi = \hat{H} \psi,
\end{equation}
where \(i\) is the imaginary unit, \(\hbar\) is the reduced Planck constant, and \(\psi\) is the wave function of the quantum system. The Hamiltonian operator,  \(\hat{H}\), represents the total energy of the system, i.e., the sum of kinetic and potential energies.

In \cref{sec:methods}, we describe the numerical methods used to solve the two-dimensional Schrödinger equation. In \cref{sec:results_and_discussion}, we present and discuss the results obtained from our numerical simulations. Finally, in \cref{sec:conclusion}, we summarize our findings and discuss potential future work.
