19 lines
1.1 KiB
TeX
19 lines
1.1 KiB
TeX
\documentclass[../schrodinger_simulation.tex]{subfiles}
|
|
|
|
\begin{document}
|
|
\begin{abstract}
|
|
We have simulated the two-dimensional time-dependent Schrödinger equation, to study
|
|
variations of the double-slit experiment. To solve the partial differential equations
|
|
we have applied the Crank-Nicolson scheme in 2+1 dimensions, to derive a discretized
|
|
equation. In addition, we have used Dirichlet boundary conditions to express the
|
|
equation in matrix form and solve it using the sparse matrix solver \verb|superlu|.
|
|
Our implementation, and choice of solver method, resulted in conserved total
|
|
probability $\sum_{\ivec, \jvec} p_{\ivec, \jvec}^{n}=1$ for both the single and
|
|
double slit setup. To illustrate the time evolution of the probability function,
|
|
we created colormap plots at time steps $t = [0, 0.001, 0.002]$. We also included
|
|
separate plots for each time step of Re$(u_{\ivec, \jvec})$ and Im$(u_{\ivec, \jvec})$.
|
|
In addition, we determined the normalized particle detection probability $p(y \ | \ x=0.8, t=0.002)$,
|
|
for single-, double- and triple-slit.
|
|
\end{abstract}
|
|
\end{document}
|