26 lines
1.3 KiB
TeX
26 lines
1.3 KiB
TeX
\documentclass[../schrodinger_simulation.tex]{subfiles}
|
|
|
|
\begin{document}
|
|
\section{Conclusion}\label{sec:conclusion}
|
|
% Rewrite this section to differ from the 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, and derived 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 $p_{\ivec, \jvec}^{n} = u_{\ivec, \jvec}^{n*} u_{\ivec, \jvec}^{n}$,
|
|
we created colormap plots at time steps $t = [0, 0.001, 0.002]$. Since we are working
|
|
with complex numbers, we included separate plots for each time step of Re$(u_{\ivec, \jvec})$
|
|
and Im$(u_{\ivec, \jvec})$.
|
|
% We observed something...
|
|
|
|
In addition, we determined the normalized particle detection probability $p(y \ | \ x=0.8, t=0.002)$,
|
|
for single-, double- and triple-slit.
|
|
% We observed something here as well...
|
|
|
|
\end{document}
|