Report ready for proofreading and review.

latex/schrodinger_simulation.tex

@@ -102,7 +102,7 @@
 \subfile{sections/results}
 
 % Conclusion
-% \subfile{sections/conclusion}
+\subfile{sections/conclusion}
 
 % Notes
 % \subfile{draft}

latex/sections/appendices.tex

@@ -3,7 +3,8 @@
 \begin{document}
 \appendix
 \section{The Crank-Nicholson method}\label{ap:crank_nicolson}
-The Crank-Nicolson (CN) approach consider both the forward difference, an explicit scheme,
+The Crank-Nicolson (CN) approach consider both the forward difference, an explicit 
+scheme,
 \begin{equation*}
     \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} = F_{\ivec, \jvec}^{n} \ ,
 \end{equation*}
@@ -20,9 +21,11 @@ The parameter $\theta$ is introduced for a general approach, however, for CN $\t
     \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} &= \frac{1}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \\
 \end{align*}
 
-We need the first derivative in respect to both time and position, as well as the second derivative in respect to position. Taylor expanding will result in a discretized version, assume this is known...
+We need the first derivative in respect to both time and position, as well as the 
+second derivative in respect to position. Taylor expanding will result in a discretized 
+version.
 
-Schrödinger contain $i$ at the lhs, factor it as
+The Schrödinger equation contains $i$ on the left hand side, we rewrite it as
 \begin{align*}
     \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} &= \frac{1}{2i} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \\
     &= -\frac{i}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \ ,
@@ -44,8 +47,7 @@ the left hand side, and the terms containing $n$ time step on the right hand sid
     &= u_{\ivec, \jvec}^{n} + \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\
     & \quad + \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} 
 \end{align*}
-In addition, since we will use an equal step size $h$ in both $x$ and $y$ direction, 
-we can use 
+Since we will use an equal step size $h$ in both $x$ and $y$ direction, we can use 
 \begin{align*}
     \frac{i \Delta t}{2 \Delta h^{2}} = \frac{i \Delta t}{2 \Delta x^{2}} = \frac{i \Delta t}{2 \Delta y^{2}} \ ,
 \end{align*}
@@ -53,7 +55,7 @@ and define
 \begin{align*}
     r \equiv \frac{i \Delta t}{2 \Delta h^{2}}
 \end{align*}
-Now, the discretized Schrödinger equation can be written as 
+Now, the discretized Schrödinger equation can be rewritten as 
 \begin{align*}
     & u_{\ivec, \jvec}^{n+1} - r \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\
     & - r \big[ u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1} \big] + \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\
@@ -97,6 +99,8 @@ $(M-2)^{2} \times (M-2)^{2} = 9 \times 9$ given by
 
 
 \section{Figures}\label{ap:figures}
+We created colormap plots of the real and imaginary part of $u_{\ivec, \jvec}$, 
+in Figure \ref{fig:colormap_real_imag}.
 \begin{figure*}
     \centering
     \begin{subfigure}[b]{0.3\textwidth}
@@ -119,9 +123,7 @@ $(M-2)^{2} \times (M-2)^{2} = 9 \times 9$ given by
         \caption{Re$(u_{\ivec, \jvec})$ at time $t=0.002$.}
         \label{fig:colormap_2_real}
     \end{subfigure}
-
     \newline
-
     \begin{subfigure}[b]{0.3\textwidth}
         \centering
         \includegraphics[width=\textwidth]{images/color_map_0_imag.pdf}
@@ -142,7 +144,17 @@ $(M-2)^{2} \times (M-2)^{2} = 9 \times 9$ given by
         \caption{Im$(u_{\ivec, \jvec})$ at time $t=0.002$.}
         \label{fig:colormap_2_imag}
     \end{subfigure}
-    \caption{The time evolution of the probability function $p_{\ivec, \jvec}^{n}$.}
-    \label{fig:colormap}
+    \caption{The time evolution of Re($u_{\ivec, \jvec}$) and Im($u_{\ivec, \jvec}$), 
+    for time steps $t=[0, 0.001, 0.002]$.}
+    \label{fig:colormap_real_imag}
 \end{figure*}
+% \begin{figure*}
+%     \centering
+%     \includegraphics[width=0.9\textwidth]{images/color_map_all.pdf}
+%     \caption{The time evolution of the probability function $p_{\ivec, \jvec}^{n}$ (top row), 
+%     Re($u_{\ivec, \jvec}^{n}$) (middle row), and Im($u_{\ivec, \jvec}^{n}$) (bottom row). 
+%     Time step $t=0$ in the left column, $t=0.001$ in the middle column, and $t=0.002$ 
+%     in the right column.}
+%     \label{fig:colormap_all}
+% \end{figure*}
 \end{document}

latex/sections/methods.tex

@@ -273,8 +273,8 @@ and for the simulations, we used the parameter settings in Table \ref{tab:sim_se
 To check if the total probability is conserved over time, and that the implementation 
 was correct, we computed the deviation from $1.0$ given by
 \begin{align*}
-    s^{n} &= |\sum_{\ivec , \jvec} p_{\ivec , \jvec}^{n} - 1| \\
-    &= |\sum_{\ivec , \jvec} u_{\ivec , \jvec}^{n*} u_{\ivec , \jvec}^{n} - 1| \ .
+    s^{n} &= |1.0 - \sum_{\ivec , \jvec} p_{\ivec , \jvec}^{n}| \\
+    &= |1.0 - \sum_{\ivec , \jvec} u_{\ivec , \jvec}^{n*} u_{\ivec , \jvec}^{n}| \ .
 \end{align*}
 
 \subsection{Tools}\label{ssec:tools} %

latex/sections/results.tex

@@ -74,6 +74,20 @@ areas at both sides of the barries. Some of the probability function is reflecte
 by the barrier, while the the rest spread out after passing the barrier. This is 
 a consequence of the wave-particle duality. 
 
+To compare the probability function $p_{\ivec, \jvec}$ for all the time steps in 
+a colormap plot, with the real and imaginary part of $u_{\ivec, \jvec}$, we created 
+Figure \ref{fig:colormap_all}. Where we excluded all x- and y-ticks, and labels, to 
+better visualize the diffraction pattern.
+\begin{figure}
+    \centering
+    \includegraphics[width=\linewidth]{images/color_map_all.pdf}
+    \caption{The time evolution of the probability function $p_{\ivec, \jvec}^{n}$ (top row), 
+    Re($u_{\ivec, \jvec}$) (middle row), and Im($u_{\ivec, \jvec}$) (bottom row). 
+    Time step $t=0$ in the left column, $t=0.001$ in the middle column, and $t=0.002$ 
+    in the right column.}
+    \label{fig:colormap_all}
+\end{figure}
+
 
 \subsection{Particle detection}\label{ssec:particle_detection}
 % Problem 9: Plot detection probability for single-, double- and triple-slit