Finish method and results, and add script for including all colormaps in one figure.

latex/schrodinger_simulation.tex

@@ -122,14 +122,14 @@
 
 \end{document}
 
-% Abstract: OK
-% Introduction: 
-% Methods: Wave packet, Heisenberg, Hyugen
-% Results: 
-% Conclusion: OK
+% Abstract: OK!
+% Introduction: Hyugen, Heisenberg
+% Methods: OK!
+% Results: OK
+% Conclusion: OK!
 % Appendices: 
 
 % Results
-% P7: OK
-% P8:
-% P9:
+% P7: OK!
+% P8: OK!
+% P9: OK

latex/sections/methods.tex

@@ -46,7 +46,14 @@ As a result of working in position space, the Born rule is given by
     p(x, y \ | \ t) &= |u(x, y, t)|^{2} = u^{*}(x, y, t) u(x, y, t) \ ,
     \label{eq:born_rule_scaled}
 \end{align}
-where we assume a normalized wave function $u(x, y, t)$.
+where we assume a normalized wave function $u(x, y, t)$. We will initialize the wave 
+function, using a Gaussian wavepacket, given by 
+\begin{align*}
+    u(x, y, t=0) &= e^{- \frac{(x-x_{c})^{2}}{2 \sigma_{x}^{2}} - \frac{(y-y_{c})^{2}}{2 \sigma_{y}^{2}} + ip_{x}x + ip_{y}y} \ .
+\end{align*}
+$x_{c}$ and $y_{c}$ are the coordinates of the center of the wavepacket, $\sigma_{x}$ 
+and $\sigma_{y}$ are the width of the wavepacket. The wave packet momenta are 
+given by $p_{x}$ and $p_{y}$.
 
 
 \subsection{The Crank-Nicolson scheme}\label{ssec:crank_nicolson} %
@@ -101,9 +108,10 @@ of Thomas Young's setup can be found in Figure \ref{fig:youngs_double_slit}.
 
 After the wave passes through the barrier, the pattern observed is determined by 
 the path difference given by 
-\begin{align*}
+\begin{align}
     \delta = d \sin (\theta) = m \lambda \ ,
-\end{align*}
+    \label{eq:interference}
+\end{align}
 where $\lambda$ is the wavelength and $m$ is called the order number. $d$ is the 
 distance between the center of the two slits, while assuming that the distance between 
 the wall and the detector screen $L >> \delta$ \cite[p. 6]{mit:2004:physics}. In 
@@ -216,13 +224,6 @@ An example of filled matrices can be found in Appendix \ref{ap:matrix_structure}
 
 For the general setup of the barrier, we used the values in Table \ref{tab:barrier_setup}, 
 and for the simulations, we used the parameter settings in Table \ref{tab:sim_settings}. 
-In addition, we initialized the wave function using a Gaussian wavepacket, given by 
-\begin{align*}
-    u(x, y, t=0) &= e^{- \frac{(x-x_{c})^{2}}{2 \sigma_{x}^{2}} - \frac{(y-y_{c})^{2}}{2 \sigma_{y}^{2}} + ip_{x}x + ip_{y}y} \ .
-\end{align*}
-$x_{c}$ and $y_{c}$ are the coordinates of the center of the wavepacket, $\sigma_{x}$ 
-and $\sigma_{y}$ are the width of the wavepacket, when it is initialized. The wave 
-packet momenta are given by $p_{x}$ and $p_{y}$. 
 % Insert Heisenberg uncertainty here? Or refer to it?
 \begin{table}[H]
     \centering

latex/sections/results.tex

@@ -20,24 +20,28 @@ from $1.0$.
     \caption{Deviation of total probability, for time $t \in [0, T]$ where $T=0.008$.}
     \label{fig:deviation}
 \end{figure}
-In Figure \ref{fig:deviation}, we observe a larger deviation of total probability 
-for a barrier with double slits compared to no barrier. Interaction with the 
-barrier result in a change ... of values, as the light passes through, 
-The result is more prone to computational errors. Using no barrier, the light is not 
-affected by obstacles resulting in a more stable deviation from the total probability. 
+We simulated the wave equation with the barrier switched off, using setting 1 in 
+Table \ref{tab:sim_settings} found in Section \ref{ssec:implementation}. When the 
+barrier was switched on, we used setting 2 in \ref{tab:sim_settings}. We observed 
+a larger deviation of total probability for a barrier with double slits compared 
+to no barrier, the result is showed in Figure \ref{fig:deviation}. The wave interacts  
+with the barrier resulting in a change in kinetic energy. The result is more prone 
+to computational errors, than if the wave propagates without interacting with a 
+barrier. No interaction results in a more stable deviation from the total probability. 
 In addition, we have to consider the limitation of a computer, some computational 
 error is to be expected.
 
+
 \subsection{Time evolution}\label{ssec:time_evolution}
 % Problem 8: Colormap, include plot of both Re and Im for different time steps
 %            Account for color scale
-We ran the simulation using the values in column 2 of Table \ref{tab:sim_setup}, found 
-in Section \ref{ssec:implementation}. To study the time evolution of the probability 
-function, we created colormap plots for different time steps. Figure \ref{fig:colormap_0_prob}, 
+We studied the time evolution of the probability function, using setting 2 in 
+Table \ref{tab:sim_settings}, found in Section \ref{ssec:implementation}. To visualize 
+the time evolution, we created colormap plots for different time steps. Figure \ref{fig:colormap_0_prob}, 
 Figure \ref{fig:colormap_1_prob}, and Figure \ref{fig:colormap_2_prob} show the 
-result for time steps $t=[0, 0.001, 0.002]$, respectively. In addition, we created 
+results for time steps $t=[0, 0.001, 0.002]$, respectively. In addition, we created 
 separate plots for the real and imaginary part of $u_{\ivec, \jvec}$, for the same 
-time steps. The result can be found in Appendix \ref{ap:figures}, in Figure \ref{fig:colormap}.
+time steps. The results can be found in Appendix \ref{ap:figures}, in Figure \ref{fig:colormap}.
 \begin{figure}
     \centering
     \includegraphics[width=\linewidth]{images/color_map_0_prob.pdf}
@@ -56,17 +60,30 @@ time steps. The result can be found in Appendix \ref{ap:figures}, in Figure \ref
     \caption{The probability function $p_{\ivec, \jvec}^{n}$, at time $t=0.002$.}
     \label{fig:colormap_2_prob}
 \end{figure}
-In Figure \ref{fig:colormap_1_prob}, the ... interacts with the double 
-slitted barrier, which result in a wavelike pattern. However, the waves are more 
-visible when we observe the real and imaginary part separately in Figure \ref{fig:colormap}.
+At time step $t=0.001$, Figure \ref{fig:colormap_1_prob}, when the wave interacts 
+with the double slit barrier, we observe a clear diffraction pattern in the 
+probability function. At time step $t=0$ (Figure \ref{fig:colormap_0_prob}) and 
+$t=0.002$ (Figure \ref{fig:colormap_2_prob}), the diffraction pattern is not as 
+clear. It is, however, more visible when we observe the real and imaginary part 
+separately in Figure \ref{fig:colormap}, found in Appendix \ref{ap:figures}. Since 
+the probability function is a product of $u_{\ivec, \jvec}$ and its conjugate $u_{\ivec, \jvec}^{*}$, 
+initialized by a Gaussian wavepacket, the result is a sum of the real and imaginary part. 
+% This can be found using Euler's formula, and the diffraction pattern is determined by interference given by \eqref{eq:interference}
+In Figure \ref{fig:colormap_2_prob}, the probability function result in positive 
+areas at both sides of the barries. Some of the probability function is reflected 
+by the barrier, while the the rest spread out after passing the barrier. This is 
+a consequence of the wave-particle duality. 
 
 
 \subsection{Particle detection}\label{ssec:particle_detection}
 % Problem 9: Plot detection probability for single-, double- and triple-slit
-We used the simulation from the previous section, assumed a detector screen 
-located at $x=0.8$, and plotted the detection probability along the screen at time 
-$t=0.002$. We adjusted the parameters to include single-, double-, and triple-slitted 
-barrier. The results is found in Figure \ref{}, Figure \ref{}, and Figure \ref{fig:}.
+We simulation the wave equation using setting 2 in Table \ref{tab:sim_settings}, 
+and assumed a detector screen located at $x=0.8$. To visualize the pattern of constructive 
+and destructive interference, we plotted the probability of particle detection, 
+along the screen, at time $t=0.002$. We adjusted the parameters to include single-, double-, and triple-slit 
+barriers. The results is found in Figure \ref{fig:particle_detection_single}, 
+Figure \ref{fig:particle_detection_double}, and Figure \ref{fig:particle_detection_triple}, 
+respectively.
 \begin{figure}
     \centering
     \includegraphics[width=\linewidth]{images/single_slit_detector.pdf}
@@ -88,5 +105,10 @@ barrier. The results is found in Figure \ref{}, Figure \ref{}, and Figure \ref{f
     when using a triple-slit barrier.}
     \label{fig:particle_detection_triple}
 \end{figure}
-
+When the barrier has a single slit, there is no destructive interference and we 
+observe a single peak in the probability of particle detection. Adding another slit 
+result in more peaks, as there are both constructive and destructive interference. 
+When we use a triple-slit barrier, we observe an increase in interference which 
+result in narrow peaks. In addition, the probability of detecting a particle at 
+the ends of the screen increase with number of slits.
 \end{document}

python_scripts/colormap.py

@@ -19,6 +19,7 @@ plt.rcParams.update(params)
 
 
 def plot():
+    ticks = [0, 0.25, 0.5, 0.75, 1.0]
     with open("data/color_map.txt") as f:
         lines = f.readlines()
         size = int(lines[0])
@@ -38,16 +39,19 @@ def plot():
                 np.multiply(arr, arr.conj()).real,
                 interpolation="nearest",
                 cmap=sns.color_palette("mako", as_cmap=True),
+                extent=[0, 1.0, 0, 1.0]
             )
             color_map2 = ax2.imshow(
                 arr.real,
                 interpolation="nearest",
                 cmap=sns.color_palette("mako", as_cmap=True),
+                extent=[0, 1.0, 0, 1.0]
             )
             color_map3 = ax3.imshow(
                 arr.imag,
                 interpolation="nearest",
                 cmap=sns.color_palette("mako", as_cmap=True),
+                extent=[0, 1.0, 0, 1.0]
             )
 
             # Create color bar
@@ -60,10 +64,26 @@ def plot():
             ax2.grid(False)
             ax3.grid(False)
 
+            # Set custom ticks
+            ax1.set_xticks(ticks)
+            ax1.set_yticks(ticks)
+            ax2.set_xticks(ticks)
+            ax2.set_yticks(ticks)
+            ax3.set_xticks(ticks)
+            ax3.set_yticks(ticks)
+
+            # Set labels
+            ax1.set_xlabel("x-axis")
+            ax1.set_ylabel("y-axis")
+            ax2.set_xlabel("x-axis")
+            ax2.set_ylabel("y-axis")
+            ax3.set_xlabel("x-axis")
+            ax3.set_ylabel("y-axis")
+
             # Save the figures
-            fig1.savefig(f"latex/images/color_map_{i}_prob.pdf")
-            fig2.savefig(f"latex/images/color_map_{i}_real.pdf")
-            fig3.savefig(f"latex/images/color_map_{i}_imag.pdf")
+            fig1.savefig(f"latex/images/color_map_{i}_prob.pdf", bbox_inches="tight")
+            fig2.savefig(f"latex/images/color_map_{i}_real.pdf", bbox_inches="tight")
+            fig3.savefig(f"latex/images/color_map_{i}_imag.pdf", bbox_inches="tight")
 
             # Close figures
             plt.close(fig1)

python_scripts/colormap_all.py

@@ -0,0 +1,93 @@
+import ast
+
+import matplotlib.pyplot as plt
+import numpy as np
+import seaborn as sns
+
+sns.set_theme()
+params = {
+    "font.family": "Serif",
+    "font.serif": "Roman",
+    "text.usetex": True,
+    "axes.titlesize": "large",
+    "axes.labelsize": "large",
+    "xtick.labelsize": "large",
+    "ytick.labelsize": "large",
+    "legend.fontsize": "medium",
+}
+plt.rcParams.update(params)
+
+
+def plot():
+    fig, axes = plt.subplots(nrows=3, ncols=3)
+    with open("data/color_map.txt") as f:
+        lines = f.readlines()
+        size = int(lines[0])
+        for i, line in enumerate(lines[1:]):
+            # Create figures for each plot
+            # fig1, ax1 = plt.subplots()
+            # fig2, ax2 = plt.subplots()
+            # fig3, ax3 = plt.subplots()
+
+            arr = line.strip().split("\t")
+            arr = np.asarray(list(map(lambda x: complex(*ast.literal_eval(x)), arr)))
+            # Reshape and transpose array
+            arr = arr.reshape(size, size).T
+
+            # Plot color maps
+            color_map1 = axes[i,0].imshow(
+                np.multiply(arr, arr.conj()).real,
+                interpolation="nearest",
+                cmap=sns.color_palette("mako", as_cmap=True)
+            )
+            color_map2 = axes[i,1].imshow(
+                arr.real,
+                interpolation="nearest",
+                cmap=sns.color_palette("mako", as_cmap=True)
+            )
+            color_map3 = axes[i,2].imshow(
+                arr.imag,
+                interpolation="nearest",
+                cmap=sns.color_palette("mako", as_cmap=True)
+            )
+
+            # Create color bar
+    fig.colorbar(color_map1, ax=axes)
+            # fig2.colorbar(color_map2, ax=ax2)
+            # fig3.colorbar(color_map3, ax=ax3)
+
+            # # Remove grids
+    axes.grid(False)
+            # ax2.grid(False)
+            # ax3.grid(False)
+
+            # # Set custom ticks
+            # ax1.set_xticks(ticks)
+            # ax1.set_yticks(ticks)
+            # ax2.set_xticks(ticks)
+            # ax2.set_yticks(ticks)
+            # ax3.set_xticks(ticks)
+            # ax3.set_yticks(ticks)
+
+            # # Set labels
+            # ax1.set_xlabel("x-axis")
+            # ax1.set_ylabel("y-axis")
+            # ax2.set_xlabel("x-axis")
+            # ax2.set_ylabel("y-axis")
+            # ax3.set_xlabel("x-axis")
+            # ax3.set_ylabel("y-axis")
+
+            # # Save the figures
+    fig.savefig(f"latex/images/color_map_all.pdf", bbox_inches="tight")
+            # fig2.savefig(f"latex/images/color_map_{i}_real.pdf", bbox_inches="tight")
+            # fig3.savefig(f"latex/images/color_map_{i}_imag.pdf", bbox_inches="tight")
+
+            # # Close figures
+            # plt.close(fig1)
+            # plt.close(fig2)
+            # plt.close(fig3)
+
+
+if __name__ == "__main__":
+    plot()
+

python_scripts/detector.py

@@ -27,6 +27,7 @@ def plot():
         "latex/images/double_slit_detector.pdf",
         "latex/images/triple_slit_detector.pdf",
     ]
+    colors = sns.color_palette("mako", n_colors=2)
     for file, output in zip(files, outputs):
         with open(file) as f:
             lines = f.readlines();
@@ -43,8 +44,10 @@ def plot():
             slice *= norm
             slice = np.asarray([i * i.conjugate() for i in slice])
 
-            ax.plot(x, slice)
-            plt.savefig(output)
+            ax.plot(x, slice, color=colors[0])
+            ax.set_xlabel("Detector screen (y-axis)")
+            ax.set_ylabel("Detection probability")
+            fig.savefig(output, bbox_inches="tight")
             plt.close(fig)