FYS3150/Project-5
3
0
Fork 0
Code Issues Pull Requests Packages Projects 1 Releases Wiki Activity

Report ready for proofreading and review.

Browse Source
This commit is contained in:
Janita Willumsen 2024-01-01 15:34:19 +01:00
parent c51bcb4369
commit 70057de961
6 changed files with 75 additions and 37 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

BIN
latex/images/color_map_all.pdf
View File

Binary file not shown.

2
latex/schrodinger_simulation.tex
View File

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

32
latex/sections/appendices.tex
View File

@ -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}

4
latex/sections/methods.tex
View File

@ -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} %

14
latex/sections/results.tex
View File

@ -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

60
python_scripts/colormap_all.py
View File

@ -9,17 +9,23 @@ 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",
"xtick.bottom": False,
"xtick.labelbottom": False,
"ytick.left": False,
"ytick.labelleft": False,
"axes.grid": False,
# "figure.autolayout": True
# "figure.constrained_layout.use": False,
# "figure.constrained_layout.h_pad": 0.04167,
# "figure.constrained_layout.w_pad": 0.04167
# "figure.subplot.wspace": 0,
# "figure.subplot.hspace": 0
}
plt.rcParams.update(params)
def plot():
fig, axes = plt.subplots(nrows=3, ncols=3)
fig, axes = plt.subplots(figsize=(6,6), nrows=3, ncols=3)
with open("data/color_map.txt") as f:
lines = f.readlines()
size = int(lines[0])
@ -35,39 +41,45 @@ def plot():
arr = arr.reshape(size, size).T
# Plot color maps
color_map1 = axes[i,0].imshow(
color_map1 = axes[0, i].imshow(
np.multiply(arr, arr.conj()).real,
interpolation="nearest",
cmap=sns.color_palette("mako", as_cmap=True)
)
color_map2 = axes[i,1].imshow(
color_map2 = axes[1, i].imshow(
arr.real,
interpolation="nearest",
cmap=sns.color_palette("mako", as_cmap=True)
)
color_map3 = axes[i,2].imshow(
color_map3 = axes[2, i].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)
fig.subplots_adjust(wspace=0, hspace=0)
# fig.colorbar(color_map1, ax=axes[0, 2])
# fig.colorbar(color_map2, ax=axes[1, 2])
# fig.colorbar(color_map3, ax=axes[2, 2])
# # Remove grids
axes.grid(False)
# ax2.grid(False)
# ax3.grid(False)
# Remove grids
# for i in range(3):
# axes[i, 0].set_xticks([])
# axes[i, 0].set_yticks([])
# axes[i, 0].
# axes[i, 1].set_xticks([])
# axes[i, 1].set_yticks([])
# axes[i, 2].set_xticks([])
# axes[i, 2].set_yticks([])
# # 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)
# axes[0].set_xticks([])
# axes[0].set_yticks([])
# axes[1].set_xticks([])
# axes[].set_yticks([])
# axes[].set_xticks([])
# axes[].set_yticks([])
# # Set labels
# ax1.set_xlabel("x-axis")
@ -78,7 +90,7 @@ def plot():
# ax3.set_ylabel("y-axis")
# # Save the figures
fig.savefig(f"latex/images/color_map_all.pdf", bbox_inches="tight")
plt.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")