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latex/sections/methods.tex
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@@ -3,11 +3,8 @@
\begin{document}
\section{Methods}\label{sec:methods} %
\subsection{The Schrödinger equation}\label{ssec:schrodinger} %
% Add something that takes Planck to Schrödinger
% In classical mechanics, we have Newton laws and conservation of energy. In quantum
% mechanics, we have Schrödinger equation.
Erwin Schrödinger wanted to find a mathematical description of the wave characteristics
of matter, supporting the wave-particle idea. He postulated a wave function which varies
of matter, which supported the wave-particle idea. He postulated a wave function which varies
with position, where the function squared can be interpreted as the probability
of finding an electron at a given position. This resulted in the Schrödinger equation,
a wave eqution of the energy levels for a hydrogen atom. It also shows how a quantum
@@ -17,21 +14,20 @@ has a general form
i \hbar \frac{\partial}{\partial t} | \Psi \rangle &= \hat{H} | \Psi \rangle \ ,
\label{eq:schrodinger_general}
\end{align}
where $i$ is the imaginary unit, and $\hbar$ is Plancks constant. $\hat{H}$ is
where $i$ is the imaginary unit, and $\hbar$ is the reduced Planck's constant. $\hat{H}$ is
a Hamiltonian operator, which represents the energy for the system, and $| \Psi \rangle$
is the quantum state. In two-dimensional position space, the quantum state can
be expressed using the time-dependent complex-valued wave function $\Psi (x, y, t)$.
Using Born rule, the square modulus of the wave function is proportional to the
Using the Born rule, the square modulus of the wave function is proportional to the
probability density of detecting a particle at position $(x, y)$ at time $t$. The
relation is given by
\begin{align}
p(x, y \ | \ t) &= |\Psi(x, y, t)|^{2} = \Psi^{*}(x, y, t) \Psi(x, y, t) \ ,
\label{eq:born_rule}
\end{align}
where $\Psi^{*}$ denotes the complex conjugated wave function.
% Add something about kinetic and potential energy, to introduce the potential V
When the potential is time-independent, and the particle is non-relativistic,
the Schrödinger equation can be expressed as
where $\Psi^{*}$ denotes the complex conjugate of the wave function. When the potential
is time-independent, and the particle is non-relativistic, the Schrödinger equation
can be expressed as
\begin{align*}
i \hbar \frac{\partial}{\partial t} \Psi (x, y, t) &= - \frac{\hbar^{2}}{2m} \bigg( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \bigg) \Psi (x, y, t) \\
& \quad + V(x, y, t) \Psi (x, y, t) \numberthis \ .
@@ -39,9 +35,9 @@ the Schrödinger equation can be expressed as
\end{align*}
The partial derivatives are expressions of the kinetic energy, and the potental $V$
encodes the external environment. In this experiment we will only consider the case where
the potential is time-independent, resulting in $V = V(x, y)$
the potential is time-independent, resulting in $V = V(x, y)$.
When we scale Schrödinger equation by the dimensionful variables, we are left with
When we scale the Schrödinger equation by the dimensionful variables, we are left with
the wave function $u$ and the potential $v$. The dimensionless equation is given by
\begin{align}
i \frac{\partial u}{\partial t} &= - \frac{\partial^{2} u}{\partial x^{2}} - \frac{\partial^{2} u}{\partial y^{2}} + v(x, y) u \ .
@@ -63,9 +59,9 @@ given by $p_{x}$ and $p_{y}$.
\subsection{The Crank-Nicolson scheme}\label{ssec:crank_nicolson} %
When we evaluate a particles position, we have to consider partial differential
equations (PDE). To solve these numerically, we have to discretize Equation \eqref{eq:schrodinger_dimensionless}.
We use the $\theta$-rule \footnote{Using the $\theta$-rule, we can derive Forward Euler using $\theta = 1$, and Backward Euler using $\theta = 0$},
When we evaluate a particle's position, we have to consider partial differential
equations (PDEs). To solve these numerically, we have to discretize Equation \eqref{eq:schrodinger_dimensionless}.
We use the $\theta$-rule \footnote{We can derive Forward Euler using $\theta = 1$, and Backward Euler using $\theta = 0$, in the $\theta$-rule.},
to combine the forward (explicit) and backward (implicit) finite difference methods.
The result is a linear combination of the explicit and implicit scheme, given by
\begin{align}
@@ -99,7 +95,7 @@ can be found in Appendix \ref{ap:crank_nicolson}.
\subsection{The double-slit experiment}\label{ssec:double_slit} %
Thomas Young first performed the double-slit experiment in 1801 to demonstrate the
principle of interference of light \cite{britannica:2023:young}, while postulating
principle of the interference of light \cite{britannica:2023:young}, while postulating
light as waves rather than particles. The double-slit experiment results in a diffraction
pattern on a detector screen, where constructive interference of light result in
bright spots, and destructive interference result in dark spots. An illustration
@@ -133,14 +129,14 @@ and destructive interference when
\subsection{Implementation}\label{ssec:implementation} %
In this experiment, we set up the grid with an equal step size in x- and y-direction $h$,
and step size in t-direction $\Delta t$, such that
In this experiment, we set up the grid with an equal step size in the x- and y-direction $h$,
and step size in the t-direction $\Delta t$, such that
\begin{align*}
x \in [0, 1] && x \rightarrow x_{\ivec} = \ivec h && \ivec = 0, 1, \dots, M-1 \\
y \in [0, 1] && y \rightarrow y_{\jvec} = \jvec h && \jvec = 0, 1, \dots, M-1 \\
t \in [0, T] && t \rightarrow t_{n} = n \Delta t && n = 0, 1, \dots, N_{t}-1 \ .
\end{align*}
In addition, we simplified the indices such that
In addition, we simplify the indices such that
\begin{align*}
u(x, y, t) \rightarrow u(\ivec h, \jvec h, n \Delta t) \equiv u_{\ivec, \jvec}^{n} \\
v(x, y) \rightarrow u(\ivec h, \jvec h) \equiv v_{\ivec, \jvec} \ ,
@@ -152,7 +148,7 @@ conditions, given by
u(x=0, y, t) &= 0 & u(x=1, y, t) &= 0 \\
u(x, y=0, t) &= 0 & u(x, y=1, t) &= 0 \ ,
\end{align*}
which allowed us to express Equation \eqref{eq:schrodinger_discretized} as a matrix
which allows us to express Equation \eqref{eq:schrodinger_discretized} as a matrix
equation
\begin{align}
A u^{n+1} = B u^{n} \ .
@@ -221,12 +217,12 @@ with the following pattern
\end{matrix}
\end{bmatrix} \ .
\end{align*}
To fill the matrices $A$ and $B$, we used
To fill the matrices $A$ and $B$, we use
\begin{align*}
a_{k} &= 1 + 4r + \frac{i \Delta t}{2} v_{\ivec, \jvec} \\
b_{k} &= 1 - 4r - \frac{i \Delta t}{2} v_{\ivec, \jvec} \ .
\end{align*}
An example of filled matrices can be found in Appendix \ref{ap:matrix_structure}.
An example of a pair 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}.
@@ -265,7 +261,7 @@ and for the simulations, we used the parameter settings in Table \ref{tab:sim_se
\hline
\end{tabular}
\caption{Simulation settings used in the double slit experiment. Setting 1 is
used when the barrier is switched off and setting 2 is used when the barrier
applied when the barrier is switched off and setting 2 is applied when the barrier
switched on.}
\label{tab:sim_settings}
\end{table}
@@ -279,7 +275,7 @@ was correct, we computed the deviation from $1.0$ given by
\subsection{Tools}\label{ssec:tools} %
The double-slit experiment is implemented in C++. We use the Python library
\verb|matplotlib| \cite{hunter:2007:matplotlib} to produce all the plots, and
\verb|NumPy| \cite{harris:2020:numpy}, \verb|matplotlib| \cite{hunter:2007:matplotlib} to produce all the plots, and
\verb|seaborn| \cite{waskom:2021:seaborn} to set the theme in the figures.
\end{document}