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chapters/fundamentals/examples.tex

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+% !TeX root = ../../script.tex
+\documentclass[../../script.tex]{subfiles}
+
+\begin{document}
+\section[Examples of the Stationary Schrödinger Equation]{Examples of the Stationary Schrödinger Equation}
+
+We now want to solve the Schrödinger equation
+\[
+	\frac{-\hbar^2}{2m} \dv[2]{\psi}{x} + \epot\psi = E\psi
+\]
+for a few simple, one-dimensional problems. These examples will illustrate the description of classical particles as waves and the following physical consequences.
+
+\subsection{The Free Particle}
+
+A particle is said to be free, if it is moving in a constant potention $\phi_0$, because then $\vec{F} = -\grad{\epot}$ means that no forces are acting on the particle. 
+Through a suitable choice of the zero point energy we can set $\phi_0 = 0$, i.e. $\epot = 0$, and thus get the Schrödinger equation for a free particle 
+\begin{equation}\label{eq:onedschroedinger}
+	\frac{-\hbar^2}{2m} \dv[2]{\psi}{x} = E\psi
+\end{equation}
+The total energy $E = \ekin + \epot$ is because of $\epot$ now 
+\[
+	E = \frac{p^2}{2m} = \frac{\hbar^2 k^2}{2m}
+\]
+Thus~\eqref{eq:onedschroedinger} gets reduced to 
+\[
+	\dv[2]{\psi}{x} = -k^2 \psi
+\]
+which has the general solution 
+\begin{equation}\label{eq:onedsolution}
+	\psi(x) = Ae^{ikx} + Be^{-ikx}
+\end{equation}
+The time-dependent wave function 
+\begin{equation}
+	\psi(x, t) = \psi(x) \cdot e^{-i\omega t} = Ae^{i(kx - \omega t)} + Be^{-i(kx + \omega t)}
+\end{equation}
+represents the superposition of a planar wave travelling in the $+x$ and $-x$ direction.
+
+The coefficients $A$ and $B$ are the amplitudes of those waves, which are determined by the boundary conditions.
+For example, the wave function of electrons which are emitted from a cathode in $+x$ direction towards a detector, will have $B = 0$, since there are no particles moving in $-x$.
+From this experimental setup we know that the electrons are found along the length $L$ of the path between cathode and detector. This means their wave function can only be different from zero in this region of space.
+Using the normalization condition we get 
+\begin{align*}
+	&\int_0^L \abs{\psi(x)}^2 \dd{x} = 1 \\
+	&\implies A^2 \cdot L = 1 \implies A = \frac{1}{\sqrt{L}}
+\end{align*}
+To determine the location of a particle at time $t$ more accurately, we will have to construct \textit{wave packets} in place of planar waves~\eqref{eq:onedsolution}
+\begin{equation}
+	\psi(x, t) = \int_{k_0 - \Delta k / 2}^{k_0 + \Delta k / 2} A(k) e^{i(kx - \omega t)} \dd{k}
+\end{equation}
+The location uncertainty of this packet at $t = 0$ is 
+\[
+	\Delta x \ge \frac{\hbar}{2 \Delta p_x} = \frac{1}{2\Delta k}
+\]
+and depends on the pulse width $\Delta p_x = \hbar \Delta k$. The larger $k$ is, the more certainly $\Delta x(t = 0)$ can be determined, but the faster the wave packet spreads.
+
+Experimentally, this can be illustrated as follows: If we apply a short voltage pulse to the cathode at time $t = 0$, then electrons can start travelling towards the detector at this instance.
+The emitted electrons have a velocity distribution $\Delta v$, such that electrons with differing velocities $v$ will not necessarily be in the same location $x$ at a later point in time $t$.
+Instead they are spread over the interval $\Delta x(t) = t \cdot \Delta v$. The velocity distribution is described by $\Delta v \propto \Delta k$ of the wave packet, such that the location uncertainty $\Delta x$ 
+\[
+	\dv{(\Delta x(t))}{t} = \Delta v(t = 0) = \frac{\hbar}{m} \Delta k(t = 0)
+\]
+changes proportionally to the initial impulse uncertainty.
+\end{document}

chapters/fundamentals/fundamentals.tex

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+% !TeX root = ../script.tex
+\documentclass[../../script.tex]{subfiles}
+
+\begin{document}
+    \chapter{Fundamentals of Quantum Physics}
+    \minitoc
+    \vspace*{\fill}\par
+    \pagebreak  
+
+    Due to the uncertainty principle, location and impulse of an atomic particle cannot be both stated with arbitrairy precision.
+    The classic trajectory, represented in the model of mass points by a well-defined curve in space $\vec{r}(t)$, is replaced by the probability
+    \begin{equation}
+        W(x, y, z, t) \dd{v} = \abs{\psi(x, y, z, t)}^2 \dd{V}
+    \end{equation}
+    to find the particle in the volume element $\dd{V} = \dd{x}\dd{y}\dd{z}$ at the time $t$. This probability depends on the absolute square of the matter wave function $\psi(x, y, z, t)$.
+
+    In this chapter we wnat to show how this wave function can be calculated for simple examples. These examples will also demonstrate the physical fundamentals of quantum mechanics and its
+    differences to classical particle mechanics, elaborate on the concept of \textit{quantum numbers} and show under which conditions quantum mechanical results can be transitioned into classical physics.
+    This is supposed to clarify that classic (i.e.\ pre-quantum) mechanics are contained in quantum mechanics as a limiting case for very small de Broglie wavelengths $\lambda_{dB} \rightarrow 0$.
+
+    These examples should also demonstrate that almost all insights of quantum mechanics are already known in classic wave optics. This means: the actual novel concepts in quantum mechanics
+    is the description of classic particles with matter waves. The deterministic description of the temporal development of location and impulse of a particle is thus replaced with a statistical
+    treatment, by which we can only discuss probabilities of the results of a measurement. A fundamental uncertainty occurs when we observe location and impulse at the same time.
+
+    \subfile{schroedinger.tex}
+    \subfile{examples.tex}
+\end{document}

chapters/fundamentals/schroedinger.tex

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+% !TeX root = ../../script.tex
+\documentclass[../../script.tex]{subfiles}
+
+\begin{document}
+\section[The Schrödinger Equation]{The Schrödinger Equation}
+
+In this section we will outline the fundamental equation of quantum mechanics, which was established by \textit{Erwin Schrödinger} (1887--1961) in 1926.
+The solutions of this equation are the desired wave functions $\psi(x, y, z, t)$. However, these solutions can only be derived in analytical form for a few very simple physical problems.
+Very fast computers are usually able to numerically compute solutions for complex problems.
+
+First, we will consider the mathematically simplest case.A free particle of mass $m$ which moves at a constant velocity $\vec{v}$ in direction $x$.
+With $\vec{p} = \hbar\vec{k}$ and $E = \hbar \omega = \ekin$ (because $\epot = 0$), we know the wave function must be of the form 
+\begin{equation}\label{eq:wavefuncform}
+	\psi(x, t) = Ae^{i(kx-\omega t)} = Ae^{(i/\hbar)(px-\ekin t)}
+\end{equation}
+Here we use the fact that $\ekin = p^2 / 2m$, which is the kinetic energy of the particle. Since the mathematical representation is absolutely identical to that of an electromagnetic wave,
+it makes sense to start with the wave equation 
+\begin{equation}\label{eq:1dwaveeq}
+	\pdv[2]{\psi}{x} = \frac{1}{u^2} \pdv[2]{\psi}{t}
+\end{equation}
+for waves propagating with phase velocity $u$ in direction $x$. For \textit{stationary} problems where $\vec{p}$ and $E$ are time-independent the wave function can be split into a 
+strictly location-dependent factor $\psi(x) = Ae^{ikx}$, and a strictly time-dependent phase factor $e^{-i\omega t}$. Thus we can write 
+\begin{equation}\label{eq:splitwave}
+	\psi(x, t) = \psi(x) \cdot e^{-i\omega t} = A e^{ikx} \cdot e^{-i\omega t}
+\end{equation}
+If we use the ansatz~\eqref{eq:splitwave} in the wave equation~\eqref{eq:1dwaveeq}, and the fact that $k^2 = p^2 / \hbar^2 = 2m\ekin/\hbar^2$, we get the equations
+\begin{equation}\label{eq:values}
+	\begin{split}
+		&\pdv[2]{\psi}{x} = -k^2\psi = -\frac{2m}{k} \cdot \ekin \cdot \psi \\
+		&\pdv[2]{\psi}{t} = -\omega^2 \psi
+	\end{split}
+\end{equation}
+Comparison with~\eqref{eq:1dwaveeq} gives us 
+\[
+	u^2 = \frac{\omega^2}{k^2} \implies u = \frac{\omega}{k}
+\]
+Note that the particle velocity $v_\text{P} = v$ 
+\[
+	v = \frac{p}{m} = \frac{\hbar k}{m} = \pdv{\omega}{k}
+\]
+is different from the phase velocity $u = v_{\text{ph}} = \omega / k$.
+
+In the general case the particle can move in a force field. If it is conservative then we can assign each point a potential energy, with the condition that the total energy $E = \ekin + \epot$ remains constant.
+Using $\ekin = E - \epot$ and~\eqref{eq:values} we then receive the one-dimensional stationary Schrödinger equation
+\begin{tcolorbox}[ams equation]
+	\frac{-\hbar^2}{2m} \pdv[2]{\psi}{x} + \epot\psi = E\psi
+\end{tcolorbox}
+For the general case where the particle is moving freely in three-dimensional space we can use the three-dimensional wave equation 
+\[
+	\laplacian\psi = \frac{1}{u^2} \pdv[2]{\psi}{t}	
+\]
+and the ansatz $\psi(x, y, z, t) = \psi(x, y, z) \cdot e^{-i\omega t}$, we can establish the three-dimensional stationary Schrödinger equation 
+\begin{tcolorbox}[ams equation]\label{eq:stationaryschroedinger}
+	\frac{-\hbar^2}{2m} \laplacian \psi = \epot \psi = E \psi
+\end{tcolorbox}
+If we differentiate~\eqref{eq:wavefuncform} partially for time we receive 
+\[
+	\pdv{\psi}{t} = -\frac{i}{h} \ekin \cdot \psi	
+\]
+and with~\eqref{eq:values} we can find the time-dependent equation for a free particle with $\epot = 0$ (i.e. $\ekin = \const$)
+\begin{equation}
+	-\frac{\hbar^2}{2m} \pdv[2]{\psi(x, t)}{x} = i\hbar \pdv{\psi(x, t)}{t}
+\end{equation}
+The three-dimensional representation is then 
+\begin{tcolorbox}[ams equation]
+	-\frac{\hbar^2}{2m} \laplacian \psi(\vec{r}, t) = i\hbar \pdv{\psi(\vec{r}, t)}{t}
+\end{tcolorbox}
+There are some remarks to be made:
+\begin{itemize}
+	\item In this ``derivation'' we have used the de Broglie-relationship $\vec{p} = \hbar \vec{k}$, which is only supported by experiments and has no mathematical justification.
+	\item The law of conservation of energy of quantum mechanics is $E\psi = \ekin\psi + \epot\psi$. Like in classical mechanics, there is no derivation for this law, and is accepted as truth from experience.
+	\item While electromagnetic waves have a linear dispersion relation $\omega(k) = kc$, the matter wave $\psi(\vec{r}, t)$ of a free particle has a \textit{quadratic} dispersion relation $\omega(k) = (\hbar/2m) \cdot k^2$. This results from $E = \hbar\omega = p^2/2m$.
+	\item The Schrödinger equations are a \textit{linear} homogeneous differential equation. Because of this, different solutions of the equation can be superpositioned. This means, if $\psi_1$ and $\psi_2$ are solutions to the equation, then $\psi_3 = a\cdot\psi_1 + b\cdot\psi_2$ is also a solution.
+	\item Since the time-dependent Schrödinger equation is a complex equation, the wave functions $\psi$ may also be complex. The absolute square $\abs{\psi}^2$ however, which represents the probability of the presence of a particle, is always real.
+\end{itemize}
+For non-stationary problems (i.e. $E = E(t)$ and $p = p(t)$), the dispersion relation $\omega(t)$ also becomes time-dependent. This means that $\partial^2\psi/\partial t^2$ can no longer be written as $-\omega^2 \psi$, and cannot be derived from the wave equation for matter waves of particles.
+
+Schrödinger postulated (!), that even for time-dependent potential energy $\epot(\vec{r}, t)$ the equation 
+\begin{tcolorbox}[ams equation]\label{eq:schroedinger}
+	\frac{-\hbar^2}{2m} \laplacian \psi(\vec{r}, t) + \epot(\vec{r}, t) \psi(\vec{r}, t) = i\hbar\pdv{\psi(\vec{r}, t)}{t}
+\end{tcolorbox}
+holds. The general time-dependent Schrödinger equation has since been verified in numerous experiments, and is generally considered correct, even if no mathematical justification exists.
+This equation is the fundamental equation of quantum mechanics.
+
+For stationary problems we can separate $\psi(\vec{r}, t)$ into $\psi(\vec{r}, t) = \psi(\vec{r}) \cdot e^{-i(E/\hbar) \cdot t}$. Inserting this into~\eqref{eq:schroedinger} yields the 
+stationary Schrödinger equation~\eqref{eq:stationaryschroedinger} for $\psi(\vec{r})$.
+\end{document}

script.tex

@@ -12,6 +12,7 @@
 \usepackage{tikz, pgfplots}
 \usepackage{kbordermatrix}
 \usepackage{fancyhdr}
+\usepackage[most]{tcolorbox}
 \usepackage{pdfpages}
 \usepackage[arrowdel]{physics}
 \usetikzlibrary{calc,trees,decorations.markings,positioning,arrows,fit,shapes,angles,patterns}
@@ -37,8 +38,22 @@
     version={4.0},
 ]{doclicense}
 
+\tcbset{colback=yellow!10!white, colframe=red!50!black, 
+        highlight math style= {enhanced, %<-- needed for the ’remember’ options
+            colframe=red,colback=red!10!white,boxsep=0pt}
+        }
+
 \usepackage{subfiles}
 
+\DeclareMathOperator{\const}{const}
+
+\newcommand{\energy}[1]{E_{\text{#1}}}
+\newcommand{\ekin}{\energy{kin}}
+\newcommand{\epot}{\energy{pot}}
+
+\renewcommand{\vec}{\vb*}
+\renewcommand{\laplacian}{\Delta}
+
 \begin{document}
 \begin{titlepage}
   \begin{center}
@@ -61,6 +76,8 @@
     urlcolor=black
 }
 
+\subfile{chapters/fundamentals/fundamentals.tex}
+
 \pagestyle{headings}
 
 \end{document}