diff --git a/chapters/fundamentals/examples.tex b/chapters/fundamentals/examples.tex new file mode 100644 index 0000000..e2ae95b --- /dev/null +++ b/chapters/fundamentals/examples.tex @@ -0,0 +1,63 @@ +% !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} \ No newline at end of file diff --git a/chapters/fundamentals/fundamentals.tex b/chapters/fundamentals/fundamentals.tex new file mode 100644 index 0000000..b01d2cb --- /dev/null +++ b/chapters/fundamentals/fundamentals.tex @@ -0,0 +1,27 @@ +% !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} \ No newline at end of file diff --git a/chapters/fundamentals/schroedinger.tex b/chapters/fundamentals/schroedinger.tex new file mode 100644 index 0000000..14e3e39 --- /dev/null +++ b/chapters/fundamentals/schroedinger.tex @@ -0,0 +1,87 @@ +% !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} \ No newline at end of file diff --git a/script.pdf b/script.pdf index 9a8eef6..c9063d3 100644 Binary files a/script.pdf and b/script.pdf differ diff --git a/script.tex b/script.tex index 43842b1..41ec251 100644 --- a/script.tex +++ b/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}