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diff --git a/presentation/KugelBSc.tex b/presentation/KugelBSc.tex new file mode 100644 index 0000000..7128908 --- /dev/null +++ b/presentation/KugelBSc.tex @@ -0,0 +1,684 @@ +\documentclass[xetex, onlymath]{beamer} +\usefonttheme{serif} +\usetheme{hsr} + +% use lmodern for math +\usepackage{lmodern} + +% math packages +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{bm} + +\renewcommand{\vec}[1]{\mathbf{\bm{#1}}} + +% use plex font for monospaced, roboto for the rest +\usepackage[T1]{fontenc} +\usepackage{plex-otf} % monospaced +% \usepackage{roboto} % other +\renewcommand*\familydefault{\sfdefault} + +\usepackage{graphicx} +\usepackage{booktabs} +\usepackage{array} + +% biblopgraphy +\usepackage[backend=bibtex, style=ieee]{biblatex} +\addbibresource{KugelBSc.bib} + +% links +\usepackage{hyperref} +\hypersetup{ + % Remove ugly boxes + hidelinks, + % Set colors + colorlinks = true, + anchorcolor = black, + citecolor = black, + filecolor = black, + linkcolor = black, + menucolor = black, + runcolor = black, + urlcolor = {black!50!blue}, +} + +% pretty drawings +\usepackage{tikz} +\usetikzlibrary{calc} +\usepackage{xcolor} + +% source code +\usepackage{listings} +%% create a lstlisting style +\lstdefinestyle{samplestyle}{ + belowcaptionskip=\baselineskip, + breaklines=true, + frame=none, + inputencoding=utf8, + % margin + xleftmargin=\parindent, + % background + backgroundcolor=\color{hsr-lightgrey20}, + % default language: + language=[LaTeX]TeX, + showstringspaces=false, + % font + basicstyle=\ttfamily\small, + identifierstyle=\color{hsr-black}, + keywordstyle=\color{hsr-blue}, + commentstyle=\color{hsr-black40}, + stringstyle=\color{hsr-mauve80}, +} + +%% and set the chosen style +\lstset{style=samplestyle, escapechar=`} + +% metadata +\title{Spherical Harmonics} +\author[NaoPross]{\texttt{Naoki Pross, Manuel Cattaneo}} +\date{Spring Semester 2022} + +\institute[OST]{OST FHO Campus Rapperswil} +% \logo{\includegraphics[width=3cm]{figs/hsr-logo}} + +\AtBeginSection[] +{ + \begin{frame} + \frametitle{Table of Contents} + \tableofcontents[currentsection] + \end{frame} +} + + +\begin{document} + +\frame{ + \maketitle +} + +\begin{frame}{Goals for Today} + \Large \uncover<1->{\textbf{Spherical Harmonics}} \uncover<2->{\,\textit{and}\, \textbf{Electron Orbitals}} + \begin{tikzpicture} + \uncover<1->{ + \node (i1) { + \includegraphics[height=8cm, trim=200 100 50 50, clip]{figures/buchcover} + }; + } + + \uncover<2->{ + \node (i2) at ($(i1) + (2cm, 0)$) { + \nocite{minutephysics_better_2021} + \includegraphics[height=65mm]{figures/orbitals-minutephysics} + }; + } + \end{tikzpicture} +\end{frame} + +\section{Fourier on \(\mathbb{R}^2\)} + +\begin{frame}{Nice Periodic Functions} + \begin{definition} + A function + \[ + f : \mathbb{R}^2 \to \mathbb{C} + \] + is a ``nice periodic function'' when it is + \begin{itemize} + \item smooth, + \item differentiable, + \item \textcolor{gray}{(abs.)} integrable, + \item periodic on \([0, 1] \times [0, 1]\), i.e. + \[ + f(\mu, \nu) = f(\mu + 1, \nu) = f(\mu, \nu + 1). + \] + \end{itemize} + \end{definition} +\end{frame} + +\begin{frame}{Function Space} + \begin{block}{Basis Functions} + The space of nice periodic functions is spanned by the (also nice) functions + \[ + B_{m, n}(\mu, \nu) = e^{i2\pi m\mu} e^{i2\pi n\nu}. + \] + \end{block} +\end{frame} + +\begin{frame} \centering + \includegraphics[height=.9\paperheight]{figures/flat-basis-functions} +\end{frame} + +\begin{frame}{Inner Product} + \begin{definition}<1-> + Let \(f(\mu, \nu)\) and \(g(\mu, \nu)\) be nice periodic functions. Their inner product is + \[ + \langle f, g \rangle = \iint_{[0, 1]^2} f g^* \, d\mu d\nu. + \] + \end{definition} + + \begin{definition}<2-> + For a nice periodic function \(f(\mu, \nu)\): the numbers + \[ + c_{m, n} = \langle f, B_{m, n} \rangle + \] + are the \emph{Fourier coefficients} or \emph{spectrum} of \(f\). + \end{definition} +\end{frame} + +\begin{frame}{Fourier Series} + \begin{theorem} + For nice periodic functions: + \[ + f(\mu, \nu) = \sum_{m \in \mathbb{Z}} \sum_{n \in \mathbb{Z}} + c_{m, n} B_{m, n} (\mu, \nu) + \] + where + \[ + c_{m, n} = \langle f, B_{m, n} \rangle. + \] + \end{theorem} +\end{frame} + +\begin{frame}{Why exponentials?} + + \centering + + {\huge\bfseries\itshape Why \(B_{m, n} = e^{i2\pi m\mu} e^{i2\pi n\nu}\)?} + \vspace{3em} + + {\huge\bfseries\itshape Because + {\Huge \(\nabla^2\)} + } + +\end{frame} + +\begin{frame}{The Problem} + \begin{block}{Fourier's Problem}<1-> + \[ + \nabla^2 f(\mu, \nu) + = \frac{\partial^2 f}{\partial \mu^2} + \frac{\partial^2 f}{\partial \nu^2} + = \lambda f(\mu, \nu) + \] + \end{block} + \begin{alertblock}{Solution}<2-> + Separation ansatz: + \[ + f(\mu, \nu) = M(\mu) N(\nu) + \] + Resulting ODEs: + \begin{align*} + \frac{d^2 M}{d \mu^2} &= \kappa M(\mu), & \frac{d^2 N}{d \nu^2} &= (\lambda - \kappa) N(\nu) + \end{align*} + \end{alertblock} +\end{frame} + +\section{The functions \(Y_{m, n}(\varphi, \vartheta)\)} + +\begin{frame}{Spherical Coordinates} + \begin{columns} + \begin{column}{.6\linewidth} + \includegraphics[width=\linewidth]{figures/spherical-coordinates} + \end{column} + \begin{column}{.4\linewidth} + \noindent + Variables + \begin{align*} + r &\in \mathbb{R}^+ \\ + \vartheta &\in [0, \pi] \\ + \varphi &\in [0, 2\pi) + \end{align*} + To cartesian + \begin{align*} + x &= r\cos\varphi \sin\vartheta \\ + y &= r\sin\varphi\sin\vartheta \\ + z &= r\cos\vartheta + \end{align*} + \end{column} + \end{columns} +\end{frame} + +\begin{frame}{Spherical Laplacian} + \uncover<1->{ + Cartesian Laplacian + \[ + \nabla^2 \equiv \frac{\partial^2}{\partial \mu^2} + \frac{\partial^2}{\partial \nu^2} + \] + } + + \uncover<2->{ + Spherical Laplacian + \[ + \nabla^2 \equiv + \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \right) + + \frac{1}{r^2} \onslide<3-> \underbrace{ \onslide<2-> \left[ + \frac{1}{\sin\vartheta} \frac{\partial}{\partial \vartheta} + \left( \sin\vartheta \frac{\partial}{\partial\vartheta} \right) + + \frac{1}{\sin^2 \vartheta} \frac{\partial^2}{\partial\varphi^2} + \right] + \onslide<3-> }_{\text{Surface Spherical Laplacian}~ \nabla^2_s} \onslide<2-> + \] + } + + \uncover<4->{ + Surface Spherical Laplacian + \[ + \nabla^2_s \equiv r^2 \nabla^2 + - \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \right) + \] + } +\end{frame} + +\bgroup +\setbeamercolor{background canvas}{bg=black} +\setbeamertemplate{navigation symbols}{} +\begin{frame}{Geometrical Intuition} +\end{frame} +\egroup + +\begin{frame}{Where is \(\nabla^2_s\) useful?} + To do brain scans, apparently \cite{carvalhaes_surface_2015} + \only<1>{ + \begin{figure} + \includegraphics[width=.8\linewidth, clip=100 0 0 0]{figures/eeg-photo} + \caption{Electroencephalogram (EEG). Image from Wikimedia \cite{baburov__2009}.} + \end{figure} + } + \only<2>{ + \begin{figure} \centering + \includegraphics[width=\linewidth]{figures/surface-laplacian-eeg} + \caption{Surface Laplacian in EEG. Taken from \cite{ries_role_2013}.} + \end{figure} + } +\end{frame} + +\begin{frame}{Brain Scans} + \begin{columns} + \begin{column}{.6\linewidth} + Electrodynamics + \begin{align*} + \nabla^2 \phi + &= \bm{\nabla \cdot} \bm{\nabla} \phi \qquad + \color{lightgray} \left( \phi = \int_\mathsf{A}^\mathsf{B} \vec{E} \bm{\cdot} d\vec{s} \right) \\ + &= \bm{\nabla \cdot} \vec{E} \\ + &\color{lightgray}= \int_{\Omega} (\bm{\nabla \cdot} \vec{E}) \bm{\cdot} d\vec{s} + = \oint_{\partial \Omega} \vec{E} \bm{\cdot} d\vec{s} \\ + &= \frac{\rho}{\varepsilon} + \end{align*} + So over the scalp + \[ + \nabla^2_s \phi + = \frac{\rho_s}{\varepsilon} + = \text{Current flow in the brain} + \] + \end{column} + \begin{column}{.4\linewidth} + \centering + \includegraphics[width=\linewidth]{figures/flux} + \nocite{maschen_divergence_2013} + \end{column} + \end{columns} +\end{frame} + +\begin{frame}{New Hard Problem} + \begin{block}{The Problem}<1-> + \only<1>{ + \[ + \nabla^2_s f(\varphi, \vartheta) = \lambda f(\varphi, \vartheta) + \] + } + \only<2->{ + \[ + \frac{1}{\sin\vartheta} \frac{\partial}{\partial \vartheta} + \left( \sin\vartheta \frac{\partial f}{\partial\vartheta} \right) + + \frac{1}{\sin^2 \vartheta} \frac{\partial^2 f}{\partial\varphi^2} + = \lambda f(\varphi, \vartheta) + \] + } + \end{block} + \begin{alertblock}{Idea}<3-> + Separation ansatz: + \[ + f(\varphi, \vartheta) = \Phi(\varphi) \Theta(\vartheta) + \] + From the ``easy'' part: + \[ + \frac{d^2\Phi}{d\varphi^2} = \kappa \Phi(\varphi) + \implies \Phi(\varphi) = e^{im\varphi}, + \quad m \in \mathbb{Z} + \] + \end{alertblock} +\end{frame} + +\begin{frame}{Associated Legendre Differential Equation} + \begin{alertblock}{Separation (cont.)}<1-> + The hard part is the ODE for \(\Theta(\vartheta)\): + \[ + \sin^2\vartheta \frac{d^2 \Theta}{d (\cos\vartheta)^2} - 2\cos\theta \frac{d \Theta}{d \cos\vartheta} + + \left[ n(n+1) - \frac{m^2}{\sin^2 \vartheta} \right] \Theta(\cos\vartheta) = 0 + \] + \end{alertblock} + + \uncover<2->{ + Substituting \(x = \cos\vartheta\) and \(y = \Theta\): + } + + \begin{definition}<2->[Associated Legendre Differential Equation] + \[ + \left( 1 - x^2 \right) \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + + \left[ n(n+1) - \frac{m^2}{1 - x^2} \right] y(x) = 0 + \] + \end{definition} +\end{frame} + +\begin{frame}{Legendre Polynomials} + \begin{definition}[Legendre Polynomials] + The polynomials + \begin{align*} + P_n(x) + &= \sum_{k=0}^{\lfloor n/2 \rfloor} + \frac{(-1)^k (2n-2k)!}{2^n k! (n-k)!(n-2k)!} x^{n-2k} \\[1em] + &= {}_2F_1 \left( \begin{matrix} + n + 1, & -n \\ \multicolumn{2}{c}{1} + \end{matrix} ; \frac{1 - x}{2} \right) \\[1em] + &= \frac{1}{n!2^n}\frac{d^n}{dx^n}(x^2-1)^n + \end{align*} + are a solution to the associated Legendre differential equation when \(m = 0\). + \end{definition} +\end{frame} + +\begin{frame} + \centering + \includegraphics[width=\linewidth]{figures/legendre-polynomials} +\end{frame} + +\begin{frame}{Associated Legendre Polynomials} + \begin{lemma} + For \(x \in [-1, 1]\) the polynomials + \[ + P_{m, n} (x) = \left( 1 - x^2 \right)^{m/2} \frac{d^{m}}{dx^{m}} P_n (x) + \] + solve the associated Legendre differential equation. + \end{lemma} + + \begin{alertblock}{Observation}<2-> + If \(m > n\) then \(P_{m, n}(x) = 0\) for all \(x\). + \end{alertblock} +\end{frame} + +\begin{frame} + \centering + \includegraphics[width=\linewidth]{figures/associated-legendre-polynomials} +\end{frame} + + +\begin{frame}{Putting it back together} + \begin{block}{The Problem} + \[ + \nabla^2_s f(\varphi, \vartheta) = \lambda f(\varphi, \vartheta) + \] + \end{block} + \begin{alertblock}{Current solution} + For \(m \in \mathbb{Z}\) and \(m < n\): + \[ + \tilde{Y}_{m, n}(\varphi, \vartheta) + = \Phi(\varphi) \Theta(\vartheta) + = e^{im\varphi} P_{m, n}(\cos\vartheta) + \] + \end{alertblock} +\end{frame} + +\bgroup +\setbeamercolor{background canvas}{bg=black} +\setbeamertemplate{navigation symbols}{} +\begin{frame}{Intuition of conditions for \(m\) and \(n\)} +\end{frame} +\egroup + +\section{Fourier on \(S^2\)} + +\begin{frame}{Basis functions?} + The functions \(\tilde{Y}_{m, n}\) span the space of nice functions \(S^2 \to \mathbb{C}\). + + \begin{definition}<2-> + The inner product of nice functions \(f(\varphi, \vartheta)\) and \(g(\varphi, \vartheta)\) from \(S^2\) to \(\mathbb{C}\) is + \[ + \langle f, g \rangle + = \iint_{S^2} f g^* \, d\Omega + \uncover<3->{ + = \int\limits_0^{2\pi} \int\limits_0^{\pi} + f(\varphi, \vartheta) g^*(\varphi, \vartheta) + \sin\vartheta \, d\vartheta d\varphi + } + \] + \end{definition} +\end{frame} + +\begin{frame}{Orthonormality} + \begin{definition}<1-> + A set of basis functions are \emph{orthonormal} if + \[ + \langle B_{m, n}, B_{m', n'} \rangle = \begin{cases} + 1 & m = m' \wedge n = n' \\ + 0 & \text{else} + \end{cases} + \] + \end{definition} + + \begin{alertblock}{Problem}<2-> + \[ + \langle \tilde{Y}_{m, n}, \tilde{Y}_{m', n'} \rangle = \begin{cases} \displaystyle + \frac{4 \pi}{2n+1} \frac{(n+m)!}{(n-m)!} & m = m' \wedge n = n' \\ + 0 & \text{else} + \end{cases} + \] + \end{alertblock} +\end{frame} + +\begin{frame}{Spherical Harmonics} + \begin{definition}<1-> + The orthonormal spherical harmonics are + \[ + Y_{m, n}(\varphi, \vartheta) = N_{m, n} e^{im\varphi} P_{m, n}(\cos\vartheta) + \] + where the normalisation constant + % FIXME: (-1)^m + \[ + N_{m, n} = \sqrt{\frac{2n+1}{4 \pi} \frac{(n-m)!}{(n+m)!}} + \] + \end{definition} + \begin{alertblock}{Fixed}<1-> + \[ + \langle Y_{m, n}, Y_{m', n'} \rangle = \begin{cases} + 1 & m = m' \wedge n = n' \\ + 0 & \text{else} + \end{cases} + \] + \end{alertblock} +\end{frame} + +\begin{frame}{Fourier Series} + \begin{theorem} + For nice periodic functions on \(S^2\): + \[ + f(\varphi, \vartheta) = \sum_{m \in \mathbb{Z}} \sum_{n \in \mathbb{Z}} + c_{m, n} Y_{m, n} (\varphi, \vartheta) + \] + where + \[ + c_{m, n} = \langle f, Y_{m, n} \rangle. + \] + \end{theorem} +\end{frame} + +\section{Quantum Mechanics} + +\begin{frame}{Linear and Rotational Kinetic Energy} + \begin{columns} + \begin{column}{.5\linewidth} + \begin{block}{Momentum and KE}<1-> + \[ + \vec{p} = m \vec{v}, + \quad + E_k = \frac{\vec{p}^2}{2m} + \] + \end{block} + \begin{alertblock}{QM Formulation}<3-> + \[ + \vec{\hat{p}} = -i\hbar \bm{\nabla}, + \quad + \hat{E}_k = -\frac{\hbar^2}{2m} \nabla^2 + \] + \end{alertblock} + \end{column} + \begin{column}{.5\linewidth}<2-> + \begin{block}{Angular M. and KE} + \[ + \vec{L} = \vec{r}\bm{\times}{\vec{p}}, + \quad + E_{k, a} = \frac{\vec{L}^2}{2m r^2} + \] + \end{block} + \begin{alertblock}{QM Formulation}<4-> + Pretty long derivation yields: + \begin{align*} + % \hat{L}_z &= -i \hbar \frac{\partial}{\partial \varphi}, \\[1em] + \hat{E}_{k, a} &= -\frac{\hbar^2}{2mr^2} \nabla^2_s + \end{align*} + \end{alertblock} + \end{column} + \end{columns} +\end{frame} + + +\bgroup +\setbeamercolor{background canvas}{bg=black} +\setbeamertemplate{navigation symbols}{} +\begin{frame}{Intuition for the Operators} +\end{frame} +\egroup + +\begin{frame}{Schrödinger Equation} + \begin{block}{Time independent SE} + \[ + % hamiltonina + \only<1>{ + \mathrm{\hat{\mathcal{H}}} | \Psi \rangle = E | \Psi \rangle + } + % KE + U + \only<2>{ + \left( + \hat{E}_k + U + \right) | \Psi \rangle = E | \Psi \rangle + } + % KE with p + \only<3>{ + \left( + \frac{\vec{\hat{p}}^2}{2m} + U + \right) | \Psi \rangle = E | \Psi \rangle + } + % KE with p as 1D derivative + \only<4>{ + \text{Meili} \qquad + \left[ + - \frac{\hbar^2}{2m} \frac{d^2}{d x^2} + U(x) + \right] \Psi(x) = E \Psi(x) + } + % KE with p as 3D derivative + \only<5>{ + \text{3D} \qquad + \left[ + - \frac{\hbar^2}{2m} \nabla^2 + U(\vec{x}) + \right] \Psi(\vec{x}) = E \Psi(\vec{x}) + } + % Decompose laplacian + \only<6>{ + \left\{ + - \frac{\hbar^2}{2m} \frac{1}{r^2} \left[ + \nabla^2_s - \frac{\partial}{\partial r} \left( + r^2 \frac{\partial}{\partial r} + \right) + \right] + U(\vec{r}) + \right\} \Psi(\vec{r}) = E \Psi(\vec{r}) + } + % rewrite using L + \only<7>{ + \left[ + \frac{\vec{\hat{L}}^2}{2mr^2} + + \frac{1}{r^2} \frac{\partial}{\partial r} \left( + r^2 \frac{\partial}{\partial r} + \right) + + U(\vec{r}) + \right] \Psi(\vec{r}) = E \Psi(\vec{r}) + } + % rewrite using E_ka + \only<8>{ + \left[ + \hat{E}_{k,a} + + \frac{1}{r^2} \frac{\partial}{\partial r} \left( + r^2 \frac{\partial}{\partial r} + \right) + + U(\vec{r}) + \right] \Psi(\vec{r}) = E \Psi(\vec{r}) + } + % What is KE + \only<9>{ + \Bigg[ + \underbrace{\hat{E}_{k,a} + + \frac{1}{r^2} \frac{\partial}{\partial r} \left( + r^2 \frac{\partial}{\partial r} + \right)}_\text{Kinetic Energy} + + U(\vec{r}) + \Bigg] \Psi(\vec{r}) = E \Psi(\vec{r}) + } + % Introduce E_kr + \only<10>{ + \Bigg[ + \hat{E}_{k,a} + + \underbrace{\frac{1}{r^2} \frac{\partial}{\partial r} \left( + r^2 \frac{\partial}{\partial r} + \right)}_{\text{Radial KE } \hat{E}_{k, r}} + + U(\vec{r}) + \Bigg] \Psi(\vec{r}) = E \Psi(\vec{r}) + } + \only<11->{ + \left\{ + \hat{E}_{k,a} + \hat{E}_{k,r} + U(\vec{r}) + \right\} \Psi(\vec{r}) = E \Psi(\vec{r}) + } + \] + \end{block} + \begin{columns} + \begin{column}{.6\linewidth} + \Large + \uncover<11->{ + \Large + \textit{But why?} \\[2em] + } + + \uncover<12->{ + \bfseries + Hydrogen atom has radial symmetry! + } + \end{column} + \begin{column}{.35\linewidth} + \uncover<11->{ + \includegraphics[width=\linewidth]{figures/hydrogen} + \nocite{depiep_electron_2013} + } + \end{column} + \end{columns} +\end{frame} + +\begin{frame}{Electron Orbitals} +\end{frame} + +% \section{Other applications} + +\begin{frame}{Bibliography} + \renewcommand*{\bibfont}{\tiny} + \printbibliography +\end{frame} + +\end{document} + +% vim:et:ts=2:sw=2:wrap:nolinebreak: |