119 lines
4.9 KiB
TeX
119 lines
4.9 KiB
TeX
\documentclass[english,notitlepage]{revtex4-1} % defines the basic parameters of the document
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%For preview: skriv i terminal: latexmk -pdf -pvc filnavn
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\let\tablenum\relax % to avoid clash with siunitx
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% if you want a single-column, remove reprint
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% allows special characters (including æøå)
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\usepackage[utf8]{inputenc}
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%\usepackage[english]{babel}
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%% note that you may need to download some of these packages manually, it depends on your setup.
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%% I recommend downloading TeXMaker, because it includes a large library of the most common packages.
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\usepackage{amssymb} % mathematical symbols (physics imports amsmath)
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\usepackage{amsmath}
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\usepackage{graphicx} % include graphics such as plots
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\usepackage{xcolor} % set colors
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\usepackage{hyperref} % automagic cross-referencing (this is GODLIKE)
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\usepackage{listings} % display code
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\usepackage{subfigure} % imports a lot of cool and useful figure commands
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\usepackage{float}
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%\usepackage[section]{placeins}
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\usepackage{algorithm}
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\usepackage[noend]{algpseudocode}
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\usepackage{subfigure}
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\usepackage{tikz}
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\usepackage{siunitx}
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\usetikzlibrary{quantikz}
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% defines the color of hyperref objects
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% Blending two colors: blue!80!black = 80% blue and 20% black
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\hypersetup{ % this is just my personal choice, feel free to change things
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colorlinks,
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linkcolor={red!50!black},
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citecolor={blue!50!black},
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urlcolor={blue!80!black}}
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%% Defines the style of the programming listing
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%% This is actually my personal template, go ahead and change stuff if you want
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%% USEFUL LINKS:
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%%
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%% UiO LaTeX guides: https://www.mn.uio.no/ifi/tjenester/it/hjelp/latex/
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%% mathematics: https://en.wikibooks.org/wiki/LaTeX/Mathematics
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%% PHYSICS ! https://mirror.hmc.edu/ctan/macros/latex/contrib/physics/physics.pdf
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%% the basics of Tikz: https://en.wikibooks.org/wiki/LaTeX/PGF/Tikz
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%% all the colors!: https://en.wikibooks.org/wiki/LaTeX/Colors
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%% how to draw tables: https://en.wikibooks.org/wiki/LaTeX/Tables
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%% code listing styles: https://en.wikibooks.org/wiki/LaTeX/Source_Code_Listings
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%% \includegraphics https://en.wikibooks.org/wiki/LaTeX/Importing_Graphics
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%% learn more about figures https://en.wikibooks.org/wiki/LaTeX/Floats,_Figures_and_Captions
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%% automagic bibliography: https://en.wikibooks.org/wiki/LaTeX/Bibliography_Management (this one is kinda difficult the first time)
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%% REVTeX Guide: http://www.physics.csbsju.edu/370/papers/Journal_Style_Manuals/auguide4-1.pdf
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%%
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%% (this document is of class "revtex4-1", the REVTeX Guide explains how the class works)
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%% CREATING THE .pdf FILE USING LINUX IN THE TERMINAL
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%%
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%% [terminal]$ pdflatex template.tex
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%%
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%% Run the command twice, always.
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%% If you want to use \footnote, you need to run these commands (IN THIS SPECIFIC ORDER)
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%%
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%% [terminal]$ pdflatex template.tex
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%% [terminal]$ bibtex template
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%% [terminal]$ pdflatex template.tex
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%% [terminal]$ pdflatex template.tex
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%%
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%% Don't ask me why, I don't know.
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\newcommand{\tderiv}[2]{\frac{\mathrm{d} #1}{\mathrm{d} #2}} % defines a command for total derivatives
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\newcommand{\pderiv}[2]{\frac{\partial #1}{\partial #2}} % defines a command for partial derivatives
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\newcommand{\ptderiv}[3]{\frac{\mathrm{d}^{#3} #1}{\mathrm{d} #2^{#3}}} % defines a command for total derivatives of arbitrary power
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\newcommand{\ppderiv}[3]{\frac{\partial^{#3} #1}{\partial #2^{#3}}} % defines a command for partial derivatives of arbitrary power
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\begin{document}
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\title{Solutions to Project 2} % self-explanatory
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\author{Lars Bogner} % self-explanatory
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\date{\today} % self-explanatory
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\noaffiliation % ignore this, but keep it.
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\maketitle
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\textit{Code hosted at \url{https://github.uio.no/larsbog/FYS4150}}
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\section*{Problem 1}
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Using the rescaling of the positions $\hat x \equiv \frac{x}{L}$ we define the following relations:
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\begin{equation}
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u_x \equiv u, \quad u_{\hat x} \equiv \hat u, \quad \text{with} \quad u(x) = \hat u(\hat x) = u (L \hat x).
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\end{equation}
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Now we can rewrite the equation as
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\begin{equation}
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\begin{aligned}
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\gamma \ptderiv{u(x)}{x}{2} &= - F u(x)\\
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\gamma \ptderiv{u(L \hat x)}{(L \hat x)}{2} &= - F u(L \hat x).
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\end{aligned}
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\end{equation}
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In this step we simply replaced $x$ by the equivalent expression $L \hat x$. Now we can use the chain rule to rewrite the left hand side:
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\begin{equation}
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\gamma \ppderiv{u(L \hat x)}{\hat x}{2} \underbrace{\ppderiv{\hat x}{(L \hat x)}{2}}_{=L^{-2}} = - F u(L \hat x).
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\end{equation}
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Rearranging the factors gives the final result:
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\begin{equation}
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\ptderiv{\hat u (\hat x)}{\hat x}{2} = - \frac{F L^2}{\gamma} \hat u(\hat x) \equiv - \lambda \hat u(\hat x).
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\end{equation}
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\section*{Problem 2}
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\bibliography{include/citations}
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\end{document}
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