\documentclass[english,notitlepage]{revtex4-1}  % defines the basic parameters of the document
%For preview: skriv i terminal: latexmk -pdf -pvc filnavn
\let\tablenum\relax % to avoid clash with siunitx


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% allows special characters (including æøå)
\usepackage[utf8]{inputenc}
%\usepackage[english]{babel}

%% note that you may need to download some of these packages manually, it depends on your setup.
%% I recommend downloading TeXMaker, because it includes a large library of the most common packages.

\usepackage{amssymb}  % mathematical symbols (physics imports amsmath)
\usepackage{amsmath}
\usepackage{graphicx}         % include graphics such as plots
\usepackage{xcolor}           % set colors
\usepackage{hyperref}         % automagic cross-referencing (this is GODLIKE)
\usepackage{listings}         % display code
\usepackage{subfigure}        % imports a lot of cool and useful figure commands
\usepackage{float}
%\usepackage[section]{placeins}
\usepackage{algorithm}
\usepackage[noend]{algpseudocode}
\usepackage{subfigure}
\usepackage{tikz}
\usepackage{siunitx}
\usetikzlibrary{quantikz}
% defines the color of hyperref objects
% Blending two colors:  blue!80!black  =  80% blue and 20% black
\hypersetup{ % this is just my personal choice, feel free to change things
    colorlinks,
    linkcolor={red!50!black},
    citecolor={blue!50!black},
    urlcolor={blue!80!black}}

%% Defines the style of the programming listing
%% This is actually my personal template, go ahead and change stuff if you want



%% USEFUL LINKS:
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%%   UiO LaTeX guides:        https://www.mn.uio.no/ifi/tjenester/it/hjelp/latex/
%%   mathematics:             https://en.wikibooks.org/wiki/LaTeX/Mathematics

%%   PHYSICS !                https://mirror.hmc.edu/ctan/macros/latex/contrib/physics/physics.pdf

%%   the basics of Tikz:       https://en.wikibooks.org/wiki/LaTeX/PGF/Tikz
%%   all the colors!:          https://en.wikibooks.org/wiki/LaTeX/Colors
%%   how to draw tables:       https://en.wikibooks.org/wiki/LaTeX/Tables
%%   code listing styles:      https://en.wikibooks.org/wiki/LaTeX/Source_Code_Listings
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%%   learn more about figures  https://en.wikibooks.org/wiki/LaTeX/Floats,_Figures_and_Captions
%%   automagic bibliography:   https://en.wikibooks.org/wiki/LaTeX/Bibliography_Management  (this one is kinda difficult the first time)
%%   REVTeX Guide:             http://www.physics.csbsju.edu/370/papers/Journal_Style_Manuals/auguide4-1.pdf
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%% CREATING THE .pdf FILE USING LINUX IN THE TERMINAL
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%% Don't ask me why, I don't know.

\newcommand{\tderiv}[2]{\frac{\mathrm{d} #1}{\mathrm{d} #2}} % defines a command for total derivatives
\newcommand{\pderiv}[2]{\frac{\partial #1}{\partial #2}} % defines a command for partial derivatives
\newcommand{\ptderiv}[3]{\frac{\mathrm{d}^{#3} #1}{\mathrm{d} #2^{#3}}} % defines a command for total derivatives of arbitrary power
\newcommand{\ppderiv}[3]{\frac{\partial^{#3} #1}{\partial #2^{#3}}} % defines a command for partial derivatives of arbitrary power

\begin{document}

\title{Solutions to Project 2}      % self-explanatory
\author{Lars Bogner}          % self-explanatory
\date{\today}                             % self-explanatory
\noaffiliation                            % ignore this, but keep it.


\maketitle 
    
\textit{Code hosted at \url{https://github.uio.no/larsbog/FYS4150}}
    
\section*{Problem 1}
Using the rescaling of the positions $\hat x \equiv \frac{x}{L}$ we define the following relations:
\begin{equation}
    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). 
\end{equation}
Now we can rewrite the equation as
\begin{equation}
    \begin{aligned}
        \gamma \ptderiv{u(x)}{x}{2} &= - F u(x)\\
        \gamma \ptderiv{u(L \hat x)}{(L \hat x)}{2} &= - F u(L \hat x).
        \end{aligned}
\end{equation}
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:
\begin{equation}
        \gamma \ppderiv{u(L \hat x)}{\hat x}{2} \underbrace{\ppderiv{\hat x}{(L \hat x)}{2}}_{=L^{-2}} = - F u(L \hat x).
\end{equation}
Rearranging the factors gives the final result:
\begin{equation}
        \ptderiv{\hat u (\hat x)}{\hat x}{2} = - \frac{F L^2}{\gamma} \hat u(\hat x) \equiv - \lambda \hat u(\hat x).
\end{equation}

\section*{Problem 2}


\bibliography{include/citations}
\end{document}