\documentclass[onecolumn,fleqn]{revtex4}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%% Please use this template.
%%% Edit it using e.g. Notepad
%%% Ignore the header (do not change it)
%%% Process the file in the Latex site 
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% Special 
\usepackage{ifthen}
\usepackage{ifpdf}

% Fonts
\usepackage{latexsym}
\usepackage{amsmath} 
\usepackage{amssymb} 
\usepackage{bm}

\ifpdf
\usepackage{graphicx}
\usepackage{epstopdf}
\newcommand{\rang}{90}
\else
\usepackage{graphicx}
\usepackage{epsfig}
\newcommand{\rang}{0}
\fi



% Page setup
\setlength{\parindent}{0cm} 
\setlength{\parskip}{0cm} 


% Math symbols I
\newcommand{\sinc}{\mbox{sinc}}
\newcommand{\const}{\mbox{const}}
\newcommand{\trc}{\mbox{trace}}
\newcommand{\intt}{\int\!\!\!\!\int }
\newcommand{\ointt}{\int\!\!\!\!\int\!\!\!\!\!\circ\ }
\newcommand{\ar}{\mathsf r}
\newcommand{\im}{\mbox{Im}}
\newcommand{\re}{\mbox{Re}}

% Math symbols II
\newcommand{\eexp}{\mbox{e}^}
\newcommand{\bra}{\left\langle}
\newcommand{\ket}{\right\rangle}

% Mass symbol
\newcommand{\mass}{\mathsf{m}} 
\newcommand{\Mass}{\mathsf{M}} 

% More math commands
\newcommand{\tbox}[1]{\mbox{\tiny #1}}
\newcommand{\bmsf}[1]{\bm{\mathsf{#1}}} 
\newcommand{\amatrix}[1]{\begin{matrix} #1 \end{matrix}} 
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}

% Other commands
\newcommand{\hide}[1]{}
\newcommand{\drawline}{\begin{picture}(500,1)\line(1,0){500}\end{picture}}
\newcommand{\bitem}{$\bullet$ \ \ \ }
\newcommand{\Cn}[1]{\begin{center} #1 \end{center}}
\newcommand{\mpg}[2][1.0\hsize]{\begin{minipage}[b]{#1}{#2}\end{minipage}}
\newcommand{\Dn}{\vspace*{3mm}}

% Figures
\newcommand{\putgraph}[2][0.30\hsize]{\includegraphics[width=#1]{#2}}

% heading
\newcommand{\heading}[1]{\begin{center} \Large {#1} \end{center}}
\newcommand{\auname}[1]{\begin{center} \bf Submitted by: #1 \end{center}}


\begin{document} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\heading{E604: First Order Perturbation Theory for an one dimension box with a displacing wall} 
\auname{Liana Diesendruck}


{\bf The problem:}

\Dn

A particle is placed in a one dimensional box of length L, such that ${0 < x < L}$.
The purpose of this problem is to find the first order correction for the particle's energies, when we have a ${dL}$ displacement of the wall, using the solution of the previous exercise and afterwards compare it to the exact solution. 

\Dn


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf The solution:}

\Dn
(1)From the previous exercise, the perturbed Hamiltonian can be writen as
\[
	\mathcal{H}= {{H}_{0}}+{V}dL 
\]

where  ${{V_{nm}}=-{\frac{{\pi^{2}}{{\hbar}^{2}}}{ML^{3}} }nm}$.
\newline
\newline
For the first order perturbation theory, we have
\[
	{E_{n}}={E^{(0)}}+{E^{(1)}}dL+{O({{dL}^2})}
\]
where ${{E^{(0)}}={\frac{{\pi^{2}}{\hbar^{2}}}{2M{{L}^{2}}}}{n^{2}}}$ is the eigenvalue of the unperturbed Hamiltonian of a particle in a one dimensional box.
\newline
\newline
The first order correction of the energy is given by

\[
	{E^{(1)}}={-{\frac{\pi^{2}{\hbar^{2}}}{ML^{3}}}{n^{2}}}
\]

Therefore

\[	{E_{n}}={{\frac{{\pi^{2}}{\hbar^{2}}{n^{2}}}{2M{{L}^{2}}}}\left({1-{\frac{2dL}{L}}}\right)}+{O({{dL}^2})}
\]

%%%%%%%%%%%%%%%
\Dn
(2)The exact solution for the energy levels a particle in an one dimensional box is 
\[	
{{E_{n}}}={\frac{{\pi^{2}}{\hbar^{2}}{n^{2}}}{2M{{L}^{2}}}}
\]
As we have ${L\rightarrow{L}+dL}$, we use Taylor expansion around ${dL=0}$
\[	
{{{\left({L}+dL\right)}^{-2}}}={\frac{1}{{L}^2}}-{\frac{2dL}{{L}^3}}+{O({{dL}^2})}
\]

We receive that
\[
{E_{n}}={{\frac{{\pi^{2}}{\hbar^{2}}{n^{2}}}{2M{{L}^{2}}}}\left({1-{\frac{2dL}{L}}}\right)}+{O({{dL}^2})}
\]

As one can see, we receive the same solution in both cases.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document} 

