\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{E106: Base translation} 
\auname{Harel Frish}


{\bf The problem:}

\Dn
Given two different bases:
\[
	\mathcal{ \mathrm{\left\{|{{\eta}_{i}}\rangle\right\}}\hspace{1em}  \mathrm{and}\hspace{1em} \mathrm{\left\{|{{\zeta}_{i}}\rangle\right\}}}
\]

with the relation:
\[
	\mathcal{\mathrm{|{{\eta}_{1}}\rangle} = \mathrm{a|{{\zeta }_{1}}\rangle}+ \mathrm{b|{{ \zeta }_{2}}\rangle}
	\hspace{1em} \mathrm{and} \hspace{1em} \mathrm{|{{\eta}_{2}}\rangle} = \mathrm{a|{{\zeta }_{1}}\rangle}- \mathrm{b|{{ \zeta }_{2}}\rangle}}
\]

(1) What should be the condition on a and b so that the base will be ortonormaly.
     
\Dn       
     
(2) Express this $ |{\mathrm{a}}_{{1}}\rangle  = 3|  $$ {\mathrm{\zeta}}_{{1}}\rangle + 7 | {\mathrm{\zeta}}_{{2}}\rangle\hspace{0.5em}  $ relation in the $\hspace{0.5em}|$$ {\mathrm{\eta}}_{{i}}\rangle \hspace{0.5em} $ base.


\Dn\Dn


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

\Dn

(1) 
\[
	\mathcal{\mathrm{|{{\eta}_{1}}\rangle} = \mathrm{a|{{\zeta }_{1}}\rangle}+ \mathrm{b|{{ \zeta }_{2}}\rangle}}
\]
\[
	\mathcal{\mathrm{|{{\eta}_{2}}\rangle} = \mathrm{a|{{\zeta }_{1}}\rangle}- \mathrm{b|{{ \zeta }_{2}}\rangle}}
\]

\Dn

We will suppose that $ {|{\mathrm{\zeta}}_{{1}}\rangle} \hspace{0.5em}\mathrm{and}\hspace{0.5em} {|{\mathrm{\zeta}}_{{2}}\rangle}
\hspace{0.5em} $are unit vectors (it's the base) so:

\[
|\eta_{\alpha}\rangle =\sum_j T_{j\alpha}|\zeta_{j}\rangle 
\]


	

means: 

\[
T_{j,\alpha} = \langle \zeta_{j}|\eta_{\alpha}\rangle = \left( \begin{matrix} {a} & {a}\\
            {b} & {-b} \end{matrix} \right)  
\]

\Dn

For vector $\vec{\eta_{1}}$ and $\vec{\eta_{2}}$ to be ortonormaly, we should 

demand that $T^{+}T=I$.

\Dn

From this condition we get:

\[\left|a\right|^2=\frac{1}{2}\]
\[\left|b\right|^2=\frac{1}{2}\]


(2)
The transformation matrix is:
\Dn

\[
S=T^{-1}=T^{\dag}=\left( \begin{matrix} {a^{*}} & {b^{*}}\\
            {a^{*}} & {-b^{*}} \end{matrix} \right)  
\]

The coefficient vector of $|a_{1}\rangle $ is:
\[
\Psi = \left( \begin{matrix} {3} \\ {7} \end{matrix} \right) 
\]

Then the coefficient vector in the new basis will be:
\[
\tilde{\Psi}=S\Psi=\left( \begin{matrix} {3a^{*}+7b^{*}} \\ {3a^{*}-7b^{*}} \end{matrix} \right)
\]





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