\documentclass[11pt,fleqn]{article}

% Page setup
\topmargin -1.5cm      
\oddsidemargin -0.04cm   
\evensidemargin -0.04cm  
\textwidth 16.59cm
\textheight 24cm 
\setlength{\parindent}{0cm} 
\setlength{\parskip}{0cm} 

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

% 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{E712: Transitions between energy levels due to wall movment} 
\auname{Ofek Asban and Yuval Shilon}


{\bf The problem:}

\Dn

A particle with mass M in 1D box ${x\in [0,a]}$ with length a. 

At time ${t=-\infty}$ the particle is in ground state.

The movment of the wall is ${x=-\epsilon(t)}$.

The speed in which the wall is moving is  ${\dot{\epsilon}=\frac{\epsilon_0}{\sqrt{2\pi}\tau_0}\exp[-\frac{1}{2}(\frac{t}{\tau_0})^2]}$.

You may assume that the displacement of the wall is small so that the perturbation is
\Dn

(*)${<\varphi|V|\psi>=\mp(\frac{\epsilon}{2M})[\frac{\partial \varphi}{\partial x}][\frac{\partial \psi}{\partial x}]}$.
\Dn

- The derivatives are calculated at the point where the wall is located.

- The sign +/- refer to the case of enlargement/reduction (respectively) of the box's length.

\Dn

(1) Find the probability ${P(n)}$,at time ${t=\infty}$, to find the particle at an excited 

    energy level n for abrupt displacement.

(2) Using first order Perturbation theory, find the probability ${P(n)}$ for finite displacement.

(3) Answer paragraph (2) only this time both of the walls are displaced outward in the next fashion [${-\epsilon(t)}$, ${a+\epsilon(t)}$].

(4) Define two different conditions that can ensure the Validity of the calculation of first order Perturbation theory.

    -Small displacement of the wall (small ${\epsilon_0}$) or a slow movment of the wall (large   ${\tau_0)}$.

(5) Write the condition needed for the process to be adiabatic and note if the adiabatic area is contained within the validity area of the Perturbation theory.  
    
    
\Dn\Dn


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

\Dn

(1)
For 1D potential well the energy levels and wave function are:
\Dn

  ${E_n=\frac{\pi^2}{2Ma^2}n^2}$ , ${\psi_n=\sqrt{\frac{2}{a}}sin(\frac{\pi nx}{a})}$.
\Dn

The probability of a particle to pass from energy level m to evergy level n is:
\Dn

  ${P(n|m)=\frac{|V_n_m|^2}{(E_n-E_m)^2}}$, for ${n\ne1}$.
\Dn

using equation (*) we see that :
\Dn

  ${V_n_1=\frac{-\epsilon(t) \pi^2}{Ma^3}n}$.
\Dn

the overall displecment of the wall is
\Dn

  ${\int\limits_{-\infty}^{\infty}\dot{\epsilon}(t)dt=\frac{\epsilon_0}{\sqrt{2\pi}\tau_0}\int\limits_{-\infty}^{\infty}e^{-\frac{1}{2}(\frac{t}{\tau_0})^2}dt=\epsilon_0}$
\Dn

thus at ${t=\infty}$
\Dn

  ${V_n_1=\frac{-\epsilon_0 \pi^2}{Ma^3}n}$.
\Dn

In our case 
\Dn

  ${P(n|1)=\frac{|V_n_1|^2}{(E_n-E_1)^2}}$
\Dn

  ${E_n-E_1=\frac{\pi^2}{2Ma^2}(n^2-1)}$,
\Dn

Therefore
\Dn

  ${P_0(n)=P(n|1)=4\frac{\epsilon_0}{a^2}(\frac{n}{n^2-1})^2}$ for ${n\ne1}$.

%%%%%%%%%%%%%%%
\Dn\Dn

(2) using the formula for the transition probability:
\Dn

 ${P(n)=|w_n_m|^2|FT(f(t))|^2}&
\Dn

${|w_n_1|=\frac{\pi^2}{Ma^3}n}&
\Dn

${FT(f(t))=FT(\epsilon}(t))=\int\limits_{-\infty}^{\infty}\epsilon}(t)e^{-iw_nt}dt=\frac{\epsilon(t)e^{-iw_nt}}{iw_nt}\mid_{-\infty}^{\infty}+\frac{1}{iw_n}\int\limits_{-\infty}^{\infty}\dot{\epsilon}(t)\e^{-iw_nt}
\Dn

where : 
\Dn

&{w_n=E_n-E_1}$
\Dn

the first term vanishes (${\epsilon(-\infty)=0}$) and from sloving the integral by parts we get:
\Dn

${\frac{\epsilon_0}{\sqrt{2\pi}\tau_0}\int\limits_{-\infty}^{\infty}e^{-\frac{1}{2}(\frac{t}{\tau_0})^2}e^{-iw_nt}dt=\frac{\epsilon_0}{\sqrt{2\pi}\tau_0}e^-\frac{(\tau_0w_n)^2}{2}\int\limits_{-\infty}^{\infty}
e^{-\frac{(t+iw_n\tau_0^2)^2}{2\tau_0^2}}dt=\epsilon_0}e^-\frac{(\tau_0w_n)^2}{2}}$
\Dn

so finaly we get:
\Dn

${FT(\epsilon}(t))=\frac{1}{iw_n}\epsilon_0e^{-\frac{(\tau_0w_n)^2}{2}}$
\Dn

${P(n)=\frac{\epsilon_0^2|w_n_1|^2}{w_n^2}e^{-(\tau_0w_n)^2}=P_0(n)e^{-(\tau_0w_n)^2
\Dn\Dn
 
(3) using (*) on both walls we get:
\Dn

${V_n_1=\frac{-\epsilon(t) \pi^2}{Ma^3}n+\frac{-\epsilon(t) \pi^2}{Ma^3}n(-1)^n=-2\epsilon(t)\frac{\pi^2}{Ma^3}n}$ 
\Dn

n=3,5,7...
\Dn

from here it is easy to see that by following the same prosses as in the last section we will get:
\Dn

${P`(n)=4P(n)}$ ,for n=3,5,7...
\Dn

${P`(n)=0}$ , for  n=2,4,6...
\Dn\Dn

(4) in order for the perturbation theory to be valid P(n) has to be small ,from here we get the conditions:
\Dn

${\epsilon_0<<a}$
\Dn

${\tau_0w_n>>1\Rightarrow\tau_0>>ma^2}$

\Dn\Dn

(5) the adiabatic condition is : 
\Dn

${\dot{\epsilon}<<\frac{\triangle^2}{|w|}}$

\Dn

${|w|=\frac{1}{ma}K_nK_m=\frac{m}{a}V_E^2}$

\Dn

${\triangle=V_E\triangle p=\frac{\pi}{a}V_E}$ , ${(\pi\simeq1)}$
\Dn

where we have used the aproximations:
\Dn
 
${K_n\simeq K_m\simeq K_E=mV_E}$ 
\Dn

${\Rightarrow\dot{\epsilon}<<\frac{1}{ma}
\Dn

and substituting the expression for ${\dot{\epsilon}(t)}$ we get:
\Dn

${\epsilon_0<<\frac{1}{ma}\tau_0}&
\Dn

which is contained within the validity area of the Perturbation theory




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