\documentclass{article}[12pt]
\topmargin=0cm
\textwidth=16cm
\textheight=26cm
\oddsidemargin=0mm
\evensidemargin=0mm
\tolerance=10000
\usepackage{latexsym,amssymb,amsfonts,epsfig,graphicx,color}
%\usepackage{amssymb}
%\usepackage{amsfonts}
\usepackage{bm,exscale, amsmath, mathrsfs}
\usepackage{amsfonts, amssymb}
\pagestyle{empty}
\begin{document}

\begin{center}
\textbf{Application of time-independent Schr$\ddot{o}$dinger's equation}
\end{center}
 \section*{Potential Barrier}
 
\begin{figure}[h]
 \centering
    \includegraphics[width=.6\textwidth]{sqp}
     \end{figure}

{ \bf Potential:} \begin{align*}V(x) = \begin{array}{ll} 
                    V_0 & \text{if } 0<x<a\\ 
                    0 & \text{otherwise}
                                \end{array}\end{align*}
                                
                               { \bf Solutions:}
                                \begin{eqnarray*}
                                 \psi_1&=&A e^{ikx}+Be^{-ikx},\,\qquad region\, I,\, x\leq0 \\
                                 \psi_2&=&Fe^{k_2x}+Ge^{-k_2x}, \qquad region\, II, \,0<x<a\\
                                 \psi_3&=&Ce^{ikx},\qquad region\, III,\, x\geq a
                                \end{eqnarray*}
                                
Here $A$ is incident wave in region I, $B$ is transmitted wave from barrier at $x=0$, $C$ is transmitted wave in region III ($x>a$). $F$ and $G$ are the wave transmitted and reflected at $x=a$ respectively in region II.                                 
\begin{equation*}
 k^2=\frac{2mE}{\hbar^2};\qquad k_2=\frac{\sqrt{2m(V_0-E)}}{\hbar},
\end{equation*}
and we are dealing with the case when $E<V_0$.

{\bf Boundary conditions:}

$\psi_1, \psi_1^{'}, \psi_2, \psi_2^{'}$ and $\psi_3, \psi_3^{'}$ are continuous at $x=0$ and $x=a$. These will give rise four equations of 5 unknowns $A, B, C, F, G$. Solve these and write B, F, C, G in terms of A. 

The transmission co-efficient is given by

\[  T=\frac{C^{*}C}{A^{*}A}=\left[1 + \frac{(e^{k_2a}-e^{-k_2a})^2}{16\left(\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)\right)}\right]^{-1}\]

For large $k_2a$ or for $k_2a>>1$ one can write :

\[T\backsimeq 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)e^{-2k_2a}\]

There exists a finite probability for the particle to tunnel through the barrier!

The reflection co-efficient is defined as :

\[R=\frac{B^{*}B}{A^{*}A}\]

Check $T+R=1$.

\end{document}