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\begin{center}
\textbf{Application of time-independent Schr$\ddot{o}$dinger's equation}
\end{center}
 \section*{Potential Barrier}
 
\begin{figure}[h]
 \centering
    \includegraphics[width=1\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}\\
                                 \psi_2&=&Fe^{-k_2x}+Ge^{-ik_2x}\\
                                 \psi_3&=&Ce^{ikx}
                                \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*}

{\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 $T+R=1$.

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