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\begin{document}

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\begin{center}
\large{\textbf{Indian Institute of Information Technology Allahabad}}
\end{center}
\vspace{-24pt}
\begin{center}
\large{\textbf{Recapitulation Test on Quantum Mechanics}}
\end{center}

\hfill \footnotesize{Date: 17/09/2018}
%\vspace{0.05cm}
\vspace{-15pt}

\noindent
\hrulefill
\vspace{-16pt}
\begin{center}
\footnotesize{B.Tech. IT, ECE - Semester I}
\end{center}
\vspace{-20pt}

\begin{center}
\footnotesize{Paper: Engineering Physics}
\end{center}
\vspace{-20pt}

%\begin{center}
%\footnotesize{Paper Setter: Abdullah Bin Abu Baker \& Sumit Kumar Upadhyay}
%\end{center}
\vspace{-10pt}
\noindent
\footnotesize{Max Marks: 15} \hfill \small{Duration: 1 hour}

\noindent
\footnotesize{Attempt \textbf{all} the questions. There is \textbf{no credit} for an answer if proper justification is not given, even if the answer is correct. Notations are standard.}

\noindent
\hrule

\singlespacing
\thispagestyle{empty}
\begin{enumerate}

 \item An electron is described by the wave function \hfill[2+4+3+6]\\
 \begin{center}
 $\psi(x) = \left\{\begin{array}{ll} 
                    0 & \text{if }x < 0\\ 
                    C e^{-x}(1- e^{-x}) & \text{if }x \geq 0
            \end{array}\right. ,$ \end{center}
            where $C$ is a constant and $x$ is measured in nm.
    \begin{enumerate}
    \item Determine the value of $C$ that normalizes $\psi(x)$. 
    
    \item Where is the electron most likely to be found? That is, for what value of $x$ is the probability of finding the electron the largest? Plot the wave function.
    
    \item Calculate the average position $<x>$ for the electron. Compare this result with the most likely position, and comment on the difference. Calculate $<x^2>$ and then the dispersion $(\Delta x)^2=<x^2> - <x>^2$.
    
    \item Calculate the dispersion in momentum and verify whether the given system satisfies the Heisenberg's uncertainty principle or not. 
\end{enumerate}

\end{enumerate}
\vspace{2.5cm}
\parskip = 8pt

\begin{center}
\large{\textbf{Indian Institute of Information Technology Allahabad}}
\end{center}
\vspace{-24pt}
\begin{center}
\large{\textbf{Recapitulation Test on Quantum Mechanics}}
\end{center}

\hfill \footnotesize{Date: 17/09/2018}
%\vspace{0.05cm}
\vspace{-15pt}

\noindent
\hrulefill
\vspace{-16pt}
\begin{center}
\footnotesize{B.Tech. IT, ECE - Semester I}
\end{center}
\vspace{-20pt}

\begin{center}
\footnotesize{Paper: Engineering Physics}
\end{center}
\vspace{-20pt}

%\begin{center}
%\footnotesize{Paper Setter: Abdullah Bin Abu Baker \& Sumit Kumar Upadhyay}
%\end{center}
\vspace{-10pt}
\noindent
\footnotesize{Max Marks: 15} \hfill \small{Duration: 1 hour}

\noindent
\footnotesize{Attempt \textbf{all} the questions. There is \textbf{no credit} for an answer if proper justification is not given, even if the answer is correct. Notations are standard.}

\noindent
\hrule

\singlespacing
\thispagestyle{empty}
\begin{enumerate}

 \item An electron is described by the wave function \hfill[2+4+3+6]\\
 \begin{center}
 $\psi(x) = \left\{\begin{array}{ll} 
                    0 & \text{if }x < 0\\ 
                    C e^{-x}(1- e^{-x}) & \text{if }x \geq 0
            \end{array}\right. ,$ \end{center}
            where $C$ is a constant and $x$ is measured in nm.
    \begin{enumerate}
    \item Determine the value of $C$ that normalizes $\psi(x)$. 
    
    \item Where is the electron most likely to be found? That is, for what value of $x$ is the probability of finding the electron the largest? Plot the wave function.
    
    \item Calculate the average position $<x>$ for the electron. Compare this result with the most likely position, and comment on the difference. Calculate $<x^2>$ and then the dispersion $(\Delta x)^2=<x^2> - <x>^2$.
    
    \item Calculate the dispersion in momentum and verify whether the given system satisfies the Heisenberg's uncertainty principle or not. 
\end{enumerate}

\end{enumerate}
\vspace{2.5cm}
%\vspace{3.5cm}
\end{document}