\documentclass{article}[11pt]
\topmargin=-1cm
\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}

\begin{document}
 

\begin{enumerate}

 \item Show that $<p>$ is always zero for stationary states. What is the standard deviation of Hamiltonian (energy) in a stationary state? interpret your answer. 
 
 \vspace{3cm}
 
 \item Show that if $V(x)$ is an even function then every solution of time-independent Schr\"{o}dinger's equation can be expressed as even or odd function. What is a stationary state? Expectation value of $\hat{H}$ is time independent in stationary state, true or false?
 
 \vspace{3cm}
 
 \item Show that $\frac{d}{dx}$ is not a Hermitian operator.  Show that particle in an one dimensional box possesses orthonormal energy eigenfunctions. What is the expectation value of $<p^2>$ in $n$th state? \vspace{3cm}
 

  
 \item Show that no acceptable solution exist to Schr\"{o}dinger equation of the particle in a box problem with energy $E=0$ and $E<0$. What is the expectation value of $x$ in the $n$th stationary state?
 
 \vspace{3cm}
 
 \item Let the initial wave function of a particle in a box is the superposition of first two stationary states $\Psi(x,0)=A[\psi_1(x)+\psi_2(x)]$. Find $A$. Find $\Psi(x,t)$. Find $<x>$ and show it oscillates. 

\vspace{3cm}
\item For 1-d particle in a box, find out the uncertainty in position and momentum in the $n$th stationary state. In which state the uncertainty is the minimum? 
\end{enumerate}

\end{document}