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\begin{enumerate}
\item A particle of mass $m$ in a rigid, one dimensional box, can move freely between $-L/2$ to $L/2$ at constant speed. The boundaries of the box are at $x=-L/2$ and $x=L/2$. The region $x<-L/2$ and $x>L/2$ are forbidden. Plot the wave function $\psi(x)$ i.e. $\psi (x)$ vs $x$ and $|\psi(x)|^{2}$ i.e. $|\psi(x)|^{2}$ vs $x$ for $n=1,2,3$. 
\vspace{2cm}

\item Consider the Schr\"{o}dinger equation for an electron confined to a two dimensional box, $0<x<a$ and $0<y<b$. Potential inside the box is zero i.e. $V(x,y)=0$. This 2-D box is non-penetrable. The wave function to this problem is given below-
$$ \psi(x,y)=N\sin(\frac{n_{x}\pi x}{a})\sin(\frac{n_{y}\pi y}{b}) $$
Find $N$ and establish the expression for allowed values of the total energy?
\vspace{2cm}
\item Show that for a one dimensional bound particle ($\psi$ need not be a stationary state)-
$$ \frac{d}{dt}\int_{-\infty}^{\infty}\psi^{*}(x,t)\psi(x,t)dx=0$$
After establishing the above relation show that if the particle is in a stationary state at a given time $t_{0}$, then it will always remain in a stationary state.
\vspace{2cm}
\item Normalised wave-function of a particle is,
\begin{align*}
\psi(x)=& 2\alpha\sqrt{\alpha}xe^{-\alpha x} \qquad for \,\,x>0\\
=&0 \qquad\qquad\qquad\quad for\,\, x<0
\end{align*}
Calculate $<x>$ and $<x^{2}>$. Estimate the probability that the particle is found between $x=0$ and $x=\frac{1}{\alpha}$. Consider the limit from $0$ to $\frac{1}{\alpha}$.
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\item At time $t=0$ a particle in the potential $V(x)=\frac{1}{2}m\omega^{2}x^{2}$ is described by the wave function,
\begin{center}
$ \psi(x,0)=A\Sigma_{n}(\frac{1}{\sqrt{2}})^{n}\psi_{n}(x) $
\end{center}
Where $\psi_{n}(x)$ are eigenstates of the energy with eigenvalues $E_{n}=(n+\frac{1}{2})\hbar\omega$. You are given that ($\psi_{n},\psi_{n'})=\delta_{nn'}$. Find the normalisation constant $A$. Show that $|\psi(x,t)|^2$ is a periodic function of time. Also find the average energy at $t=0$.
\end{enumerate}



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