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\begin{document}
\begin{center}
{\bf QM I: Homework 2}\\
Please hand in {\bf before} Friday, 27 November, 12 noon in the departmental
office, H floor
\end{center}

\Example
a) Calculate the expectation value of $x$ and $x^2$ in the normalised 
wave function\hspace*{\fill}[10 Marks]
\be
\phi(x) = \sqrt{\half} e^{-|x|/2}
\ee
{\bf Hint:}
\be |x| = \left\{ \begin{array}{l} -x\qquad\textrm{if}\ x<0\\
x\qquad\textrm{if}\ x>0
\end{array}\right.\quad.
\ee
b) 
Sketch the function $\phi$, its derivative and the second derivative.
How do they behave
at zero? 
Calculate the derivative of this wave function. 
What conclusion can you draw about the behaviour of the
potential in the Hamiltonian of which $\phi(x)$ is an eigenfunction?\hspace*{\fill}[10 Marks]\\
c) Argue that the potential $V(x)$ in the Schr\"odinger
equation must be proportional to $\delta(x-0)$.\hspace*{\fill}[5 Marks]\\
{\bf Bonus marks:} Find the proportionality constant.\hspace*{\fill}[5 Marks]\\

\newpage
\Example
We study the reflection and transmission coefficients for a square barrier
extending from $x=-a$ to $x=a$, with height $V_0$. The Schr\"odinger equation
is
\be
-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \phi(x) + V(x)\phi(x) = E \phi(x),
\ee
with the potential\\
\be V(x) =\left\{ \begin{array}{lll} 
0 & x < -a &{\rm (region\ I)}\\
V_0 & -a < x < a &{\rm (region\ II)}\\
0 & x> a&{\rm (region\ III)}
\end{array}\right. .
\ee
\begin{figure}[htb]
\begin{center}\includegraphics[width=5cm]{Figures/square_barrier.eps}
\end{center}
\caption{The square barrier.}
\end{figure}


 Assume that the energy $E$
is {\em larger} than the height of the barrier.\\
a) What are the wave functions?\hspace*{\fill}[9 Marks]\\
b) What are the matching conditions?\hspace*{\fill}[5 Marks]\\
c) What is their explicit form?\hspace*{\fill}[5 Marks]\\
d) Sketch a strategy to find the solution for the transmission coefficient.\hspace*{\fill}[6 Marks]\\[\fill]
\hspace*{\fill}P.T.O.
\newpage
\Example
Consider the Schr\"odinger equation
\be
-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \phi(x) + V(x)\phi(x) = E \phi(x),
\ee
with the potential\\
\be V(x) =\left\{ \begin{array}{lll} 
0 & x < -a &{\rm (region\ I)}\\
-V_0 & -a < x < 0 &{\rm (region\ II)}\\
\infty & x> 0&{\rm (region\ III)}
\end{array}\right. .
\ee
%\end{minipage}
%\\ \begin{minipage}[b]{6cm}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=5cm]{Figures/half_hard.eps}
\end{center}
\caption{The potential.}
\end{figure}
%\end{minipage}\\

\begin{enumerate}
\item[a)] Assume $-V_0<E<0$, where we look for bound (normalisable) states.
 What is the form of the wave function in
regions I, II? Explain why the wave function in region III is zero.
{}[5 Marks]
\item[b)] What are the (continuity) conditions satisfied by the wave function?
Show that the wave function in region II can be written in the form
$\phi(x)=A_2 \sin(\kappa x)$. Find an implicit relation defining the
eigen energies.
{}[8 Marks]
\item[c)] Assume $E>0$. What are the values of the transmission and reflection
coefficients for a plane wave coming in from the the left?
{}[2 Marks]
\item[d)] For $E>0$ we send in an incoming plane wave, $\phi(x)=\exp(ikx)$.
Assume that the reflected wave can be written as $\phi_R(x)=A\exp(-ikx))$,
and evaluate $A$ as a function of $V_0$ and $E$.
{}[10 Marks]
%\item Sketch the function $\delta$ as a function of $E/V_0$ for
%$(2m V_0)a^2 = \hbar^2$.
%{}[ Marks]
\end{enumerate}

%\noindent{\Large \bf 4.}
%Use the so-called ladder operators
%\be
%\hat a = \frac{1}{\sqrt{2}} \left(y+\frac{d}{dy}\right),\;\;\;
%{\hat a}^\dagger = \frac{1}{\sqrt{2}} \left(y-\frac{d}{dy}\right).
%\ee
%\begin{enumerate}
%\item[a)]Show that 
%\be y^2 = \half \left(\left({\hat a}^\dagger\right)^2+{\hat a}^2
%+2 {\hat a}^\dagger \hat a +1 \right). \ee
%\hspace*{\fill}[8 Marks]
%\item[b)]Express $\frac{d}{dy}$ in terms of $\hat a$ and ${\hat a}^\dagger$.
%\hspace*{\fill}[5 Marks]
%\item[c)]Find an expression for $\frac{d^2}{dy^2}$.
%\hspace*{\fill}[5 Marks]
%\item[d)]Use the two results a) and c) to find an expression for
%\be
%-\frac{1}{2}\frac{d^2}{dy^2} + \frac{1}{2} y^2.
%\ee
%Explain your answer.\hspace*{\fill}[7 Marks]
%\end{enumerate}


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