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%{Phys 623, Introduction to Quantum Mechanics II, Spring 1998, Dr.~Yakovenko}
%{Phys 623, Introduction to Quantum Mechanics II, Spring 1998, Dr.~Yakovenko}
\begin{document}
\begin{center}
{\sl Phys 623, Introduction to Quantum Mechanics II, Spring 1998,
Dr.~Yakovenko}
\bigskip%\bigskip
\large {\bf Final Examination} \\
Wednesday, March 20, 1998, 10:30 a.m.--12:30 p.m.
\end{center}
%\bigskip
\begin{enumerate}
\item A particle of mass $m$ is in the ground state of the
one-dimensional potential $V(x)=-\lambda\delta(x)$, where
$\lambda>0$.
\begin{enumerate}
\item {\bf [5 points]} Suddenly, the depth of the potential
doubles: $\lambda\to2\lambda$. What is the probability
that the particle will stay in the ground state of the new
potential?
\item {\bf [5 points]} Suddenly, the potential
$V(x)=-\lambda\delta(x)$ starts to move with a velocity
$v$. What is the probability that the particle will stay
in the ground state of the moving potential?
\end{enumerate}
\item {\bf [5 points]} A particle experiences the one-dimensional
potential $V(x)=-\lambda\delta(x-a)-\lambda\delta(x+a)$
($\lambda>0$) of the two delta-functions located initially at a
very long distance $a$. The initial wave function of the
particle is the ground-state wave function $\psi_0(x+a)$ of the
left delta-function potential $-\lambda\delta(x+a)$.
Slowly, adiabatically, the distance $a$ between the
delta-functions reduces to zero, so that the potential finally
becomes $V(x)=-2\lambda\delta(x)$. What is the probability that
the particle remains in the ground state of the final potential,
and what is the probability of ionization (transition of the
particle into the continuous spectrum)?
\item {\bf [5 points]} {\em Qualifier, January 1993, Fall 1986, Fall
1979, II-2.}
A beam of neutrons travels with velocity 2000 meters/second. The
spins of the neutrons are parallel to the $z$-axis.
Initially, the beam travels in a region 1 of uniform magnetic
field $B_0\hat{z}$ parallel to the $z$ direction. Then, the beam
enters another region 2 where the magnetic field acquires a
component parallel to the $x$-axis: ${\bf
B}=B_0\hat{z}+B_1\hat{x}$. After traveling for 20 meters in this
region, the beam enters a third region where the magnetic field
is once again $B_0\hat{z}$.
Given that $B_1=0.01B_0$ and $B_0\mu_n/\hbar=60$ sec$^{-1}$,
where $\mu_n$ is the magnetic moment of the neutron, calculate
the probability of spin-flip of the neutrons caused by the
travel through the region 2. Use a perturbation theory.
\item {\bf [7 points]} {\em Adapted from Qualifier II-3, August 1997,
January 1993, Fall 1986, Fall 1979.}
A hypothetical particle of mass $m$ interacts with electrons via
the contact potential
\begin{equation}
U({\bf r}-{\bf r}\,')=\lambda\delta({\bf r}-{\bf r}\,'),
\label{delta}
\end{equation}
where ${\bf r}$ and ${\bf r}\,'$ are the coordinates of the
particle and an electron, and $\delta({\bf r})$ is the
three-dimensional Dirac delta-function.
Consider elastic scattering of the particle on a hydrogen atom
where the electron is in its ground state. (The word ``elastic''
means that the electron remains in its ground state after the
particle has scattered.) Assume that the particle does not
interact with the nucleus.
In the Born approximation, calculate the differential and the
total cross sections of scattering of the particle.
{\em Hint:} The scattering potential is proportional to the
electron density, distributed in space according to the wave
function of the ground state.
\end{enumerate}
\end{document}