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\markboth{Homework \#4, Phys623, Spring 1998, Prof.~Yakovenko}
{Homework \#4, Phys623, Spring 1998, Prof.~Yakovenko}
\begin{document}
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\noindent
\begin{minipage}[t]{3.5in}
{\bf Homework \#4} --- Phys623 --- Spring 1998 \\
{\bf Deadline: 5 p.m., Monday, March 2, 1998.} \\
Return homework in class, by e-mail, or \\
put in the box on the door of Phys 2314.
\end{minipage}
\hfill
\begin{minipage}[t]{2.9in}
\raggedleft
Victor Yakovenko, Assistant Professor \\
Office: Physics 2314 \\
Phone: (301)--405--6151 \\
E-mail: yakovenk@physics.umd.edu
\end{minipage}
\medskip
\centerline{\bf Do not forget to write your name and the homework
number!}
\centerline{Equation numbers with the period, like (3.25), refer to the
equations of Schwabl.}
\centerline{Equation numbers without period, like (5), refer to
the equations of this homework.}
\begin{center}
\section*{Identical Particles (Chapter 13.1)}
\end{center}
\begin{enumerate}
\item Consider two particles in one dimension. One particle occupies a
state described by a wave function $\psi_1(x)$; the other
particle the state described by a wave function $\psi_2(x)$. The
wave functions $\psi_1(x)$ and $\psi_2(x)$ are normalized,
orthogonal, and have opposite parities:
\begin{equation}
\psi_1(-x)=\psi_1(x),\quad\quad\quad\quad\psi_2(-x)=-\psi_2(x).
\label{psi12}
\end{equation}
($\psi_1(x)$ and $\psi_2(x)$ could be the first and the second
energy eigenfunctions of an oscillator.)
Answer the following questions in the cases where the particles
are
\begin{enumerate}
\begin{enumerate}
\item not identical,
\item identical bosons,
\item identical fermions.
\end{enumerate}
\end{enumerate}
Express your answers in terms of some integrals of $\psi_1(x)$
and $\psi_2(x)$.
\begin{enumerate}
\item {\bf [2 points]} Write the normalized wave function of the
system, $\psi(x_1,x_2)$, in terms of $\psi_1(x)$ and
$\psi_2(x)$. \label{x1x2}
\item {\bf [5 points]} Integrating the wave function
$\psi(x_1,x_2)$ found in Problem \ref{x1x2}, calculate the
probabilities that
\begin{enumerate}
\item both particles are located in the right semi-space, $x>0$;
\item both particles are located in the left semi-space, $x<0$;
\item the particles are located in different semi-spaces.
\end{enumerate}
Make sure that these three probabilities add up to unity.
What is the difference in the behavior of bosons and
fermions? \label{><}
\item {\bf [3 points]} Using the solution of Problem \ref{><},
calculate the probability to find one particle in the
\begin{enumerate}
\item right semi-space, $x>0$,
\item left semi-space, $x<0$,
\end{enumerate}
whereas we don't care where the other particle is. Do
these two probabilities add up to unity?
\item {\bf [5 points]} Calculate the probability density
$P(l)\,dl$ to find the two particles at a distance $l$
from each other.
\end{enumerate}
\item {\bf [5 points]} Two identical particles of mass $m$ interact
with each other via the three-dimensional harmonic potential
$V({\bf r}_1-{\bf r}_2)=k({\bf r}_1-{\bf r}_2)^2/2$. Find the
energy spectrum and the eigenfunctions of the system, when the
particles are
\begin{enumerate}
\begin{enumerate}
\item bosons,
\item fermions.
\end{enumerate}
\end{enumerate}
\item {\bf [7 points]} {\em Adapted from Physics Qualifier, Fall 1983,
Problem II-1.}
Find all energy eigenvalues and eigenfunctions for a particle of
mass $m$ moving in two dimensions in a triangular box {\bf(see
Hints)}: \label{triangle}
\begin{equation}
U(x,y)=\left\{\begin{array}{ll}
0, & \quad\mbox{when $x>0$ and $y>0$ and $x+yp_F$ and $E>E_F$ are empty, where $p_F$ is the Fermi
momentum, and $E_F$ is the Fermi energy. The electrons have the
electric charge $e$, which is neutralized by a uniform
background of positive electric charge.
\begin{enumerate}
\item {\bf [5 points]} \label{D} Suppose the Fermi energy
slightly changes by the amount $dE$. That would cause the
electron concentration to change by $dn$. Calculate the
so-called energy density of states $D=dn/dE$.
\item {\bf [5 points]} \label{Pauli} Suppose a magnetic field
$B$ is applied to the system. Find the magnetic moment
$\mu$ per unit volume induced by the magnetic field and
the magnetic susceptibility $\chi=\mu/B$ (called the
\emph{Pauli spin susceptibility} in this case). Take into
account the Zeeman effect of the magnetic field on
electrons' spins, but neglect the orbital effect of the
magnetic field (the Landau levels). Express your answer in
terms of the energy density of states $D$ found in Problem
\ref{D} {\bf(see Hints)}
\item {\bf [5 points]} Suppose the system is placed in a weak,
slowly varying in space electric potential $\phi({\bf
r})$. The electric potential changes the local Fermi
energy by the amount $-e\phi({\bf r})$. According to
Problem \ref{D}, this produces a local electric charge
density $-e^2D\phi({\bf r})$. This charge feeds back into
the Poisson equation governing the distribution of the
electrostatic potential $\phi({\bf r})$. If an external
charge $q$ is placed in the electron gas at the point
${\bf r}=0$, then the Poisson equation is
\begin{equation}
\Delta\phi({\bf r})=4\pi e^2D\phi({\bf r}) - 4\pi q\delta({\bf r}),
\label{Poisson}
\end{equation}
where $\Delta$ is the Laplacian differential operator, the
right hand side represents the electric charge
concentration, and $\delta({\bf r})$ is the 3D
delta-function.
Solve Eq.\ (\ref{Poisson}) and find the characteristic
length $\Lambda$ over which the electron density is
perturbed and beyond which the electric potential
essentially vanish. This length $\Lambda$ is called the
{\em screening length.}
\item {\bf [3 points]} In copper, $n=8.5\times10^{22}$ cm$^{-3}$
and $m=9.1\times10^{-28}$ grams. Find the numerical value
of $\Lambda$ for copper and compare it with the average
distance between the electrons.
\end{enumerate}
\end{enumerate}
%\newpage
\label{Hints}
\section*{\centerline{Hints}}
\begin{description}
\item[\ref{triangle}] Map the problem onto the problem of two
one-dimensional fermions moving in a box of the length $L$.
\item[\ref{R1R2}] Two photons may come to the detectors $a$ and $b$
either both from the source $A$, or both from the source $B$, or
one from the source $A$ and another from the source $B$. In
quantum mechanics, when there are alternatives, we add the
amplitudes of probabilities, when the events are
indistinguishable, and the probabilities, when the events are
distinguishable.
\item[\ref{Pauli}] Because of the Zeeman effect, the magnetic field
increases (decreases) the Fermi energy of the electrons with the
spins parallel (antiparallel) to the magnetic field. Thus, the
concentrations of the spin-up and spin-down electrons become
different, which produces net magnetization of the system.
\end{description}
\end{document}