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\line{Physics 352 \hfill Spring, 1995 \hfill Problem Set \chapno \hfill Due
date: 
Th. April 28}\smallskip\hrule\bigskip

\problem{ Most random spin:}  A population of atoms with $J=1$ is prepared so that 
$\expectation{\hat J_z^2} = 0.2$ and $\expectation{\hat J_x} = 0.1$.  This can be
done by applying appropriate magnetic fields and electric field gradients.  
The atoms are subject to random environmental forces and thus their state becomes 
as random as possible consistent with these $\expectation{...}$'s.  The only
degree of freedom is the orientation.  It is convenient 
to use eigenstates of $J_z$, so that 
$$\hat J_z^2 = \left ( \matrix{
1&0 &0 \cr
0&0 &0 \cr
0&0 &1 \cr}\right )
\indent
\hat J_x = 2^{-1/2}\left ( \matrix{
0&1 &0 \cr
1&0 &1 \cr
0&1 &0 \cr}\right )$$
\item{a)} What is the density matrix $\hat \rho$?
\item{b)} What is the entropy of an atom relative to that of an unconstrained
atom?

\problem{ Monte-Carlo pendulum:}  A
particle sits on a one-dimensional lattice of points $x$ indexed 1, 2, 3, ...100
.  It experiences a force $F(x)=-sin(\pi (x-50)/50)$ attracting it towards
$x=50$.  We wish to find the probability $p(x)$ that the particle is at $x$ in
thermal equilibrium at temperature $\beta^{-1}$.  Accordingly, we assume that
this particle moves randomly to the left or right by one lattice site.  As
described in the Monte-Carlo handout, its probability of moving left or right is influenced
by the change of energy $F(x)\times 1$.  The {\it Metropolis algorithm} consists
of i) making a step to a random neighboring site, ii) finding the change of energy $\Delta E$
for 
this step, and iii) retracting the step with probability $1- \exp(-\beta \Delta E)$.  
(If $\Delta E$ is negative, the probability of retraction is zero and the 
step is taken with certainty.)

\item{a)} Show that if the particle is at $x$ with a Boltzmann probability 
distribution $p(x) \proportionalto
e^{-\beta E(x)}$, the distribution remains unchanged by the Metropolis process.

\item{b)} Devise a Monte-Carlo program to
simulate this process and make a histogram of the number of times $N(x)$ each
site is visited.  If the particle stays at the same $x$, that still counts as a
visit.  Use $\beta=1, .3$ and $.1$, or three other values of your choice.  A
biased random step with \eg .6 probability of moving to the right is made by
first choosing a uniform random number from 0 to 1.  (Use any simple
random-number generator.)  If the random number is less than .6, move the
particle a step to the right; otherwise, don't.  Note that there may be
attempts to step to $x$ values less than 1 or more than 100.  Such attempts
must be checked for and thwarted.  To get good statistics you may need $10^5$
time steps or more.  Make a graph of your results.  
\item{c)} Compare your simulation
results with the exact equilibrium distribution with continuous $x$ for these
three values of $\beta$.

\problem{ Monte-Carlo equilibration time:}  The ultimate probability distribution
in the previous problem is independent of the starting position $x_0$.  Thus
after many timesteps $t$ $C(t)\definedas \expectation{(x_0 - 50)(x(t)-50)}
\goesto \expectation{x_0 -50}\expectation{x(t)-50} = 0$.  The average
here is over runs with different $x_0$, all sampled at the $t$'th time
step.  Measure this correlation function $C(t)$ as a function of $t$ by averaging
over 100 runs of length $t=300$ with $\beta=.3$.  (Each run gives one data point
for each $t$.)  To begin each run, place $x_0$ randomly between 40 and 60. 
Check whether for long times the correlation function decays to zero
exponentially: $C(t)\goesto (const.)\exp(-t/\tau)$.  Estimate the decay time
$\tau$.

\problem{ Excitations of diatomic molecule:}  The vibrational
energy levels of a diatomic molecule can be approximated by a one-dimensional
quantum-mechanical harmonic oscillator.  The fundamental vibrational frequency
$\nu=\omega/2\pi = \Delta E/(2\pi \hbar)$ is of order $10^{13}$ per second for
many diatomic molecules.  (1eV is equivalent to $1.5\times 10^{15}/sec.$ in
angular frequency $\omega$.) 
\item{a)} Calculate the fraction of molecules in the first three vibrational
levels in an ideal diatomic gas at room temperature (300K or 1/40 eV) with this
vibrational frequency.  
\item{b)}  A rotating diatomic molecule has an energy $E=J(J+1) E_0$, where
$E_0$ is of order $10^{-3}$eV for simple molecules like oxygen. Calculate the
fraction of molecules in the lowest three rotational levels at room temperature. 
Include the effects of rotational degeneracy.  See Chandler Exercise 4.14

\problem{ Surface adsorption:}  A dilute, classical gas is in
equilibrium with a solid surface, all at temperature $T$.  The surface may be
regarded as a two-dimensional lattice of sites, each of which may be occupied by
a gas molecule or not.  We define the partition function of an empty site as 1;
that of an occupied site is $q(T)>>1$.  
\item{a)} Find the partition function of a site
in equilibrium with the gas at chemical potential $\mu$.  
\item{b)} Find the fraction of
sites occupied as a function of the gas pressure $p$ for a given $q(T)$.  This
is called the Langmuir isotherm.  