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{\Large
\begin{center}\bf THE 35th\\ ---7th INTERNATIONAL---
\\ RUDOLF ORTVAY \\
PROBLEM SOLVING CONTEST IN PHYSICS \\
2004 \\
\end{center}
}
\noindent The Physics Students' Association of E{\"o}tv{\"o}s
University, Budapest, the Roland E{\"o}tv{\"o}s Physical Society
and the Hungarian Association of Physics Students proudly announce
the {\bf 35th} -- and for the seventh time {\bf international}
--
\begin{center}
Rudolf Ortvay Problem Solving Contest in Physics,
\\
from 29 October 2004, through 8 November 2004.
\end{center}
Every university student from any country can participate in the
Ortvay Contest. PhD students compete in a separate category. The
contest is for individuals: solutions sent by groups of students
are not accepted. The name, the university, the major, and the
university year should be indicated on the solutions. Pseudonyms
and passwords cannot be used: each contestant has to use his/her
own name. The problems can be {\bf downloaded} from the webpages
of the Ortvay Contest
\begin{center}
{\tt{\bf{http://ortvay.elte.hu/}}}
\end{center}
in Hungarian and English languages, in html, \LaTeX \, pdf, and
Postscript formats, from {\bf 12 o'clock (Central European Time,
11:00 GMT), Friday, 29 October 2004}. The problems will also be
distributed by local organizers at many universities outside of
Hungary.
\\
{\it
Despite all the efforts of the organizers, it may happen that some
unclear points or misprintz stay in the text. Therefore it is very
useful to visit the webpage of the contest from time to time, as
the corrections and/or modifications will appear there. \/} \\
Each contestant can send solutions for up to 10 problems. For the
solution of each problem 100 points can be given.
\medskip
{\it Each problem should be presented on (a) separate A4, or
letter-sized sheet(s). The contestants are kindly asked to use
only one side of each sheet. Solutions written by pencil or
written on thin copy-paper will not be accepted as these cannot be
faxed to the referees.} \\ Any kind of reference material may be
consulted; textbooks and articles of journals can be cited.
Computer programs appended to the solutions should be accompanied
by detailed descriptions (what computer language it has been
written in, how to use the program, which parameters can be set,
what notations are used, how to interpret the output figures and
graphs, etc.) They can be enclosed on floppy disks, or sent via
email to the addresses below. Solutions can be sent by mail, fax,
or email (in \LaTeX, \TeX \,\,or Postscript formats---or, if they
contain no formulae, in normal electronic mail). Contestants are
asked not to use very special \LaTeX \, style files unless
included in the sent file(s).
\begin{center}
Postal Address: \medskip \\
Fizikus Di\'akk{\"o}r, D\'avid Gyula, \\
ELTE TTK Atomfizika Tansz{\'e}k, \\
H-1117 Budapest, P\'azm\'any P{\'e}ter s{\'e}t\'any 1/A, HUNGARY \\
Fax: D\'avid Gyula, 36-1-3722775 {\em or} Cserti József, 36-1-3722866\\
E-mail: dgy@ludens.elte.hu \emph{or} ortvay@saas.city.tvnet.hu
\end{center}
\begin{center}
{\bf Deadline for sending the solutions:
12 o'clock CET (11:00 GMT), 8 November 2004.} \\
\end{center}
Contestants are asked to fill in the form available on our webpage
after posting their solutions. It will be used for identification
of contestants and their solutions. {\bf Without filling in the
form, the organizers cannot accept the solutions! The form is
available only on 8th and 9th November.}
The contest will be evaluated separately for each university year,
according to the total number of points. The referees reserve the
right to withhold, to multiply or to share some prizes. Beyond the
money prizes given for the first, second, and third places,
honorable mentions and special prizes for the outstanding
solutions of individual problems can be awarded. This is why it is
worthwhile sending even one or two solutions.
The announcement of the results will take place on 16 December,
2004. The detailed results will be available on the webpage of the
contest thereafter. Certificates and money prizes will be sent by
mail. We plan to publish the assigned problems and their solutions
in English language---to which the contribution of the most
successful participants is kindly asked. The volume is planned to
be distributed all over the world with the help of the
International Association of Physics Students, as well as the
contestants themselves. We hope this will help in making the
contest even more international. Wishing a successful contest to
all our participants, \bFL
the Organizing Committee: \\
\bigskip
Gyula D\'avid, Attila Pir{\'o}th,
J{\'o}zsef Cserti
\\
(E\"otv\"os University, Budapest, Hungary) \eFL
\newpage
\begin{enumerate}
\item
Go to the link
\begin{verbatim}
http://ortvay.elte.hu/2004/budapest.jpg (1.6 MB)
\end{verbatim}
to find a space photography of the inskirts of Budapest,
stretching from the southern tip of Szentendre island to the
campus of Eötvös University, in Lágymányos. \\
a) When (what month, day, hour, and minute) was the photo taken? \\
b) Determine the altitude of the clouds.
%\vspace{-5mm}
\bFL
(András Pál)
\eFL
\item
From Friday to Friday, with the help of Man Friday, Robinson
made a square-shaped pond, to spend his free afternoons swimming.
His splashing can only be disturbed by the cannibals who suddenly
pop up from the bushes around the pond---and who would certainly
appreciate if Robinson was featuring on their menu. Robinson is at
the center of the pond when a cannibal appears on the shore. The
race for life and death begins---a Robinson steak is at stake! The
cannibal cannot swim; his running speed is $u$. Ashore Robinson
runs faster than the cannibal. While Robinson is in the water, the
cannibal always runs in the direction that grants him a quicker
approach to the point where the line connecting the center of the
pond with Robinson meets the shore. Find a lower bound on
Robinson's swimming speed that guarantees he can get away.
Extra question: Does the cannibal have a better strategy than that
described above?
\bFL
(Szilárd Farkas and Zoltán Zimborás)
\eFL
\item
We wish to illustrate the principle of jet propulsion with a small
cart, moving freely on a level surface. The cart has a tank on it
that is filled with water, initially up to height $H$. A
horizontal tube of length $L$ is connected to the bottom of the
tank, through which the water can flow out ``backwards''. The
cross sectional area of the tube is $1/k$ times that of the tank.
The mass of the tank and the cart are negligible compared to that
of the water.
Describe the motion of the cart.
%\vspace{-5mm}
\bFL
(Péter Gn\"adig)
\eFL
\item
Stones are projected vertically upwards from the surface of a very rapidly
spinning planet of spherical shape that has no atmosphere. Determine where
the stones touch the ground as a function of the latitude of the place of
projection and the initial speed of the stone.
\bFL
(Gábor Veres)
\eFL
\item
Two solar type stars revolve around one another. Their largest distance is
three times as much as the sun-earth distance. The projection ellipses
of the orbits on the celestial sphere pass mutually through the center of the
other ellipse.
a) What portion of the year (ie, period of revolution) do the each star
spend inside the other star's orbit? (Give a precise numerical answer,
preferably without using a computer.)
b) How does the temperature of a small black body placed in the midpoint
of the line connecting the two stars change in the course of the year?
%\vspace{-5mm}
\bFL
(Gyula Dávid based on an MIT problem)
\eFL
\item
A practically uniform magnetic field of induction $\mbf B$ is
present within a small region of interplanetary space. Also
present are two protons: one is initially at rest at the origin,
while the other is at position $\mbf r_0$, and has an initial
velocity $\mbf v_0$. The vectors $\mbf B$, $\mbf r_0$, and
$\mbf v_0$ are mutually perpendicular.
Find the maximum distance of the two particles during their
motion, if only electromagnetic forces act. After how much time
will the distance between the protons be the same as initially?
For what value ${v}_{0} = v_c$ will the distance be a
constant as time proceeds (where $v_0 = |{\bf v}_0|$)?
Examine the special case when $v_0$ is only slightly different from $v_c$.
Are there any closed orbits?
Neglect radiation losses, as well as the magnetic field of the moving
protons compared to the external field $\mathbf B$.
%\vspace{-5mm}
\bFL
(Péter Gn\"adig)
\eFL
\item
The coefficient of restitution describes how a body bounces back from another
or from the wall. Its value is given by the ratio of the initial and final
magnitudes of the velocity component normal to the plane of rebound.
It has been shown experimentally that this value can exceed 1 for some
sufficiently soft rebounding surfaces. Model the phenomenon in two dimensions
by disks bouncing back from the wall. Explain quantitatively how this is
possible without violating the principle of energy conservation.
%\vspace{-5mm}
\bFL
(Szabolcs Borsányi)
\eFL
\item
Derive the equation of bicycle wheel tracks.
Consider a rear-wheel drive bicycle of axle distance $L$, which one rides
on a level surface. What is the relation between the tracks traced out by
the front and the rear wheel? How should the equation of the track curves
be given so that this relation take a simple form?
Apply the result for some simple cases: what track is traced out by the
rear wheel if the front wheel follows a) a straight-line path b) a circular
path? (The angle formed by the initial direction of the front wheel and the
frame of the bike can be arbitrary.) What curve does the rear wheel trace out
asymptotically if the front one follows a sinusoidal path? What are the
parameters of the curve? Does the axle distance $L$ appear among the parameters?
In a race $N$ cyclists, spaced a distance $\Delta$ apart, follow each other
with an identical speed $v$. To make use of the wind shadow, each cyclist follows
the track of the rear wheel of the cyclist in front. After a very long
straight section the road turns through 90 degrees along a circular arc of
radius $R \gg L$. How will the following distances change, if each cyclist
keeps on following the track of the rear wheel of the cyclist in front, without
changing the speed?
\vspace{-5mm}
\bFL
(Merse Előd Gáspár)
\eFL
\item
Three space stations, forming an equilateral triangle on the synchronous orbit,
revolve around an atmosphereless, spherical planet. (For simplicity, assume
that the radius of the planet, as well as its revolution and rotation periods
are the same as those of the earth, and that the central star is similar to the sun.)
The following method is proposed for revolutionizing the transportation of freight
between space stations: the packets are simply projected ``horizontally''
(ie along the tangent of the synchronous orbit) from the station; the gravity
of the planet will take care of the rest!
a) At what speed should the packet be tossed out if the destination is the
station moving in front or behind? How much time does delivery take?
Plot the path of the packet in the reference frame co-moving and co-rotating
with the station from which the packet was released. (The axes of the reference
frame are oriented towards the centre of the planet, the tangent of the circular
orbit, and the normal vector of the orbital plane). \\
b) It occurs to someone that the method can be made more efficient by
letting the packets bounce back (once) from the planet. To this end,
smooth, horizontal and perfectly elastic ``bouncers'' have to be installed
at certain points of the planet.
How do the answers given in a) change? \\
c) The captain of the station decides that the change proposed in part b) is
not worth the fuss, since the maximum attainable savings in the delivery time
are about 4\%, which does not justify the investment costs of the bouncers.
Only one question is left: why does not the planet have an atmosphere?
\bFL
(Gyula Dávid)
\eFL
\item
Inertial mass is defined by Newton's second law,
$\mathbf{F} = m \mathbf{a}$, where $m$ is a constant, independent
of the body's position and velocity. However, not only scalar masses
can satisfy the linear relationship between the two vectors.
What are the consequences of a tensorial choice of the mass $m$?
a) How are the usual conservation laws modified?
Parts b) and
c) below are concerned with motions in a two-dimensional space.
Let the mass tensor be isotropic, ie rotationally invariant.
(Attention, this is not necessarily a multiple of the unit tensor!)
\\
b) If mass is tensorial, nontrivial stationary states may exist even
in dissipative systems without external excitation.
Study the motion of a particle with the above-mentioned isotropic mass
tensor in a central potential of the form
\[
\mathbf{F} = -k \mathbf{x} \cdot {\left| \mathbf{x} \right|}^{b-1}
- \eta \mathbf{v},
\quad k, \eta > 0, \quad b\in \mathbb{R}.
\]
Under what conditions will the trajectory motion of the particle
be bounded?
What is the stationary solution? \\
c) If $N>1$ identical particles of isotropic mass move in each others'
field, the external potential can be omitted:
\[
\mathbf{F}_i = -k \sum_{j\neq i}^{N} (\mathbf{x}_i - \mathbf{x}_j) \cdot
{\left| \mathbf{x}_i-\mathbf{x}_j \right|}^{b-1} - \eta \mathbf{v}_i,
\quad k, \eta > 0, \quad b\in \mathbb{R}.
\]
When does a stationary solution exist? What is this solution?
Numerical results leading to analytical considerations are also
of interest.
\vspace{-5mm}
\bFL
(János Asbóth)
\eFL
\item
A dumb-bell shaped planet revolves around the sun. (The planet can be
considered as two mass points, connected by a rigid rod of length $d$
and negligible mass. The motion is planar; the mass of the planet can
be neglected compared to that of the sun.) When the planet's orientation
is not symmetric with respect to the vector $\mathbf r$ drawn from
the sun to the planet's center of mass, gravitational forces will tend
to turn the planet towards the direction of $\mathbf r$. If during
its revolution the planet is in its perihelion (aphelion) when a
torque in one (in the other) direction is exerted on it, an exchange
between orbital and rotational angular momenta can take place.
a) What relation holds between the two types of motion? \\
b) Can the planet's rotation increase for a long time at the expense
of its revolution? If so, under what conditions? \\
c) When can the planet fall into the sun?
(Until the planet reaches the surface of the sun, it is sufficient to
calculate gravitational forces to the same order as at the beginning of
the motion.)
\vspace{-5mm}
\bFL
(Titusz Fehér)
\eFL
\item
A rod pendulum is swinging in a uniform gravity field. At $t=0$ its
angular displacement (from equilibrium) is $\varphi_1$, while time $T$
later it is $\varphi_2 = \pi $.
Find the lowest-energy solutions, ie discard unnecessary extra turns.
How does the energy of the pendulum depend on $T$ and the initial angle
for asymptotically large values of $T$?
%\vspace{-5mm}
\bFL
(Zoltán Bajnok)
\eFL
\item
Water is rotated at a constant angular velocity $\omega$ in a cylindrical tank.
A uniform wooden ball of radius $R$, stationary in the reference frame
co-rotating with the water is placed into the water. Where will
the ball be after a long period of time? Study the case of small
balls, as well as bigger ones, for which $R \gtrsim g/\omega^2$.
%\vspace{-5mm}
\bFL
(Bence Kocsis, Győző Egri and Márton Kormos)
\eFL
\item
In a rough approximation, a rapidly spinning neutron star can
be modeled by an incompressible fluid held together by its own
gravity.
Show that in this model the star can be spheroidal, as long as
the angular speed is not too high. At most how large can its
``oblateness'' be?
\vspace{-5mm}
\bFL
(Péter Gn\"adig)
\eFL
\item
Once upon a time, when the weather was particularly windy,
Huygens, Fresnel, and Bernoulli went to the prairie to play the drum.
The instrument was with Huygens, and from time to time he hit it.
Fresnel and Bernoulli were listening from the same distance;
Bernoulli against the wind, and Fresnel in the direction of the wind.
Wind was blowing horizontally, and its speed varied with the altitude
as $v=\alpha z $ (being much smaller than the sound speed at all
heights relevant to the problem).
What did Fresnel and Bernoulli hear? (The sound emitted by the drum
can be approximated by a short delta pulse; the viscosity of air can
be neglected.)
\bFL
(Bence Kocsis and Győző Egri)
\eFL
\item
The motion of relativistic particles in a static, central Lorentz scalar
field $V(r)$ is studied. Neglect the reaction of the particle on the scalar
field and the central body producing it. Throughout the problem, do calculations
in the inertial frame fixed to the center.
For parts a) and c), examine the nonrelativistic approximation and its range
of validity. What is the physical meaning of the bounds? \\
a) Particles move in a circular orbit of radius $R$. How does the scalar
potential $V(r)$ depend on the radius $r$, if the period of revolution is
independent of the radius $R$ of the orbit? \\
b) Consider a Keplerian scalar potential $V(r) = - \alpha/r$.
Does Kepler's third law hold for relativistic motions in circular orbits? \\
c) What is the form of the scalar potential $V(r)$, if relativistic motion
is in traditional Keplerian elliptical orbits (ie closed orbits, with the
center of attraction in either focus). \\
d) What should be the impact parameter of a point particle of energy $E$,
sent towards the center of the scalar potential in part b), if we wish to have a
deflection angle that is twice as large as in the nonrelativistic case?
\bFL
(Gyula Dávid)
\eFL
\item
Chewbacca looks out of the window of the Millenium Falcon, and
suddenly spots out a very rapidly moving meteor, heading for
the base of the rebels. Of course, Chewie knows the favorite
trick of the Imperial Army: by moving large mirrors, they
deceive and draw away the rebel spaceships from their posts.
The meteor is enshrouded in a cloud of gas, therefore changes
in its shape cannot be observed.
How could Chewie verify quickly whether he sees an image reflected
from a rapidly moving mirror, or indeed a meteor is heading for the
base of the rebels? Does the method work in all cases?
\vspace{-5mm}
\bFL
(Zoltán Zimborás)
\eFL
\item
The formulae of the Doppler effect describe the properties of light
emitted by a laser moving at constant velocity relative to the observer.
What happens if the \emph{acceleration} of the emitting laser is constant?
(Calculate the results as functions of the direction and
the magnitude of the acceleration.)
\vspace{-5mm}
\bFL
(Titusz Fehér)
\eFL
\item
The elastic scattering cross section of relativistic electrons
by a Coulomb potential is well known from quantum electrodynamics
The classical ($\hbar \to 0$) limit of the formula is very simple:
\[
\frac{d\sigma (p,\theta)}{d\Omega}=
\gamma ^2(1-v ^2 \sin ^2 \theta/2)
\left. \frac{d\sigma (p,\theta)}{d\Omega}\right|_{\rm Rutherford},
\]
where $p$ is the three-momentum of the electron,
$\gamma=\frac1{\sqrt{1-v^2}}$,
$\theta$ is the scattering angle and $v$ is the electron's speed
(all measured in the frame of the Coulomb potential, in $c=1$ units).
Carry out a classical calculation to explain the
$\gamma ^2 (1-v ^2 \sin ^2 \theta/2)$ relativistic correction.
\vspace{-5mm}
\bFL
(Kálmán Szabó)
\eFL
\item
The space travel agency \emph{Black Hole Travels} organizes exotic trips
for adventurous billionaires. Travelers can select the black hole they want to
visit from a catalog. The participants are then transported into the vicinity
of the black hole on a luxurious space ship, which then starts circling the
black hole in a circular orbit. Approaching the horizon of the black hole,
the travelers may enjoy a fantastic view: the sky closes in, stars
turn blue, etc.
a) How close can they approach the horizon of a static black hole of mass
$M$?
b) How deeply can they enter into the ergosphere of a Kerr black hole of mass $M$
rotating at maximum angular momentum,
if the acceleration due to gravity should not exceed the usual terrestrial value,
and the tidal forces should not make the travelers feel \emph{too} uncomfortable?
Examine the cases $M=1 M_\odot$ and $M = 10^6 M_\odot$.
Contestants should clarify the concept of \emph{too} uneasy in their solution.
\vspace{-5mm}
\bFL
(Bence Kocsis)
\eFL
\item
Two observers are falling radially towards a black hole, one behind the other.
Their initial distance is finite, and their initial speeds are equal.
Can they see one another?
If they can, in which direction?
How does the answer change during the motion?
Is there a portion of the trajectory of one observer that can never be
seen by the other? And vice versa?
(For simplicity, assume that the observers' eyes can see the total
solid angle $4 \pi$.)
\bFL
(Gyula Dávid)
\eFL
\item
Observations establish the fact that in spiral galaxies the circumferential
velocity of the stars in circular orbits within the plane of the disk does not
depend on their distance from the galactic center. Interestingly, this relation
holds even for large distances, where only an insignificant amount of luminous
matter is present. In the standard resolution of this contradiction it is assumed
that non-luminous (dark) matter is also present in the galaxy, and the
gravitational effects of this dark matter take care of the relation at large distances.
In another possible approach the presence of dark matter is not assumed,
but rather Newton's law is modified. Suppose that the gravitational force
between two point-like particles is of the form
\[ {\bf F} = -G(r) \frac{m_1 m_2}{r^2} \frac{{\bf r}}{r}, \]
where \(G(r)\) is the distance-dependent gravitational constant. The density
distribution of a spiral galaxy can be approximated by
\[\rho ({\bf r}) = \rho _0 e^{-\alpha r} \delta (z), \]
where \(r\) is the distance from the center, and \(\alpha
\approx 4\) kpc, and $z$ is the coordinate perpendicular
to the plane of the galactic disc. How should \(G(r)\) be chosen so that the
density distribution alone may account for the observations of circumferential
velocities?
Extra question: Can the above form of \(G(r)\) be falsified through observations
concerning the solar system only?
\vspace{-3mm}
\bFL
(Bence Kocsis and Győző Egri)
\eFL
\item
Study quantum mechanically the motion of a rigid body, of moment of
inertia
\[
\mbox{\boldmath$\Theta$} = \begin{pmatrix}\Theta_1 & 0 & 0\\ 0& \Theta_2
&0 \\ 0 & 0 & \Theta_3 \end{pmatrix}
\]
($\Theta_1>0$, $\Theta_2>0$, $\Theta_3>0$),
rotating freely about its fixed center of mass.
a) What is the Hamiltonian of the system? \\
b) Find the energy eigenvalues for the spherically symmetrical case
$\Theta_1=\Theta_2=\Theta_3$. Determine the degree of degeneracy
of each level. \\
c) Rigid bodies are, however, observed to ``rotate'' macroscopically,
ie their position can be described by time-dependent angle variables.
How can this be reconciled with the quantum-mechanical description? \\
d) How are the energy levels and the degrees of degeneracy changed, if
$\Theta_1=\Theta_2 \ne \Theta_3$? \\
e) What about the general case $\Theta_1 \ne \Theta_2 \ne \Theta_3 \ne
\Theta_1$?
\vspace{-8mm}
\bFL
(Bálint Tóth)
\eFL
\item
The Hamiltonian of a quantum-mechanical system is
\[
{\cal H} = H_0^{-1}\, \left( \begin{array}{ccc}
H_0^2-\omega^2 & 2 H_0 \, \omega \cos \alpha &
2 H_0 \, \omega \sin \alpha \\
2 H_0 \, \omega \cos \alpha & \omega^2 \cos (2\alpha) -H_0^2 &
\omega^2 \sin (2\alpha) \\
2 H_0\, \omega \sin \alpha & \omega^2 \sin (2\alpha) &
- \omega^2 \cos (2\alpha) -H_0^2
\end{array} \right),
\]
($\hbar$ is set equal to 1),
where $\alpha$ is a constant scalar parameter, and $H_0$ is the well
known Hamiltonian of the one-dimensional harmonic oscillator of frequency
$\omega$ and mass $m$.
Determine (in the simplest possible way) the energy spectrum of the system
(and, if possible, the eigenstates as well).
\vspace{-5mm}
\bFL
(József Cserti and Gyula Dávid)
\eFL
\item
The method of ``varying effective mass'' is often used in the description of
inhomogeneously doped semiconductors. The mass of the electron moving inside
the solid is then considered as a given function $M({\bf r})$ of the position.
The quantum-mechanical treatment runs into difficulties, since
the operators ${\bf \hat{p}}$ and ${M(\bf \hat{r})}$---corresponding to
${\bf {p}}$ and ${M(\bf {r})}$ in the kinetic part of the Hamiltonian $H({\bf r},
{\bf p}) = \frac{{\bf p}^2}{2m} + V({\bf r})$---satisfy the usual commutation
relations and thus do not commute, therefore several different operators can be
regarded as the QM counterpart of the classical expression.
Among others, the following expressions have been proposed:
\[
a) \hspace{0.5cm} {\bf \hat{p}}\frac{1}{2 M({\bf \hat{r}})}\, {\bf \hat{p}},
\hspace{1cm}
b) \hspace{0.5cm} \frac{1}{4}\, \left( {\bf \hat{p}}^2 \frac{1}{M({\bf \hat{r}})}
+ \frac{1}{M({\bf \hat{r}})} \, {\bf \hat{p}}^2 \right),
\hspace{1cm}
c) \hspace{0.5cm} \frac{1}{2}\, \frac{1}{\sqrt{M({\bf \hat{r}}})} \, {\bf
\hat{p}}^2 \, \frac{1}{\sqrt{M({\bf \hat{r}}})}.
\]
Examine the one-dimensional case. Show that each of the three expressions
above leads to a hermitian energy operator, and that they are related by
``gauge transformations''. What ``compensating transformation'' should be
performed on the potential $V(x)$? $M(x)$ is assumed to be a continuously
differentiable function of the position.
How are the results modified if finite discontinuities may be present
in the function $M(x)$ (eg at the interface of two different substances)?
What boundary conditions should be prescribed for the stationary wave function
solution of the Schrödinger equation in coordinate representation?
\bFL
(Andor Kormányos)
\eFL
\item
The two-dimensional motion of an electron, including spin-orbit
interaction, can be described by the Hamiltonian
\[
\hat {H} = \frac{p_x^2 + p_y^2}{2m} + \frac{\alpha}{\hbar}
\left( \sigma_x p_y - \sigma_y p_x \right),
\]
where $\alpha$ is a constant related to the strength of the spin-orbit
interaction, while $\sigma_x$ and $\sigma_y$ are the Pauli matrices.
In the presence of external magnetic fields, the usual transformation
${\bf p} \rightarrow {\bf p}-e{\bf A}$ should be performed on the
Hamiltonian, in which $e$ is the electronic charge and ${\bf A}$ is
the vector potential of the external magnetic field.
The Zeeman effect is taken into account by the addition of a further term,
$\mu_B {\mathbf \upsigma} {\bf B}$, where $\mu_B $ is the Bohr
magneton and $\mathbf{\upsigma} =(\sigma_x,\sigma_y,\sigma_z)$
is the vector formed by the Pauli matrices.
Determine the electron's energy levels in a uniform magnetic field in the
$z$ direction!
\bFL
(József Cserti)
\eFL
\item
a) Show that for slowly varying rotational motions the component of the
vectors along the angular velocity is adiabatically invariant. In other
words: consider a twice continuously differentiable function
$\mbox{\boldmath $\upomega$}(t):[0,1]\to \valh$,
and then choosing an arbitrary number $a\in\valos^+$ consider the following
differential equation for the vector ${\bf r}(t):[0,1/a]\to \valh$:
\begin{equation}
\dperdt {\mathbf r}(t)=\mbox{\boldmath $\upomega$}(at)\times {\mathbf r}(t)\pont
\end{equation}
Show that, independently of the initial value ${\mathbf r}(0)$,
\begin{equation}
\lim_{a\to 0}\frac{\Big(\mbox{\boldmath $\upomega$}(1/a),{\bf r}(1/a)\Big)}
{|\mbox{\boldmath $\upomega$}(1/a)|}
=\frac{\Big(\mbox{\boldmath $\upomega$}(0),{\bf r}(0)\Big)}
{|\mbox{\boldmath $\upomega$}(0)|},
\end{equation}
where $(\cdot,\cdot)$ stands for the scalar product.
b) Demonstrate that time-dependent perturbation theory is ``adiabatically
invariant''. In other words: consider a twice continuously differentiable
function $f(t):[0,\infty)\to\valos$, for which $f(0)=1$, $\lim_{t \to
\infty} f(t)=0$, and $\lim_{t \to \infty} f'(t)=0$, and let the integral
$\int_0^\infty|f''|$ be convergent.
Let $H$ be a known Hamiltonian, and $V$ a perturbation operator.
Show that if the eigenvectors of $H+V$ are determined from time-independent
first-order perturbation theory, the same results are obtained---up to
phase factors---as when an arbitrary number $a\in\valos^+$ is chosen and
time-dependent first-order perturbation theory is applied to the operator
$H+f(at)V$ on the whole interval $t\in [0,\infty)$, and then the $a\to 0$ limit
of the obtained result is taken. (In this case the potential is switched
off, rather than on.)
What happens in second order? (Take, eg, the function $f(t)=e^{-t}$.)
What if the potential is switched off within a finite interval of time, ie
for some $b\in\valos$, $f(t)=0$, if $t>b$?
c) How are parts a) and b) related? That is: find (without proof) a general
mathematical statement of which a) and b) are special cases.
(For simplicity, the appearing vector spaces should considered as finite
dimensional.)
%\vspace{-5mm}
\bFL
(Balázs Pozsgay)
\eFL
\item
Non-commutative quantum mechanics---ie quantum mechnaics with
non-commuting coordinate components---has attracted a great deal of
attention recently. Examine one of the simplest cases of
non-commutative quantum mechanics, a quantum-mechanical model subject
to constraints.
Consider the nonrelativistic two-dimensional, planar motion of a
charged particle in a uniform and constant magnetic field perpendicular
to the plane. The Hamiltonian is
%
\[
H(\mathbf{p},\mathbf{q})=\frac1{2m}\sum_{i,j,k=1}^2\left(p_i+\frac{B}2
\epsilon_{ij} q_j\right)\left(p_i+\frac{B}2 \epsilon_{ik}
q_k\right)
\]
%
The \(m\to0\) limit is studied in a rather intuitive way:
to preserve the finiteness of the Hamiltonian, the constraints
\(C_i=p_i+\frac{B}2\epsilon_{ij} q_j \) are imposed.
Show that upon quantization, these constraints cannot be imposed
on the operator level, ie the original space of states does not
possess a (non-trivial) subspace in which the operators \(C_i\)
yield 0. Instead, proceed as follows:
a) Find a subspace \(\mathcal{M}\) and an orthogonal projector
\(P_\mathcal{M}\) (that projects onto this subspace) such that
when the operators of the constrained model
(\(\hat{\mathscr{O}}^c\)) are calculated from the constraint-free
observables \(\hat{\mathscr{O}}\) as
\(\hat{\mathscr{O}}^c=P_\mathcal{M}\hat{\mathscr{O}}P_\mathcal{M}\),
the operators \(\hat{\mathscr{O}}^c\) satisfy the Dirac commutation
relations.
b) It will be readily seen that the momentum components
\(\hat{p}_1^c\), \(\hat{p}_2^c\) do not commute any more,
and neither do the coordinates. However, the system is symmetric
under the transformations of the two-dimensional Euclidean group.
Why should this symmetry be spoilt in a suitably defined limit
\(m\to0\)? What representation of the translations is derived from
the usual representation in the original, constraint-free model?
Isn't it possible to get rid of the noncommutativity of the
momentum components by making a transition to an equivalent \emph{ray}
representation?
c) When quantizing, an operator is associated with the classical monomial
\(p_1^np_2^m\), which is a polynomial of \(\hat{p}_1^c\) and \(\hat{p}_2^c\).
What ordering rule is found for the momentum component operators?
d) Let \(f\) and \(g\) be functions of the coordinates alone.
Define a (not necessarily commutative)
\(\ast\) product that satisfies the equality
%
\[P_\mathcal{M}\,(f\ast g)(\hat{q}_1,\hat{q}_2)\,P_\mathcal{M} =
P_\mathcal{M}\,f(\hat{q}_1,\hat{q}_2)\,P_\mathcal{M}\,g(\hat{q}_1,
\hat{q}_2)\,P_\mathcal{M}.\]
%
What is the \(\ast\) product of the monomials \(q_1^kq_2^l\)\,,
\(\,q_1^mq_2^n\)?
\bFL
(Szilárd Farkas and Márton Kormos)
\eFL
\item
Consider a tiny drop of superfluid helium ($^4$He) of zero temperature.
As it is well known, sound waves propagating within the droplet
can be described by a suitable effective quantum field theory, and the
quanta of the sound waves are called phonons. According to the
uncertainty principle, a zero-point vibration is associated to each
phonon mode. What energy density is created in the helium droplet
by the zero-point vibrations? What pressure is due to this energy
density? Why does not this pressure make the droplet explode or
collapse?
(Hint: the maximum possible energy of the phonons is the Debye energy.
Perform the arising calculations for an interacting \emph{Bose gas},
and then try to deduce implications on the quantum fluid.)
\bFL
(Győző Egri)
\eFL
\item
Two small ion clouds collide. Each contains a very large number of the same
ion. Electromagnetic interactions among the particles are ignored; the
phenomenon is governed by the very short-range strong interaction.
The probability that at least one intercloud collision (ie, a collision
between one ion in one cloud and another ion in the other cloud) occurs
is known, and is denoted by $p$ ($0