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{\Large
\begin{center}\bf 28th \\ ORT\-VAY RU\-DOLF \\
PROBLEM SOLVING COMPETITION IN PHYSICS \\
1997
\end{center}
}
\bigskip
\bigskip
\noindent Deadline for solutions:
{\bf No\-vem\-ber 20 Thursday Noon, 1997.},
\smallskip
\noindent Postal address: D\'a\-vid Gyu\-la, Fi\-zi\-kus Di\-\'ak\-k\"or,
G\'o\-lya\-v\'ar, Hall\-ga\-t\'oi Iro\-da, \mbox{H-1088 Bu\-da\-pest,
M\'u\-ze\-um k\"o\-r\'ut 6-8.}
\noindent On email, \LaTeX\ format solutions should be sent to
{\bf dgy@lu\-dens.el\-te.\-hu}
\smallskip
\noindent On FAX solutions can be sent to 36-1-266-2556 .
\smallskip
\noindent Any reference is allowed to consult. Each university year will be
evaluated separately. For each problem 100 points can be given and no more
than 10 problem solution will be evaluated.
Participants are kindly asked to send each solution on a separate sheet,
indicating their names and years. In case of (nearly) identical
achievements preference will be given to contestants choosing problems
appropriate to their year.
\smallskip
\noindent The jury will give out zero, one, or more first, second and third
prize and further awards for each year.
\smallskip
\noindent For outstanding solutions of individual problems extra
rewards can be given.
Even with one solved problem prize may be won, therefore it is
worth sending
even relatively few solutions.
\smallskip
\noindent Declaration of results together with Physicists' Santa Claus will
take place in the E\"otv\"os Hall of building D, Science Faculty of
E\"otv\"os University, Budapest, on December 4, Friday 14.00, 1997.
Contestants who solve the individual
problems most successfully will kindly be asked to present their
solutions
after the declaration of results.
\noindent We wish instructive and successful problem solving for every
contestant:
\bFL Board of Physics Students of E\"otv\"os University
\medskip
Hungarian Association of Physics Students
\eFL
\vspace*{1cm}
\begin{enumerate}
\item
Oh, sunshine... Three people are standing in the middle of a field,
somewhere on the perfectly spherical earth, facing the sun, basking in the
sunshine. However, nothing lasts forever, even the sun moves in the sky. So,
our sun-worshippers start following the sun in such a way that their velocity
vectors always point in the opposite direction as their shadows. Arthur
Accelerationless never gets tired, the magnitude of his velocity is constant.
Bob Bottomheavy soon realizes that he would never catch up with the sun,
loses his heart, and as his shadow grows longer, his velocity decreases
inverse proportionally to the length of his shadow. On the other hand,
Clement Catchup turns ever angrier and faster: his velocity is proportional
to the length of his shadow.
What paths are traced out by the three sun-worshippers (A,B,C) on earth?
Discuss the solution according to the choice of starting point, the day and
minute of their set-offs, their initial velocity, and (in the case of B and
C) the coefficients of the velocity function. Examine the possible types of
motion and final points. (For simplicity, suppose that even the deep blue sea
cannot mean any problem for our wanderers.)
\bFL
(Bar\-na\-f\"ol\-di Ger\-gely -- D\'avid Gyu\-la)
\eFL
\item
Thor would like to have fish soup for dinner, so he creates a fireball on the
bottom of the lake. What will happen? A fireball is a region with an
approximate diameter of 1 metre where suddenly (in approximately 1 second)
such an amount of heat is produced that it would by far be enough for boiling
and evaporation (if it were applied slowly) $-$ however it is not enough
for molecular ionisation.
\vspace*{-0.2cm}
\bFL
(Csil\-ling \'Akos)
\eFL
\item
Sommefeld's parity violating pocket knife \newline
If a pebble is placed on the table with its convex side down and is spinned
in either direction, it will usually slow down smoothly and stop. Seldom, in
case of pebbles with special shapes it may happen that a pebble spinning in
one direction will start spinning in the other direction and stop only
afterwards. Nowadays, we can see such special shaped plastic "pebbles" in
some toy shops, and Sommerfeld's pocket knife is said to have been capable of
this trick. Give a physical explanation to the phenomenon, and examine
whether it is possible that a body spun on the table can turn back from BOTH
directions.
\bFL
(Rad\-nai Gyu\-la)
\eFL
\item
How large can be a unicelled that can serve for dinner for another cell (a
macrophag)? Macrophags are made up of approximately 60 per cent liquid and 40
per cent dry matter. Within the cell dry matter can only exist in the form of
small spheres. The volume of one sphere must exceed 0.5 per cent of the total
volume of the macrophag. Swallowing the unicelled happens by means of
pseudopodiums. The surface of the pseudopodiums cannot be more than 5 per
cent of the total surface.
\vspace*{-0.4cm}
\bFL (Horv\'ath An\-na) \eFL
\item
(The following problem is connected to one of the problems of the 1997
E\"otv\"os competition.)
A string of beads on a flexible thread is placed in a glass standing
on the brink of a table.
If a part of
this chain hangs out of the glass, it will start accelerating and will
eventually pull the rest of the chain out of the glass. According to
experiments, after a while the part of the chain falling over the rim of the
glass will be lifted and turns over well above the rim. Examine the motion,
the energy relations, the exerted forces, and explain the lifting of the
chain.
\bFL
(Gn\"adig P\'eter)
\eFL
\item
An effervescent tablet is dropped in water. Determine its size as a function
of time.
\bFL
(Horv\'ath An\-na)
\eFL
\item
Consider an inextensible chain in a uniform gravitational field suspended by
both ends. Examine the propagation of small perturbations along the chain.
Set up the wave equation in an easy-to-handle way, and derive the dispersion
relation. Study the different excitation states and give approximately the
corresponding angular frequencies. (The chain is excited only in the vertical
plane of the rest curve.)
\bFL
(Bor\-s\'a\-nyi Sza\-bolcs)
\eFL
\item
Estimate the seasonal change in the earth's angular velocity. What effects
can this be attributed to? How could this fluctuation change in the past 100
million years? How large will it be in 100-200 years' time, after the advent
of the greenhouse effect?
\vspace*{-0.4cm}
\bFL
(D\'a\-vid Gyu\-la)
\eFL
\item
Place three identical elastic balls along a straight line. Press the two
outside ones together, in the direction of the axis, so that the one in the
middle is compressed.
\begin{enumerate}
\item[a/] Examine the relation between the pressing force and the
displacement.
\item[b/] Forcing a
small displacement perpendicular to the axis on the
middle ball, shearing can be introduced by the contacts. Describe the motion
of the system. Special care should be taken of the boundary conditions at the
contacts.
\end{enumerate}
\vspace*{-0.4cm}
\bFL
(Kert\'esz J\'anos)
\eFL
\item
A rolled up water hose of radius $R$, length $L$ and mass $M$ is rolled out
on a horizontal plane with an initial velocity $v_0$ in such a way that its
end is fixed. The hose moves as a hoop with decreasing radius and mass but
increasing velocity. (Losses due to friction and resistance of media and the effects of gravitation can be
neglected, as $1/2 Mv^2 \gg MgR$.)
Use the system of units $M=1, L=1, v_0=1$.
Calculate the mass of the hoop, its radius and velocity as a function of the
covered distance $x$. Determine the external forces acting on the hoop. Show
that
$$
v=\frac{1}{\sqrt{1-x}},\qquad
\qquad F_{x}=\frac{1}{2(1-x)}
\quad \mbox{\rm and} \quad F_{y}=0,
$$
where $F_x$ and $F_y$ are the force components parallel with, and
perpendicular to the velocity.
Study the conservation of angular momentum
with respect to the fixed endpoint of the hose.
\vspace*{-0.4cm}
\bFL
(Gn\"adig P\'e\-ter)
\eFL
\item
Strong boys, after consuming enough beer, can easily fold tops of beer
bottles by compressing them between their forefinger and thumb. Estimate the
necessary force for such a feat of strength. (You may neglect the milling on
the rim of the bottle-top, or even consider it as a flat disk.)
\bFL
(Cser\-ti J\'o\-zsef)
\eFL
\item
Before hearing is established, sound waves have to be transformed into
chemical signals. How large are the increase and decrease in the intensity of
sound waves within the internal ear, if the propagation path of the waves is
air--membrane--bone--bone--bone--membrane--air?
Where should the series be cut to avoid resonance?
For the geometric arrangement consult some anatomical references. Mechanical
properties can be approximated by regarding membrane as a rubber layer, and
bone as scale.
\bFL
(Horv\'ath An\-na)
\eFL
\item
Podunk's local government is going to set up a new attraction in the local
amusement park, namely the Podunk's Swinging Lift (PSL). According
to the plans the PSL would be differ from the ordinary lifts in the respect
that:
\begin{enumerate}
\item[1/]
there is, as the name suggests, no elevator shaft, so that
the cabin can swing without restriction
apart from just moving vertically.
\item[2/] there will be no engine in it
therefore the ride would depend only on the tug--of--war
between the swinging elevator cabin and the vertically moving counterbalance.
(The vertical frame allows only one plane of movement for the cabin.)
\end{enumerate}
The problems began when all of the known lift making firms refused
to develop a detailed technical design, so that the local government
had to turn to wider range of prospective designers for completing
the necessary basic calculations that would allow a safe operation of
the system.
The most important questions are the following:
\begin{itemize}
\item[a/]
How many passangers should fit into the cabin given mass of the
counterbalance?
\item[b/]
There is a pneumatic mechanism, that can be used to start and stop the lift
and also to hold the cabin steady for boarding, mounted
on the frame of the lift. This mechanism can be approached via stairs.
The mechanism can only impart a certain (given) amount of horizontal
momentum to the vertically situated cabin, by knocking it.
What should be the range of the starting height for the point of view of
the system's safety?
(The length of the wire--rope is the same as the height of the frame.)
How does the nature of the ride change as a function of the starting height?
\item[c/] What is the longest safe ride given that braking
can be carried out only at the starting height?
\item [d/] During the round is it necessary to protect the passangers
against bumping their heads? Can the cabin turn over?
\item[e/]
Find the maximal safe value of tension in the wire-rope!
Can the designers be sure that the rope is tense all the time during
the ride?
\end{itemize}
Besides the point of view of safety, economics also have to be taken into
account: varied rounds would be needed which draw the attantion of
many people.
The smooth operation of the system would be guaranteed by one of
the representative
of the council who is a retired colonel running the local petrol station.
The local government is grateful for every kind of result.
Even partial results would be welcome!
%\item CSERELJETEK KI AZ ANGOL VALTOZATRA!
%
%A la\-posl\'api \"on\-korm\'any\-zat \'uj att\-rak\-ci\'ot k\'esz\"ul
%fel\'all\'\i ta\-ni a he\-lyi vid\'am\-park\-ban: a La\-posl\'api Leng\H o Lif\-tet.
%A ter\-vek sze\-rint az LLL ab\-ban k\"ul\"onb\"oz\-ne a ha\-gyo\-m\'anyos
%fel\-von\'okt\'ol, hogy (1) nincs lift\-akn\'aja, \'\i gy a ka\-bin -- amint az a
%n\'evb\H ol sejt\-het\H o -- a f\"ol-le lif\-tez\'es mel\-lett sza\-ba\-don
%leng\-het is, ezenk\'\i v\"ul (2) mo\-tor sem lesz ben\-ne, a me\-ne\-tek
%le\-foly\'asa a leng\H o lift\-ka\-bin \'es a csak f\"ugg\H ole\-ge\-sen mozg\'o
%el\-lens\'uly k\"oz\"ot\-ti k\"ot\'elh\'uz\'as\-ban d\H ol majd el.
%(A f\"ugg\H ole\-ges tart\'oszer\-ke\-zet csak egy leng\'esi s\'\i kot tesz
%le\-het\H ov\'e a ka\-bin sz\'am\'ara.)
%5A gon\-dok ak\-kor kezd\H od\-tek, ami\-kor a r\'esz\-le\-tes m\H usza\-ki terv
%ki\-dol\-goz\'as\'at egyik is\-mert fel\-von\'ogy\'art\'o c\'eg sem v\'al\-lal\-ta,
%ez\'ert az \"on\-korm\'any\-zat most sz\'el\-e\-sebb k\"orb\H ol v\'ar\-ja
% rend\-szer biz\-tons\'agos \"uze\-mel\-tet\'es\'ehez sz\"uks\'eges
%alap\-vet\H o sz\'am\'\i t\'asok elv\'egz\'es\'et.
%A leg\-fon\-to\-sabb k\'erd\'esek a k\"ovet\-kez\H ok:
%\begin{enumerate}
% \item[{ a/}]
% H\'any utas\-ra ter\-vezz\'ek a ka\-bint, ha az el\-lens\'uly t\"ome\-ge adott?
% \item[{ b/}]
% Az \'allv\'anyon l\'ev\H o, l\'epcs\H on megk\"ozel\'\i thet\H o
% pne\-u\-ma\-ti\-kus ind\'\i t\'o- \'es f\'ekez\H o\-szer\-ke\-zet, amely a besz\'all\'as
% idej\'ere r\"ogz\'\i ti is a rend\-szert, csak egy adott
% (v\'\i zszin\-tes ir\'any\'u)
% kezd\H ose\-bess\'eg\-gel k\'epes megl\"ok\-ni a f\"ugg\H ole\-ges hely\-zet\H u
% ka\-bint.
% Mi\-lyen tar\-tom\'any\-ban kell len\-nie az ind\'\i t\'oma\-gass\'ag\-nak a rend\-szer
% biz\-tons\'aga szem\-pontj\'ab\'ol? (A dr\'otk\"ot\'el ho\discretionary{sz-}{sz}{ssz}a meg\-e\-gye\-zik
% az \'allv\'any ma\-gass\'ag\'aval.)
% Mi\-lyen lesz a me\-net jel\-le\-ge az ind\'\i t\'oma\-gass\'ag f\"uggv\'eny\'eben?
% \item[ c/]
% Mi\-lyen hossz\'u me\-ne\-te\-ket en\-ged meg az a k\"or\"ulm\'eny, hogy a
% lef\'ekez\'est csak az ind\'\i t\'oma\-gass\'ag\-ban le\-het elv\'egez\-ni?
% \item[ d/]
% Kell-e az uta\-so\-kat \'ov\-ni a fej\-re\-es\'est\H ol a me\-net ide\-je alatt?
% \'Atp\"ord\"ul\-het-e a ka\-bin?
% \item[ e/]
% Mek\-ko\-ra ma\-xim\'alis fesz\'\i t\H oer\H ore m\'ere\-tezz\'ek a
% dr\'otk\"ote\-let? B\'\i zhat\-nak-e ab\-ban, hogy a k\"ot\'el v\'egig
% fe\-szes ma\-rad?
%\end{enumerate}
%A biz\-tons\'agi szem\-pon\-tok mel\-lett term\'esze\-te\-sen szem el\H ott kell
%tar\-ta\-ni a gaz\-das\'agoss\'agot is: sok em\-bert vonz\'o v\'al\-to\-za\-tos
%me\-ne\-tek\-re len\-ne sz\"uks\'eg.
%A rend\-szer ola\-jo\-zott m\H uk\"od\'es\'et a k\'ep\-vi\-sel\H otest\"ulet
%egyik tag\-ja, a he\-lyi ben\-zin\-ku\-tat is \"uze\-mel\-tet\H o nyu\-gal\-ma\-zott ez\-re\-des
%ga\-rant\'aln\'a.
%Az \"on\-korm\'any\-zat b\'ar\-mi\-lyen r\'esze\-redm\'enyt h\'al\'asan fo\-gad.
%
\vspace*{-0.4cm}
\bFL
(Ko\-v\'acs Zol\-t\'an)
\eFL
\item
During his breathing exercise, a yogi practises the floolwing mental
technique: "During inhalation the body expands, during exhalation it
contracts. Whenever I inhale, the surrounding air moves towards my nose, and
whenever I exhale, it moves in the other direction.
\begin{enumerate}
\item[a/] Estimate the displacement of air at a given
distance from the inhaling/exhaling yogi.
\item[b/] At what distance will it be immeasurably small?
\item[c/] How is the displacement influenced by the expansion
of the body with inhalation and its contraction with exhalation?
\end{enumerate}
\vspace*{-0.4cm}
\bFL
(M\'ark G\'e\-za)
\eFL
\item
\begin{enumerate}
\item[a/]
One wants to take a shower, so opens the hot water tap. The hot water comes
from a remote hot water reservoir through a fairly long pipe running in the
wall. Noone has taken a shower recently, so the water in the tube has assumed
the 10-degree surrounding temperature. What will the time dependence of the
water temperature be if the water current is constant?
\item[b/]
What happens if the wall (through which the pipe runs) has a nonuniform
temperature distribution, i.e. on a short section the temperature is
0.001 ${}^{\circ}$C instead of 10?
\item[c/]
One lets the hot water run for a very long time (the hot water reservoir is
supposed to be infinite). This, of course, makes the water too hot, so one
opens the cold water tap (which will not be touched again). Then one starts
playing with the hot water tap by turning it according to the following
function of time:
$\phi(t)=\phi_0+\phi_1\cdot\sin(\omega t)$.
The quantity of the water flowing from the tap is proportional to the
angular deflection. Find the time-dependence of the water flowing from the
tap.
\end{enumerate}
\vspace*{-0.4cm}
\bFL
(Ve\-res G\'abor)
\eFL
\item
Derive the Neumann equation describing the time evolution of the area of a
bubble in a two dimensional soap foam (i.e. several contacting bubbles):
$$
\frac{d A_n}{dt}=f(A_n,n,k),
$$
where $A_n$ is the surface of an $n$-faced bubble, $n$ is the number of the
faces, and $k$ is the diffusion coefficient. Give the concrete form of the
function $f$. What is the critical value of the number of sides?
\bFL (Da\-ru\-ka Istv\'an) \eFL
\item
A vessel contains liquid. The vessel is rotationally symmetric, but its wall
is not perpendicular to its bottom. Determine the equilibrium surface of the
liquid. (Consider only solutions with rotational symmetry).
\bFL
(Far\-kas Z\'en\'o)
\eFL
\item
From the bottom of a water container placed on the table a tube leads to a
cupboard sized black box next to the table. If some more water is poured in
the container, the original water level decreases. On the other hand, if
water is removed from it, the level will rise.
What is inside the black box? Set up as simple a model as possible and give
the change of the water level in terms of the water poured in or removed.
Change the parameters of the model and determine the behaviour of the system.
\bFL
(Cs\'ak\'any An\-tal)
\eFL
\item
According to Von Mises's condition, an isotropic material yields plastically
if the stress is such that
$$
\tilde{\sigma}_{ij}\tilde{\sigma}_{ij} >K^2,
$$
where $\tilde{\sigma}_{ij}$ is the traceless part of the stress tensor:
$\tilde{\sigma}_{ij}=\sigma_{ij}-1/3\sigma_{kk}\delta_{ij},$, and
$K$ is a material constant.
According to Tresca, the yield limit is determined by the eigenvalues of the
stress tensor. The yielding condition can be expressed with the smallest and
largest eigenvalues of the stress tensor as
$$
|\sigma_{max}-\sigma_{min}|>K'
$$
where $K'$ is a material constant.
Show that in two dimensions the Tresca's and Von Mises's conditions give the
same criterion. Find deformations through which it could be experimentally
decided which condition is realized.
\bFL
(Tichy G\'e\-za)
\eFL
\item
In a large vessel, far form the walls, a spherical cavity (bubble) of radius
$R_0$ is created with vacuum inside. The external air pressure is $p_0$.
\begin{enumerate}
\item[a/] Describe the behaviour of the wall of the cavity.
\item[b/] Describe the motion of a light particle inside the cavity if its
initial velocity is $v_0$, and it collides elastically with the wall.
\item[c/] How do the above change if the bubble is initially filled up with
gas of pressure $p_0$, and the external pressure varies with time as $p = p_0
\cos (\omega t)$? What is the minimal size of the bubble? How does its
temperature change?
\end{enumerate}
\bFL
(Csa\-bai Istv\'an)
\eFL
\item
When a cluster of grapes is washed, quite some water gets stuck in the
inside, which can later be removed by careful shaking (without damaging the
grapes, of course). How much water can stay in the inside (after shaking)?
The cluster of grapes can be modelled by identical fixed spheres that fill
out space as densely as possible, and we can neglect other parts of the
cluster, e.g. a frame holding the grapes together.) On what conditions, and
how much water can stay in the of this configuration? (The puddle of water
is demanded to be connected.)
Does the neglected frame has a role in case of real grapes? And the size of
the grapes?
Hint: Try to examine the phenomenon experimentally, too.
\vspace*{-0.4cm}
\bFL (Ve\-res G\'abor) \eFL
\item
It is a oft-quoted thesis that conservation laws are consequences of
symmetries. More specifically, in classical field theory for each continuous
symmetry there exists a continuity equation for a conserved quantity. What
symmetry implies the well known continuity equation of hydrodynamics?
\bFL
(D\'a\-vid Gyu\-la)
\eFL
\item
A one-dimensional rod of length $L$, Young modulus $k$, linear mass
density $\mu(x)$ is rotated about its longitudinal axis with angular velocity
$\omega$. As it is well known, over a certain angular frequency $\omega >
\omega_0$ the rod will "buckle", i.e. its equilibrium position will no longer
be the straight line joining the two endpoints. Determine the aforementioned
angular velocity $\omega_0$. \\
Hint: determine the Green function and take an inside look in the theory of
integral equations.
\bFL
(Hantz P\'eter)
\eFL
\item
Set up an experiment in which the propagation velocity of light between two
points of space is not measured in a back-and-forth way but only in one
direction. Or: show that this is impossible. Analyze the consequences of this
problem on special relativity.
\vspace*{-0.4cm}
\bFL
(Sza\-b\'o L\'asz\-l\'o)
\eFL
\item
In relativistic hydrodynamics, in the presence of dissipative forces, the
continuity equation is not satisfied by the (rest) density alone, but only by
its sum with a vector perpendicular to the four-velocity. What can be its
physical meaning? The continuity equation is usually considered as the
mathematical form of the conservation of mass. Perhaps in our case this
conservation law does not hold?
\vspace*{-0.4cm}
\bFL
(D\'a\-vid Gyu\-la)
\eFL
\item
In the era of Imre Mad\'ach (1823-64) the heat of the sun was believed to be
produced by the carbon burning in its inside. From the mass of the sun and
the outgoing heat it was calculated that the sun would be bright for another
5000 years. Give an estimate, how long the cooling of a real
body with such mass
and size would take after the energy producing processes are stopped.
\vspace*{-0.4cm}
\bFL
(Ve\-res G\'a\-bor)
\eFL
\item
A charge moves in an electrostatic field. The field is built up as a
superposition of numerous random electic fields, generated by independent
sources. Determine the probability that the lowest frequency of the light
emitted by the charge is between $\omega$ and $\omega + d \omega$. Assume that
the charge can only move in a plane.
\bFL
(Poll\-ner P\'e\-ter)
\eFL
\item
Let ${\bf E}$ and ${\bf B}$ be static, electric and magnetic source-free
fields in
${\rm I\! R}^3$, respectively, which are infinitely many times
differentiable. Furthermore, assume that they satisfy either of the following
(Bogomolny) equations
$$ {\bf E}= \pm {\bf B}.
$$
Prove that as long as the equations describe a finite energy configuraion,
${\bf E}=0$ and ${\bf B}=0$. \\
Hint: Examine the twist of the fields in infinity.
\vspace*{-0.4cm}
\bFL
(Ete\-si G\'a\-bor)
\eFL
\item
Calculate the partition function of classical perfect gas and a system of $N$
independent plane rotators. Why is it necessary to use different
normalizations in the two cases in order to get extensive potentials?
\bFL
(Poll\-ner P\'eter)
\eFL
\item
Gas is known to permeate all space available. Specifically: if half of a
vessel is filled up with gas and the other half is empty, and the separating
wall is removed, the gas molecules will soon fill up both halves of the
vessel uniformly. Statistical mechanics explains this by arguing that the
macro state in which both halves contain gas is realized by many more micro
states than the asymmetric one, therefore it is much more probable.
However, according to quantum mechanics, particles are indistinguishable. So,
the state in which only half of the vessel contains gas is only
one quantum state,
just like the other state when the vessel is uniformly filled out, as sheer
interchange among the particles will not result in a new state. Therefore,
the usual argument of statistical physics falls through. Still, despite
quantum theory and Pauli's exclusion principle, our experiences show that gas
after all permeates all space available. Why?
\bFL
(Gn\"adig P\'eter)
\eFL
\item
The virial theorem of classical mechanics establishes a relation between the
total energy and the time averages of the kinetic and potential energies for
a system performing a bound motion. The relation is especially simple in
case of potential energies that are power-law functions of distance (see
Landau and Lifshitz, Theoretical Mechanics, Volume 1).
In quantum mechanics, a similar formula holds, though not for the time
average but for the expectation value, e.g in the case of a hydrogenic atom
(see e.g.\ Constantinescu and Magyari, Quantum Mechanical Problems, problem 3/8).
Examine the case of a dielectronic helium atom, where electric interactions
are mediated by $n=-1$ power-law functions between each pair of particles.
Does the virial theorem hold? What are the effects of the interchange
interaction of typical quantum mechanics?
\bFL
(Gy\"or\-gyi G\'e\-za -- D\'avid Gyu\-la)
\eFL
\pagebreak
\item
Model the motion of a fleeing space-fly.
The latest problems of Space Station MIR have been caused by a fly. This
gives the astronauts a serious headache, as the fly disturbes their work.
They ask for the help of the Ortvay competitors. According to video records,
the motion of the fly has the following features: in the state of
weightlessness the fly cannot percieve any preferred direction (gravitational
and magnetic fields do not affect its motion). For obvious reasons, the fly
is in a state of panic, and so it flies with its final speed throughout its
motion. The direction of its motion is determined by the processes happening
in its mind $-$ and they are as clear as mud. The only thing we know about
them is that they are "clear as mud uniformly in time" (fly-psychology is not
a fully developed science yet).
\begin{enumerate}
\item[a/] Describe the motion of the fly.
\item[b/] Characterize the flight path.
\item[c/] How could the walls be incorporated in the description of motion?
\item[d/] The astronauts get fed up with the insolent animal and decide to
close it in a lock chamber (which is separated from the rest of the space
ship by a narrow slit that can be opened and closed). In the instant $t=0$
the fly was observed to be in the immediate vicinity of the lock chamber.
According to the quantitative analysis of the motion they think that in
certain instants of time the fly is more likely to be in the vicinity of the
lock chamber. Can we determine these instants of time in our present model?
\end{enumerate}
\bFL
(Al\'acs P\'eter)
\eFL
\item
There is a classical spin in each corner of a square. Assuming nearest
neighbour interaction between them the Hamiltonian of the system is
$$
H=\eta \,{\bf S}_1 \cdot {\bf S}_2 - {\bf S}_2 \cdot {\bf S}_3
- {\bf S}_3 \cdot {\bf S}_4 - {\bf S}_4 \cdot {\bf S}_1, \nonumber
$$
where $\eta \in \left[ -1, \infty \right[$,
and all spins are of unit length, i.e.
$${({\bf S}_i)}^2 =1, \quad i=1, \cdots, 4.$$
Determine the ground state of the system as a function of the parameter
$\eta$. Calculate the energy and magnetization of the system in ground state.
\bFL
(Cser\-ti J\'o\-zsef)
\eFL
\item
A one dimensional system is described by the following Hamiltonian:
$$
H(x,p)=x^2 p^2 - \frac{1}{x^2}.
$$
Examine its classical and quantum behaviour, its motion, its energy eigenvalues,
and its spectrum. Special care should be taken of the construction of the
Hamiltonian operator.
\bFL
(Baj\-nok Zolt\'an)
\eFL
\item
The method of partial waves describes the quantum mechanical problem of
elastic scattering in a central potential field by means of the $\delta_l$
phase shifts that can be read off from the asymptotic form of individual
partial waves (angular momentum eigenstates). Can such a scattering potential
$V(r)$ exist (apart from the trivial, uniformly 0 potential) in which,
on a fixed energy, each partial wave has 0 phase shift, i.e. the system is
supertransparent?
\bFL
(D\'avid Gyu\-la)
\eFL
\item
Determine the energy levels of the fullerene molecule ($C_{60}$) in
tight-binding electron approximation. Assume in the model that atomic cores
are located as if they were atoms of a fullerene molecule and {\it one}
electron is added to this system. The Hamiltonian is
$$
\hat H |i\rangle = \epsilon _0 |i\rangle - t \sum\limits_{j=0}^{60}
\alpha _{ij} |j\rangle
$$
where $|i\rangle$ designates the state in which the electron is coupled to
the $i$th atom, and
$\alpha _{ij} = 1$
if $i$ and $j$ designate neighbouring atoms, 0 otherwise.
The problem should be solved by considering the symmetries of the molecule.
What can we say about the degeneracy level of the individual energy levels?
(The solution can be tested numerically).
\bFL
(Far\-kas Z\'en\'o)
\eFL
\item
Consider the following decoherence model that tries to deduce the classical
features of macroscopic bodies from quantum mechanics.
A pointlike particle of mass $M$ moves in one dimension and it collides with
$n$ particles (each of mass $m$) moving along the same line. Let
$M \gg n m$
(e.g. gas atoms collide with a very massive particle - Brownian motion). The
light particles do not interact among each other, and the interaction of the
heavy particle and the light particles can be modelled by the rigid sphere
potential. Let the initial wave function of the light particles be
$$
\frac{1}{\sqrt[4]{2\pi \sigma^2}}\exp \left(-\frac{(x_j-x_j^{0})^2}{4\sigma^2}+i\frac{p_j^0 x_j}{\hbar}\right)\,,
$$
where $x_j^0, p_j^0, \sigma$ are constants, and $x_j$ is the coordinate of
the $j$-th particle. Let the initial wave function of the heavy particle be
$\psi(x)$.
Calculate the reduced density matrix $\rho(X,X')$ of the heavy particle (see
Landau and Lifshitz, Theoretical Physics, Volume 3). Assume that the
observable properties of the heavy particle are eigenstates of the reduced
density matrix, that is they are described by the $\varphi_j(X)$ solutions of the eigenvalue problem
$$\int_{-\infty}^\infty dX' \rho(X,X')\varphi_j(X')=p_j \varphi_j(X).$$
Determine these functions and the probability of their occurence $p_j$, if
\begin{enumerate}
\vspace*{-.1cm} \item[a/]
$$\psi(X)=\alpha
\frac{1}{\sqrt[4]{2\pi \sigma_0^2}}\exp \left(-\frac{(X+a)^2}{4\sigma_0^2}\right)
+\beta \frac{1}{\sqrt[4]{2\pi \sigma_0^2}}\exp \left(-\frac{(X-a)^2}{4\sigma_0^2}\right)\,,
$$
where $\alpha\ne \beta$ and $\sigma_0<<\frac{\sigma}{\sqrt{n}}<>\sigma$, i.e. $\psi(X)$ is one wide Gaussian function.
\end{enumerate}
Calculate the functions $\varphi_j$ in momentum representation, too. Are the
physical consequences derivable from the model in accordance with experience?
\bFL
(Be\-ne Gyu\-la)
\eFL
\item
The conductance of a clean nano-wire at low temperatures is quantized.
Suppose, we have an infinitely long straight wire. Then the conductance
is given as
$$ G=\frac{2e^2}{h}N,$$
where $e$ is the electron charge, $h$ is Planck's constant and
$N$ is the number of open modes in the wire at Fermi energy $E_F$.
Open modes are the propagating wave solutions of the Schr\"odinger
equation in the wire at energy $E_F$ such that
$$\psi(x,y,z,E_F)=\Phi_{n,m}(x,y)e^{ik_{n,m}z},$$
where $z$ is the coordinate parallel to the wire and $x$ and $y$ are
orthogonal and the wavenumber $k_{n,m}$ is a real value depending on the
$n,m$ quantum numbers.
The wave function $\psi(x,y,z,E_F)$ vanishes outside and at
the wall of the wire. Calculate the conductance as a function of the
Fermi energy if the cross section of the wire is a circle or a square.
The interaction between the electrons are taken into account by an effective
mass, so that the electrons can be treated as the free electrons.
\vspace*{-0.4cm}
\bFL (Vattay G\'abor, Cserti J\'ozsef) \eFL
\item
Consider a two-dimensional quantum mechanical system whose potential is of
the form
\mbox{$V(x,y)=\alpha y^2$}, where $x$ and $y$ are the two planar coordinates.
What is the conductivity of the system if the electrons are imagined to be
let in at one end of the potential trough (i.e. $x=-\infty$) and the number
of electrons arriving at the other end (i.e. $x=\infty$) is measured. In such
a case we can try the wave function ansatz resulting as the product of a
longitudinal (x-direction) wave and a transverse mode (y-direction):
$e^{-ik_nx}\cdot u_n(y)$. One can say that the electron is in the $r$th
channel if its transverse wave function is exactly $u_r(y)$. To obtain the
solution make use of the Landauer formula establishing a relationship between
the transmission amplitude (resulting from the transmission matrix elements
$t_{nm}$) and the conductivity. (The quantities $|t_{nm}|^2$ give the
scattering probability of an electron from the $n$th channel to the $m$th
channel.) \\
English and Hungarian references available at the homepage
{\em http://ga\-la\-had.el\-te.\-hu/$\sim$ge\-gix/publ.html}
can be used.
\bFL
(Sz\'al\-ka Ger\-gely)
\eFL
\item
How large background would be detected in the Gran
Sasso neutrino detector
if a muon accelerator were built in the tunnel of LEP?
\bFL
(Csil\-ling \'Akos)
\eFL
\item
As is well known, the equation of motion in Aristotelean mechanics... would
have been, had Greeks been familiar with differential equations
$$
{\bf \dot{r}}(t)={\bf e}({\bf r},t),
$$
where $r$ is the radius vector of the particle, and the function
${\bf e}({\bf r},t)$ is the so-called {\em hypoforce} which describes the effects
of the particle's neighbourhood. In the absence of such a hypoforce the
solution of the equation is ${\bf r}=const.$, i.e. the natural state of
bodies is equilibrium.
Ever since Galilei, Newton's law is valid:
$$
{\bf \ddot{r}}(t)={\bf f}({\bf r},{\bf \dot{r}},t),
$$
where ${\bf f}({\bf r},{\bf \dot{r}},t)$ is the ordinary {\em force}; since then
force-free state corresponds to uniform, rectilinear motion.
At the breaking of the 21st century, a new revolution is taking place. N.G.
Neer and J. Berwocky (researchers of the University of Santa Claus) have
recently published [X Fi\-les, {\bf 42} (1997) p. 137.] the so-called
Newerton law:
$$
{\bf \dddot{r}}(t)={\bf F}({\bf r},{\bf \dot{r}},\ddot{\bf r},t)
$$
where ${\bf F}({\bf r},{\bf \dot{r}},\ddot{\bf r},t)$ is the so-called
{\em hyperforce}, which describes the effects of the particle's neighbourhood.
Examine the consequences of the equation. Derive the analogues of the
well known conservation laws. Determine some simple solutions, i.e. in case
of some specific hyperforce function choices (hints: free motion,
free fall, harmonic
oscillator, hydrogenic atom). How does the derivation of the inertial forces
appearing in accelerated reference frames change in the mechanics of the new
age? Interpret the inertial hyperforces. Try to work out the new versions of
Lagrangian and Hamiltonian formulations; set up the Hamilton-Jacobi-Neer
equation and the Poisson-Berwocky brackets. Make the first steps towards the
extension of the new mechanics into special
(hyper)relativistic and quantum theoretical
directions. Those who know the ropes can set up a hyper-Schr\"odinger
equation (and can even solve it)...
\bFL
(D\'a\-vid Gyu\-la)
\eFL
\item
Camp for Freshmen... $N$ boys studying physics and $N$ girls studying {\it
liberal} arts are sitting by the flickering campfire. Just by them are $N$,
strictly coed tents (for two people each). Egon Quark has to arrange
the sleeping order.
It is known that on the average each person is willing to sleep in one tent
with $m$ members of the other sex - but only Egon knows who with
who.
How should $m(N)$ be chosen so that Egon would have a $50\%$ chance to arrange
a sleeping order that is satisfying for everyone? Give the asymptotic
behaviour of $m(N)$ as $N \rightarrow \infty$. \\
For $N$ fixed, define the width of the transition ``Egon flunks $-$ Egon does
not flunk'' $(f(N))$. How does $f(N)$ behave as $N \rightarrow \infty$?
\bFL
(Pi\-r\'oth At\-ti\-la)
\eFL
\end{enumerate}
{\tt $\backslash$end\{do\-cu\-ment\} }
\end{document}
Bi\-hary !!!!!!!!!!