EUF

Joint Entrance Examination

for Postgraduate Courses in Physics
For the first semester of 2017
October 04, 2016
Part 1

Instructions
• Do not write your name on the test.
It should be identified only by your candidate number.

• This test contains questions on:

classical mechanics, quantum mechanics, modern physics, and thermodynamics.
All questions have the same weight.

• The duration of this test is 4 hours.

Candidates must remain in the exam room for a minimum of 90 minutes.

• The use of calculators or other electronic instruments is not permitted during the exam.

• Answer each question on the corresponding sheet of the answer booklet.

The sheets with answers will be reorganized for correction. If you need more answer space, use
the extra sheets in the answer booklet. Remember to write the number of the question (Qx)
and your candidate number on each extra sheet. Extra sheets without this information will be
discarded. Use separate extra sheets for each question. Do not detach the extra sheets.

• If you need spare paper for rough notes or calculations, use the sheets marked scratch at the
end of the answer booklet. Do not detach them. The scratch sheets will be discarded and
solutions written on them will be ignored.

• Do not write anything on the list of constants and formulae.

Return both this test and the list of constants and formulae at the end of the test. The latter
will be used in the test tomorrow.

Have a good exam!

Q1. A body of mass m falls from rest along a straight line inside a fluid. The acceleration of gravity
~g can be considered as constant. The body is also subject to a drag force proportional to the
velocity: F~r = km~v , where k is a constant. Neglect the buoyant force of the fluid.
(a) Obtain the body’s speed as a function of time.
(b) What is the body’s terminal speed (speed in the limit t ! 1)?
(c) Find z(t), the body’s position as a function of time (consider z(0) = 0).
(d) Find z(v), the body’s position as a function of its speed.

Q2. The planar double pendulum consists of two particles of masses m1 and m2 and two massless
rigid rods of lengths l1 and l2 , which swing, under the action of gravity ~g , on a fixed vertical
plane, as depicted in the figure below. Considering ~g as constant and adopting as generalized
coordinates the angles ✓1 and ✓2 in the figure, obtain:

l1
θ1

m1
l2
θ2
m2

(a) The kinetic energy of the system.

(b) The potential energy of the system.
(c) The Lagrangian of the system.
(d) The equations of motion for ✓1 and ✓2 .

Q3. Consider the quantum non-relativistic dynamics of a particle of mass m in a three-dimensional

isotropic harmonic potential of angular frequency ! given by
m! 2 2
V (x,y,z) = (x + y 2 + z 2 ).
2

(a) Write the eigenstates |n1 ,n2 ,n3 i of the total Hamiltonian Ĥ in terms of the eigenstates of
one-dimensional harmonic oscillators |ni i (i = 1,2,3) and also the eigenenergies of Ĥ.
(b) One of the eigenenergies of the system is 72 ~!. What is its degeneracy?
(c) The observable Ĥ is measured when the system is in the following state (assume the
eigenstates |n1 ,n2 ,n3 i are normalized)
1 1 1
| i = p |0,0,1i + |0,1,0i + |0,1,1i.
2 2 2
What are the possible outcomes and what are their probabilities?
(d) Suppose the outcome of the measurement in item (c) was 52 ~!. Consider t = 0 the instant
immediately after the measurement. Find the state of the system | (t)i for t > 0.

1
Q4. The energy density u(T ) of electromagnetic radiation in equilibrium at temperature T can be
expressed (starting from thermodynamic arguments) as
Z 1 ⇣⌫ ⌘
u(T ) = ⌫ 3f d⌫, (1)
0 T
where ⌫ is the radiation frequency.

(a) Using only Eq. (1), find u(T ) up to a constant factor (that is independent of T ). What
is the dimension of this factor?
(b) In 1900, Planck discovered that
⇣⌫ ⌘ 8⇡⌫ 2 h⌫
⌫ 3f = 3 (h⌫/k
,
T c e BT ) 1
where h is Planck’s constant (which relates the quantum of energy to the frequency), c is the
speed of light in vacuum and kB is Boltzmann’s constant.
8⇡⌫ 2
(i) Discuss, without deriving it, the physical significance of the factor c3
d⌫.
(ii) Find the behavior of the energy distribution in the limit in which the photon energy is
much smaller than the thermal energy kB T .
(iii) What is the meaning of the result of item (ii) in the context of classical physics?

(c) The constants c, ~ = h/(2⇡) and the gravitational constant G may be used to define abso-
lute units of time (tP ), distance (lP ) and mass (mP ). Find these quantities in terms of products
of powers of ~, c and G. Find also Planck’s temperature TP . Estimate the order of magnitude
of tP , lP , mP and TP in the international system of units.

Q5. A student wants to measure the specific heat cx of an unknown substance x. For that purpose,
he can use a calorimeter, a device that ideally does not exchange heat with the environment.
The calorimeter’s heat capacity K is known. The calorimeter is initially at room temperature
Tamb . The experimental protocol consists of putting a known mass of water mH2 O at room
temperature Tamb in the calorimeter, adding a known mass of substance x, mx , initially at a
temperature
p Txp> Tamb , and measuring the final equilibrium temperature Teq . If necessary, use
2⇠= 1,4 and 26 ⇠ = 5,1.
(a) Write the equation required for determining cx from the data provided.
(b) Considering K = 30.0 cal/o C, the specific heat of water cH2 O = 1.0 cal/(g o C) and using
mH2 O = 50.0 g, mx = 200 g, Tx = 37.8 o C, Tamb = 25.0 ± 0.1 o C, the student found the final
equilibrium temperature of the system to be Teq = 27.8 ± 0.1 o C. Find cx . Express your result
with the uncertainty in the value of cx .

2
 EUF

Joint Entrance Examination

for Postgraduate Courses in Physics
For the first semester of 2017
October 05, 2016
Part 2

Instructions
• Do not write your name on the test.
It should be identified only by your candidate number.

• This test contains questions on:

electromagnetism, quantum mechanics, modern physics, and statistical mechanics.
All questions have the same weight.

• The duration of this test is 4 hours.

Candidates must remain in the exam room for a minimum of 90 minutes.

• The use of calculators or other electronic instruments is not permitted during the exam.

• Answer each question on the corresponding sheet of the answer booklet.

The sheets with answers will be reorganized for correction. If you need more answer space, use
the extra sheets in the answer booklet. Remember to write the number of the question (Qx)
and your candidate number on each extra sheet. Extra sheets without this information will be
discarded. Use separate extra sheets for each question. Do not detach the extra sheets.

• If you need spare paper for rough notes or calculations, use the sheets marked scratch at the
end of the answer booklet. Do not detach them. The scratch sheets will be discarded and
solutions written on them will be ignored.

• Do not write anything on the list of constants and formulae.

Return both this test and the list of constants and formulae at the end of the test.

Have a good exam!

Q6. A thin ring of radius R and total charge Q > 0 uniformly distributed throughout its perimeter
is fixed on the xy plane of a coordinate system with its center at the origin O. Let P be a
point with coordinates (0,0,z) (see the figure below).

z
dE
P

θ
r
z
y

R
O dQ x

(a) Summing up the contributions of all charge elements of the ring, find the electric field
~
vector E(z) (magnitude and direction) at the point P .
(b) Proceed analogously to item (a) and find the electric potential V (z) at the point P .
(c) A point particle of charge q < 0 and mass m starts at rest from a point with coordinates
(0,0,z0 ), very far from the origin (i. e., z0 R) and moves along the z axis. What is its speed
when it reaches the ring center? Neglect the e↵ects of electromagnetic radiation generated by
the particle in its journey towards the ring center.

Q7. A rigid square loop of wire of side L has total electric resistance R. The loop is on the xy
plane of a coordinate system and moves with velocity ~v away from a region where there is a
uniform magnetic field B ~ (shaded area of the figure below) pointing out of the page (in the
positive z direction). Consider the instant
p when the left-hand vertex of the loop is a distance
s inside the shaded region (0 < s < 2L/2).

y v
x L

B
R

(a) Find the magnetic flux through the loop as a function of s.

(b) Find the magnitude and the circulation direction of the electric current induced in the
loop.
(c) Find the total magnetic force (magnitude and direction) on the loop when the induced
current is flowing in it. What additional force other than the magnetic force must be applied
to the loop so that it moves with constant velocity under the sole action of these two forces?

1
Q8. Consider the quantum non-relativistic dynamics of an electron (mass m and charge e) moving
along the x axis in a one-dimensional harmonic oscillator potential with angular frequency !.
~ = E x̂ along the same axis.
The electron is also subject to an electric field E

(a) Write the total Hamiltonian of the system.

(b) Let |ni (n = 0,1,2, . . . ) be the eigenstates of the harmonic oscillator. Let us now consider
the e↵ect of the electric field as a small perturbation that changes only slightly the eigenenergies
and eigenstates of the harmonic oscillator. Let V̂E be the term of the Hamiltonian due to the
electric field. In this case, the correction to the energy of the first excited state to linear order
in E is given by the average value of V̂E in |1i. Calculate this correction.
(c) The correction to the first excited eigenstate to linear order in E is given by
r
(1) ~ eE ⇣ p ⌘
| 1 i= |0i 2 |2i . (2)
2m! ~!
The correction to the energy of the first excited state to quadratic order in E is given by the
matrix element of V̂E between |1i and the state in Eq. (2). Calculate this correction.
(d) Calculate the exact eigenenergies of the system (harmonic oscillator plus electric field)
by means of a coordinate transformation. Compare the exact result with the perturbative
calculation of items (b) and (c). Do they agree? Why?
Q9. An extra-solar planet is c T light-years distant from the Earth (c is the speed of light and T is
the time in years it takes light to reach the planet from the Earth). An expedition is planned
to send astronauts to the planet in such a way that they should age 3T /4 years in the outward
journey. The journey is made almost completely at constant speed. Neglect, therefore, the
small portions with accelerated motion.

(a) What should the astronauts’ constant speed relative to the Earth be on the outward jour-
ney?
(b) According to the astronauts, what will the traveled distance be on the outward journey?
(c) Each year (according to the spaceship’s clock) the astronauts send a light pulse back to the
Earth. What is the period of arrival of the pulses at the Earth?
(d) In the middle of the outward journey, a couple of astronauts decides to the return to Earth
in a space module. According to the astronauts that remain in the original spaceship, the
module travels at speed 5c/6. Find the total time (measured from the Earth) the couple of
astronauts will have stayed away from our planet.

Q10. A system of N one-dimensional, localized and independent quantum oscillators is in equili-

brium with a thermal reservoir at temperature T . The energies of each oscillator are given
by ✓ ◆
1
En = ~!0 n + n = 1,3,5,7, . . .
2
Note that the possible values of n are the odd positive integers only.
(a) Find an expression for the internal energy per oscillator u as a function of temperature T .
What is the expression for u in the classical limit (~!0 ⌧ kB T )? Sketch a plot of u versus T .
(b) Find an expression for the entropy per oscillator s as a function of temperature T . What
is the expression for s in the classical limit? Sketch a plot of s versus T .