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Ph. D. Qualifying Exam
Jan. 2017
Modern and Statistical Physics
You can do all the problems—the best 3 count towards your total score
Problem 1: (40 points)
Show the energy levels, ordered in energy, for the states in neutral Helium
with n  1 or 2 for the two electrons. Give the total J , L , and S plus the
degeneracy for each state, and explain why the energy ordering occurs. Give
a zeroth order estimate for the three primary energy levels (in eV) but do not
give numeric values for the shifts of those levels (just explain where the shifts
are coming from).
Problem 2: (40 points)
Assume the density of electrons in a white dwarf is 1030 /cm3. What are the
Fermi energy and average energy in eV of the degenerate electrons assuming
(a) the electrons are non-relativistic? [20 points]
(b) the electrons are ultra-relativistic? [20 points]
(in reality they are somewhere in between a and b)
Calculate numbers (in eV) using:
electron mass = 0.511 MeV/c2
hc  1240 eV nm
c  197 eV nm
Problem 3: (40 points)
A future colliding beam experiment is proposed to observe the process
e e   HZ , where H is the Higgs boson, which subsequently decays (in a
majority of events) into a bottom quark-antiquark pair: H  bb . In some
fraction of events, the Z boson can also decay into a bottom quark-antiquark
pair: Z  bb . Let the e  and e  beams each have energy E / 2 , and the
masses of the Higgs boson and the Z boson are M H and M Z . You may treat
the electron and the bottom quark as massless. The speed of light is c  1 .
(a) What are the energies and the magnitudes of the 3-momenta of the
Higgs boson and the Z boson in the center-of-momentum frame?
[20 points]
(b) For bottom quark jets produced in such events, what is the maximum
possible energy? Do they come from Z decays or H decays?
[20 points]
[You do not need to plug in numbers, but the approximate relevant ones might
be: E  300 GeV, MH  125 GeV, MZ  91 GeV.]
Problem 4: (40 points)
If a degree of freedom qi contributes a term aqi2 to the Hamiltonian, the
principle of equipartition says that its average contribution to the energy is
1
2 kT .
(a) Calculate the rms deviation (or “energy fluctuations”) of aqi2 about its
average. [35 points] (note: rms = root mean square)
N
(b) Show that if the Hamiltonian contains N such terms
aqi2 , with
i 1
average energy E  N  12 kT , the rms deviation about this average is
2
E showing that for a system with many degrees of
N
freedom, such as a macroscopic collection of molecules, the average
energy fluctuations about the mean are a very small fraction of the mean
energy. [5 points]
then  
Problem 5: (40 points)
(a) Using the Maxwell Relations (see below), evaluate an expression that
U
will enable you to calculate
for a gas given its thermodynamic
V T
equation of state f  p,T ,V   0 . [25 points]
(b) For the Van der Waals equation of state for n moles of gas

an 2 
p

(V  nb )  nRT

2 
V


derive an expression for
U
V
. [15 points]
T
The Maxwell Relations (assuming the only work done is pdV work) can be
conveniently classified according to which pair of variable is chosen to be
independent.
1. S and V independent:
 TdS  dU  pdV
dQ
U
T ,
S V
T
V

S
U
V
dU  TdS  pdV
 p
S
p
S V
--------- See Next Page ---------------
2. S and p independent:
dH  dU  pdV Vdp TdS Vdp
H
S
T
p
H
p
T ,
p

S
V
S
V
S
p
3. T and V independent:
dF  dU TdS  SdT  pdV  SdT
F
V
F
 S
T V
 p
T
p
S

T V V
T
4. T and p independent:
dG  dU TdS  SdT  pdV Vdp  SdT Vdp
G
T
G
V
p T
 S
p
S
V

p T
T
p
Ph. D. Qualifying Exam
Aug. 2016
Modern and Statistical Physics
Do
4
out of
4
problems (the best 3 count)
Problem 1: (40 points)
Ray 2
Microwaves
from Radio
Star
Detector
d
Ray 1
Lake
Microwave Detector
A microwave detector is located at the shore of a lake at a height d above the
water level. As a radio star emitting monochromatic microwaves of
wavelength  rises slowly above the horizon, the detector indicates
successive maxima and minima of signal intensity.
(a) What is the difference in phase between Ray 1 and Ray 2 arriving at the
detector (in terms of d ,  , and  )? [20 points]
(b) At what angle  above the horizon is the radio star when the first
maximum is received (in terms of d and  )? [20 points]
Problem 2: (40 points)
Classical Thermodynamics
The equation of the sublimation and the vaporization curves for a particular
material are given by
ln P  0.04  6 / T (sublimation)
ln P  0.03  4 / T (vaporization)
where the pressure P is in atmospheres (atm), and the temperature T is in
Kelvin. (The following problem involves simple arithmetic calculations.)
For the following subsections, the subscripts 1, 2, and 3 denote solid, liquid,
and vapor phases, respectively.
(a) Find the temperature of the triple point TTP . [10 points]
(b) Show that the specific latent heats of vaporization 𝑙23 and sublimation
𝑙13 are 4R and 6R , respectively. Here R is the Gas constant.
(Assume that the specific volume in the vapor phase is much larger than
the specific volume in the liquid and solid phases). [20 points]
(c) Find the latent heat of fusion 𝑙12 . [10 points]
Problem 3: (40 points)
Boltzmann Gas
Here we consider a non-interacting gas of N classical particles in three
dimensions. The gas is confined in a container of volume V . The energy of
the i th particle is given by  i  pi2 / (2m ) .
(a) Calculate the canonical partition function
cN H qi ,pi 
Z (T ,N )  f e
h
(c )
(kBT )
d
where H is the Hamiltonian of the system and d is the volume
element of the 2 f - dimensional phase-space spanned by the
components of the coordinates qi and momenta pi (i  1,..., N ) . What
is the number of spatial or momentum degrees of freedom, f , of the
gas? What is the value and meaning of the constant cN ? Use the
thermal de Broglie wavelength, B  h
2 mkBT , and the average
particle distance, a  (V / N )1/3 , in your expression. [15 points]
(b) Calculate the free energy F from Z (c ) . F is the thermodynamic
potential associated with the canonical ensemble, i.e., [5 points]
F  kBT ln Z (c ) (T ,N )
(c) Now, the particle number is not fixed anymore and we go over to the
grand canonical ensemble. Calculate the grand canonical partition
function by Laplace transformation
Z
(gc )

(T ,  )   e
 N
kBT
Z (c ) (T ,N )
N 0
What is the meaning of  ? [10 points]
(d) Show that the relation J  F  N for the thermodynamic potential
J (T ,  ) (Planck-Massieu function) of the grand canonical ensemble
holds in the thermodynamic limit using the fact that the average particle
number in the grand canonical ensemble can be obtained as [10 points]
N

ln Z (gc ) (T ,  )
( / (kBT ))
Hint: Use Stirling’s formula for large N : N !  N Ne N
Do not forget to answer the short questions in (a) and (c)!
Problem 4: (40 points)
Mandelstam Variables
Consider the relativistic two-body scattering A  B C  D with
corresponding 4-vectors pA , pB , pC , and pD . The masses of the
corresponding objects are mA , mB , mC , and mD . You can set the speed of
light as c  1 if you would like. The Mandelstam variables are useful Lorentz
invariants:
s  (pA  pB )2
t  (pA  pC )2
u  (pA  pD )2
(a) Show that s  t  u  mA2  mB2  mC2  mD2 . [10 points]
(b) Show that the total center of mass energy is given by s . [10 points]
(c) The LHC collides protons in the lab frame with an energy of 6.5 TeV
against other protons with an energy of 6.5 TeV (so that s  13 TeV).
One of the things looked for is the production of new Z  bosons. What
is the most massive Z  that the LHC could directly observe?
[10 points]
(d) Why can the photon not decay? [10 points]
Hint 1: Prove that the mass of a decaying particle must exceed the
masses of the particles produced in its decay. In other words, if A
decays into B  C , show that mA  mB  mC .
Hint 2: Evaluate in the center of mass frame.
Ph. D. Qualifying Exam
Feb. 2016
Modern and Statistical Physics
Do 3 out of 5 problems
Problem 1: (40 points)
(a) The equation of state of a van der Waals gas is
𝑎
(𝑃 + 𝑣 2) (𝑣 − 𝑏) = 𝑅𝑇
where P is pressure, T is temperature, R is gas constant, v specific
volume, a and b are characteristic constants for a given gas.
𝑎
Qualitatively explain the meaning of the terms 2, and –b.
𝑣
[10 points]
(b) A van der Waals gas undergoes an isothermal expansion from
specific volume v1 to specific volume v2. Calculate the change in
the specific Helmholtz function. [15 points]
(c) Calculate the change in the specific internal energy in terms of v1
and v2. [15 points]
Problem 2: (40 points)
A negatively charged muon stops in aluminum, which has Z=13, where it is
captured and decays down to the 1S state.
(a) What is the energy of the gamma emitted when the muon drops from
the n 2 to the n 1 atomic state? Give your answer in terms of the
ground state (1S) energy of hydrogen. [20 points]
Al
If the muon then undergoes the transition:
e
Al
(b) What normally conserved quantity is violated? [5 points]
(c) What is the total kinetic energy of the electron in terms of the masses
of the muon, electron, and aluminum? [15 points]
masses: muon 105 MeV,
electron 0.5 MeV,
Aluminum 25000 MeV
Problem 3: (40 points)
One-dimensional relativistic gas: Here we consider a non-interacting gas
of N relativistic particles in one dimension. The gas is confined in a
container of length L , i.e., the coordinate of each particle is limited to
0 qi L . The energy of the i th particle is given by i  c pi .
(a) Calculate the single particle partition function Z1(T , L) for given
energy E and particle number N . [12 points]
(b) Calculate the average energy E1 and the heat capacity C 1 per particle
from Z1(T , L) . [12 points]
(c) Calculate the Boltzmann entropy SB (E, N ) of all N particles.
Consider them as indistinguishable. [16 points]
Hint: Use Stirling’s formula for large N :
N!
2 NN Ne
N
.
Problem 4: (40 points)
A cylindrical resistor has a radius b , length L , and conductivity 1 . At the
center of the resistor is a defect consisting of a small sphere of radius a
inside which the conductivity is 2 . The input and output currents are
distributed uniformly across the flat ends of the resistor.
(a) What is the resistance of the resistor if
1
2?
[5 points]
(b) Approximate the spherical defect as:
 a cylinder of radius a and length a .
 concentric with the cylinder of the physical resistor
 and centered lengthwise on the center of the physical resistor.
What is the resistance of the resistor if
1
2?
[25 points]
(c) Estimate the relative change in the resistance to first order in 1
2
if 1
2 . (Make any assumptions needed to simplify your method
of estimation). [10 points]
Problem 5: (40 points)
Estimate the following quantities, indicating how you arrived at your
estimate (do not try to calculate exact numbers—make only estimates):
Choose only 4 of the following 7 quantities: [10 points each]
(a) the frequency of radiation used in a household microwave oven,
(b) the energy yield of a fission bomb with a 30 kg uranium core,
(c) the energy of impact of the earth with a meteorite 10 meters in
diameter,
(d) the temperature of the sun,
(e) the temperature of a 60-watt light bulb filament,
(f) the speed of sound in helium gas at STP,
(g) the total length of blood capillaries in the human body
You may find the following useful:
Speed of sound in a gas: v
( P )1/2 , where
C P / CV
The speed of sound in air is about 330 m/sec
T 4 , where B 6 10 8 W/m2·K4
Stefan’s Law: F
B
Wein’s displacement Law: maxT 2.9 10 3 m·K
The average human has about 4 liters of blood
Typical energy release in a fission process is about 200 MeV
Earth’s orbit has radius of about 8 light minutes
The radius of the sun is about 2 light seconds
The earth’s orbital velocity is about 30 km/sec
The sun’s energy flux at the earth’s surface is about 1.3 kW/m2
Ph. D. Qualifying Exam
Sept. 2015
Modern and Statistical Physics
Do
3
out of
5
problems
Problem 1: (40 points)
A parallel, thin, monochromatic laser beam of wavelength falls on a
diffraction grating at normal incidence. The grating spacing is d .
(a) Derive the grating equation for normal incidence giving the relation
between d , , and . [10 points]
(b) Next rotate the grating through an angle  90 around an axis that is
parallel to the grating lines. Derive the grating equation giving the
relation between d , , , and . [20 points]
(c) How does the interference pattern produced on a viewing screen differ
between parts (a) and (b)? (intuitive guesses are allowed). [10 points]
Problem 2: (40 points)
Classical Thermodynamics
(a) A van der Waals gas undergoes an isothermal expansion from specific
volume v1 to specific volume v2 . Calculate the change in the specific
Helmholtz function. [20 points]
Hint: The equation of state of a van der Waals gas is
a

 P  2   v  b   RT . Here P is the pressure, v the specific volume, T is
v 

the absolute temperature, and R is the gas constant, the constant a and
b are the characteristic constants of the gas.
(b) For the above problem, calculate the change in the specific internal
energy in terms of v1 and v2 . [20 points]
Problem 3: (40 points)
Statistical mechanics
There is a system of two identical particles, which may occupy any of the
three energy levels  0  0, 1   ,  2  3 .
The system is in thermal equilibrium at temperature T , which means that the
system is a canonical ensemble. For each of the following cases, determine
the partition function and the energy and carefully enumerate the
configurations.
(a) The particles are Fermions. [10 points]
(b) The particles are Bosons. [10 points]
(c) The particles obey Boltzmann statistics and now they are
distinguishable. [10 points]
(d) Discuss the conditions under which Fermions or Bosons may be
treated as Boltzmann particles. [10 points]
Problem 4: (40 points)
Electric Conductivity of Copper
The Drude model for a metal assumes that the conduction electrons can be
approximated by a gas of free electrons where the only important parameters
for the gas are n , the number density of electrons, and , the time between
collisions.
(a) Show that in this model the electrical conductivity of a metal can be
expressed as [24 points]
ne 2

m
(b) Estimate the collision time
for an electron in copper. The
resistivity of copper metal is 2 10 6 cm and the atomic density of
copper is 9 1022 atoms/cm3. [16 points]
Problem 5: (40 points)
The D0 meson , a combination of a charm quark and up anti-quark (𝑐𝑢̅), can
decay to K-mesons and Pi-mesons in several different channels. Three
possibilities are
𝐷0 → 𝐾 − + 𝜋 +
𝐷0 → 𝜋 − + 𝜋 +
𝐷0 → 𝐾 + + 𝜋 −
Where a 𝐾 + meson is composed of 𝑢𝑠̅, a 𝐾 − meson is composed of 𝑢̅𝑠, a 𝜋 +
meson is composed of 𝑢𝑑̅, and a 𝜋 − meson is composed of 𝑢̅𝑑.
(a) Draw the leading order Feynman diagrams for each of the above
decays. [10 points]
(b) Based on the quark flavors present in the initial state and the final
state, rank these decay channels from highest branching fraction to
lowest branching fraction. Explain your answer. [5 points]
(c) In a general two-body decay, particle A, at rest, decays into particles B
and C (𝐴 → 𝐵 + 𝐶). Find the energy and the magnitude of the
momentum of particles B and C. Express your answer in terms of the
particle masses (mA, mB, and mC) and the speed of light (c).
[25 points]
Ph. D. Qualifying Exam
Sept. 2014
Modern and Statistical Physics
Do
3
out of
5
problems
Problem 1: (40 points)
(a) A laser beam having a diameter D in air strikes a uniform flat piece of
glass of index of refraction n at an angle . What is the diameter of
the beam in the glass?
(b) A narrow beam of white light is incident in air at an angle on a
uniform flat sheet of glass of thickness L. The index of refraction for
red light is
and for violet light it is . Determine the diameter of
the emerging beam assuming the incident beam is exceedingly
narrow.
Problem 2: (40 points)
Find the maximum total energy and maximum transverse momentum for the
initally at rest particle after the following elastic scatter reactions:
(a) a 10 GeV positron strikes an electron at rest
(b) a 10 GeV proton strikes an electron at rest
(c) a 10 GeV electron strikes a proton at rest
Note: mass electron = 0.5 MeV mass proton=1 GeV (for this problem)
Problem 3: (40 points)
A
B
The system above is thermally isolated. It consists of two parts, a bucket
(A) containing a mixture of ice and water, and a body (B) with constant heat
capacity . The two parts are connected by a thermodynamic device which
can extract heat from part (A) and add heat to part (B).
The entire system is initially at temperature (the absolute temperature of
melting ice). The latent heat of melting of ice is per unit mass. Ignore all
effects due to changing volumes.
(a) How much heat must be removed from (A) to freeze an additional
mass of water? What is the entropy change of (A) in this process?
[10 points]
(b) In this process (B) absorbs the heat released by the thermodynamic
device, and its temperature increases to . Calculate the heat
absorbed
and the entropy change
of the body (B) in terms of
. [10 points]
(c) What is the minimum possible
? [20 points]
? What is the minimum possible
Problem 4: (40 points)
Calculate the radius of the largest atom which may be located
interstitially (without exercising stress) in a BCC lattice of a metallic system
whose atoms have a diameter of R.
Problem 5: (40 points)
(a) What are the two conditions for Fermi-Dirac statistics that the fermions
obey? [5x2 = 10 points]
(b) There is an assembly of N non-interacting fermions under the
conservation of particles and energy:
∑
∑
where Nj is the number of particles with single-particle energy εj; N and
U are fixed. There are gj quantum states at the jth energy level.
i.
Derive the thermodynamic probability ωj of this assembly of
fermions for the jth energy level εj. [10 points]
ii.
Using the above result, find the total number of microstates
corresponding to an allowable configuration ωFD. [5 points]
iii.
Express and Derive the Fermi-Dirac distribution function for a
discrete energy level εj. Hint: Find the occupation number of each
energy level when the thermodynamic probability is a maximum,
i.e., the equilibrium macrostate Ni. Apply the method of Lagrange
multipliers for a function with two variables under the conservation
of particles and energy. Set one of the multipliers to be 1/kT and
the other to be μ/kT, where μ is the chemical potential.
[3 + 12 points]
NIU Ph.D. Candidacy Examination Spring 2014 (2/22/2014)
Modern and Statistical Physics
Do 3 out of 5 problems
1. [40 points] Classical thermodynamics, Entropy of the van der Waals gas.
a) Express the combined first and second law of thermodynamics. [5 points]
b) Derive a Tds equation under the isochoric condition. Use specific heat under the isochoric
(constant volume) condition, cv. [20 points]
c) Calculate the entropy for a van der Waals gas, using the above Tds equation. Here, the
a⎞
⎛
equation of state of the van der Waals gas is ⎜ P + 2 ⎟ ( v − b ) = RT . [15 points]
⎝
v ⎠
2. [40 points] Statistics of One-dimensional lattice vibration
There is a one-dimensional lattice with lattice constant a as shown in figure below. An atom
transits from a site to a nearest neighbor site every τ seconds. The probabilities of transiting to
the right and left are p and q = 1 – p, respectively.
a) Calculate the average position x of the atom at the time t = Nτ , where N ≫ 1 . [20
points]
b) Calculate the mean-square value ( x − x ) at the time t. [20 points]
2
3. [40 points] Special relativity, a rocketship travelling back home.
A rocketship is traveling towards the Earth at 0.6c. It fires a projectile of mass M with a
velocity of 0.8c in the rocketship's frame directly at the Earth.
a) What is the energy of the projectile in the Earth's frame?
The rocketship is now traveling perpendicular to a line from the rocketship to the Earth at 0.6c
and again fires a projectile at 0.8c in the rocketship's frame and at 90 degrees from the
direction of the rocketship's motion (in the spaceship's frame).
b) What is the energy of the projectile in the Earth's frame?
c) What is the tangent of the angle of the projectile relative to the rocketship's motion in the
Earth’s frame?
You can ignore the effects of gravity as the rocketship is 2 light years away from the Earth.
4. [40 points] Higgs boson.
a) Show that the process γ → e + e − is forbidden in free space. [10 points]
b) Deduce an expression for the energy of a γ from the decay of a Higgs boson: H → γγ
in terms of the Higgs boson mass (m ), the Higgs boson energy in the lab frame (E), the
speed of the Higgs boson (β), and the emission angle (θ*) of one of the photons in the
Higgs boson rest frame. [20 points]
c) The Higgs boson does not couple directly to photons. Draw the most important, lowest
order Feynman diagram that illustrates the decay of the Higgs boson into two photons.
Remember to label all the particles in your diagram. [10 points]
H
5. [40 points] Optics.
A blade of grass standing ygrass tall is dA cm in front of a thin positive (convex) lens, A, having
a focal length, fA. A thin negative (concave) lens, b, with a focal length of –fB is placed dB
behind the first lens B. The blade of grass is illuminated by a monochromatic plane wave
parallel to the optic axis of this combined lens system.
a) Draw the ray diagram of imaging the above object by this combined lens system. [10
points]
b) Where does the first lens, A, form the image of the blade of grass? Express it with the
above given parameters. What’s its magnification? [5 x 3 = 15 points]
c) Where is the final image formed by this combined lens system? Express it with respect to
the location of the lens B. [10 points]
d) What is the total magnification of this combined lens system? [5 points]
NIU Ph.D. Candidacy Examination Fall 2013 (9/28/2013)
Modern and Statistical Physics
Do 3 out of 5 problems
1. [40 points] Classical thermodynamics, Entropy change in mixing.
Two equal quantities of water, each of mass m and at temperatures T1 and T2 are adiabatically
mixed together, the pressure remaining constant.
(a) Calculate the entropy change of the universe (thermodynamic surrounding) including cP
is the specific heat capacity of the water at constant pressure. [25 points]
(b) Show that ΔS > 0 for any finite temperatures T1 and T2. [15 points]
2. [40 points] Fraunhofer diffraction of double slits. Consider an aperture consisting of two
long parallel openings of constant width b and separation d in an opaque screen illuminated
by a coherent plane wave with a wavelength of λ. The length of the openings is L in the x
direction, perpendicular to the paper, and long enough to ignore the diffraction effects of both
ends. r0 lies in the yz-plane, and measures the distance from the geometric center of the
grating to a point on the observation screen. The coordinates on the observation screen are
(X, Y) and located far away to fulfill the Fraunhofer diffraction condition.
Y
y
r0
00
b
θ
d
z
Screen
b
Fraunhofer diffraction geometry for a multiple slit aperture.
(a) What is the condition for Fraunhofer diffraction? Express it in a mathematical form. [5 x
2 = 10 points]
(b) Calculate the electric field distribution and intensity (irradiance) along Y-axis on the
observation screen for a 2-slits (double slit) case. [20 points]
(c) Where along the Y-axis do bright fringes occur? Why? [5 points]
(d) What makes a missing order? [5 points]
1
3. [40 points] Fermi-Dirac gasses. We consider an ideal 3D Fermi gas comprising N
noninteracting fermions, each of mass m, in a container of volume V held at temperature T.
(a) Express the Fermi function at T = 0. [5 points]
(b) Find g ( ε ) dε , the number of quantum states whose energy lies in the range ε to ε + dε .
[10 points]
(c) Find the average energy per fermion at absolute zero by making a direct calculation of
U(0)/N, where U(0) is energy at T = 0 for N fermions. Express this in terms of the Fermi
energy, εF. [10 points]
(d) Similarly, calculate the average speed of a fermion gas particle at T = 0. Express this in
terms of the Fermi velocity, vF. vF is defined by ε F = (1 2 ) mvF2 . [15 points]
4. [40 points] A particle with mass M and energy E, collides with another particle of mass M
which is initially at rest. The result of the collision is two identical particles, each of mass m.
(a) Find a Lorentz transformation to the center-of-momentum frame, and find the energies of
each of the two initial particles in that frame. [20 points]
(b) Find the maximum and minimum possible energies Emax and Emin of the two final state
particle in the lab frame. [10 points]
(c) Find the energies of the two final state particles, if one of them is emitted at a right angle
to the initial direction of the incident particle in the lab frame. [10 points]
5. [40 points] Statistical mechanics. The partition function of an Einstein solid is
e−θ E 2T
,
Z=
1− e−θ E T
where θE is the Einstein temperature. Treat the crystalline lattice as an assembly of 3N
distinguishable oscillators.
(a) Calculate the Helmholtz function F. [10 points]
(b) Calculate the entropy S. [15 points]
(c) Show that the entropy approaches zero as the temperature goes to absolute zero. [15
points]
2
Ph. D. Qualifying Exam
Modern and Statistical Physics
Do
3
out of
5
problems
Problem 1: (40 points)
The specific Gibbs function of a gas is given by
g
§P·
RT ln ¨¨ ¸¸ AP
© P0 ¹
where ‫ ܣ‬is a function of ܶ. Find expressions for:
(a) the equation of state [15 points]
(b) the specific entropy [10 points]
(c) the specific Helmhotz function [15 points]
Feb. 2013
Problem 2: (40 points)
‫ܣ‬
ߙ
ߜ
‫ܥ‬
‫ܤ‬
The angle of deviation, ߜ, is defined as the angle between the direction of
the incident beam and the direction of the beam transmitted by the prism.
(a) When the incident beam strikes the prism at an angle such that the
angle of deviation is minimized, this minimum angle of deviation, ߜ௠ ,
gives a simple relation for the index of refraction of the prism. This
occurs when the angle of incidence for the 1st interface (‫ )ܤܣ‬is equal
to the angle of refraction for the 2nd interface (‫)ܥܣ‬. Find the refractive
index of refraction, ݊, of the prism in terms of the angles ߙ and ߜ௠ for
this situation. [25 points]
(b) Find the angle of deviation, ߜ, for the general case (shown above)
when it is not minimized. Find this angle in terms of the index of
refraction, ݊, of the prism, the incident angle of the 1st interface (‫)ܤܣ‬,
and ߙ. [15 points]
Problem 3: (40 points)
(a) Derive an approximate formula for the strength of the magnetic field
at the hydrogen nucleus produced by an electron in the first Bohr orbit
(using the Bohr model of the atom). [20 points]
(b) Give a rough order of magnitude estimate of the strength of the
magnetic field (in Teslas). Below are useful physical constants that
you can use: [10 points]
permeability of free space: ߤ଴ ൌ Ͷߨ ൈ ͳͲି଻ N/A2
permittivity of free space: ߝ଴ ൌ ͺǤͺͷ ൈ ͳͲିଵଶ C2/Nm2
charge of the electron:
݁ ൌ ͳǤ͸Ͳ ൈ ͳͲିଵଽ C
mass of the electron:
݉ ൌ ͻǤͳͳ ൈ ͳͲିଷଵ kg
3ODQFN¶VFRQVWDQW ݄ ൌ ͸Ǥ͸͵ ൈ ͳͲିଷସ Jsec
(c) If the nuclear Bohr mageton is ߤே ൌ ͷǤͲ ൈ ͳͲିଶ଻ J/Tesla give an
order of magnitude estimate of the hyperfine splitting (in eV) of
spectral lines. [10 points]
Problem 4: (40 points)
Solve for the threshold energy for the following examples of the production
of an electron-positron pair.
(a) Show the Feynman diagram for the process [5 points]
ߛ ൅ ܾܲ ՜ ݁ ା ൅ ݁ ି ൅ ܾܲ
(b) What is the threshold energy for the photon (expressed in terms of
electron masses) for the process in Part (a)? [5 points]
(c) What is the threshold energy for the photon (expressed in terms of
electron masses) for the process
ߛ ൅ ݁ ՜ ݁ା ൅ ݁ି ൅ ݁
where ݁ is a free electron? [15 points]
(d) Why are the threshold energies in Parts (b) and (c) different?
[5 points]
(e) In intergalactic space, protons can interact with the photons in the
cosmic microwave background via
ߛ ൅ ‫ ݌‬՜ ݁ା ൅ ݁ି ൅ ‫݌‬
Outline the steps (but don't calculate) needed to obtain the proton
threshold energy for this process. Assume the CMB photons all have
the same energy ‫ܧ‬௣௛௢௧௢௡ ൌ ݇ܶ (ܶ ൌ ͵ degrees Kelvin). [10 points]
Problem 5: (40 points)
Measurements of the entropy of a certain paramagnetic salt, as a function of
the temperature ܶ (in degrees Kelvin) and of the magnetic field ‫( ܪ‬in gauss),
have led to the following table (in units of ܴ per mole):
ܶ
‫ܪ‬
It is proposed to use this salt to produce very low temperatures by adiabatic
demagnetization. The sample can be pre-cooled to 0.8 Kelvin by pumping
on liquid He. The biggest available field is 10,000 gauss. What is the
lowest temperature which can be reached? Each step in your reasoning
should be explained very carefully (this is an essay question: numerical
computations are not necessary)
Ph. D. Qualifying Exam
Sept. 2012
Modern and Statistical Physics
Do
3
out of
5
problems
Problem 1: (40 points)
1. Classical thermodynamics, Gay-Lussac-Joule experiment.
(a) For any reversible process, calculate the dependence of the specific
!"
internal energy on volume,
, from the First law of
!" !
Thermodynamics. Here, the internal energy 𝑢 = 𝑢 𝑣, 𝑇 . Here u and
v are defined as u = U/n and v = V/n, where U, V, and n are the
internal energy, volume, and a number of kilomoles. (hint: utilize the
condition of an exact differential.) [25 points]
(b) For an ideal gas, find the dependence of 𝑢 on 𝑣 by calculating
[15 points]
!"
!" !
.
Problem 2: (40 points)
In a two-slit Young interference, the aperture-to-screen distance is 𝐿 and the
wavelength is 𝜆!"# .
(a) What slit separation, 𝑎, is required to produce a fringe spacing of ∆𝑦
at the screen? [15 points]
(b) Assume a thin glass plate of thickness 𝑑 and index of refraction 𝑛 is
placed over one of the slits (see figure below). The glass plate causes
the entire fringe pattern to laterally shift up or down on the screen.
Calculate the lateral fringe displacement. [25 points]
Screen
Glass plate
Problem 3: (40 points)
A 𝜋 meson of mass 𝑚! decays at rest into a muon (mass 𝑚! ) and a neutrino
of negligible mass. Express your answers below in terms of the masses 𝑚!
and 𝑚! .
(a) What is the momentum of the muon? [15 points]
(b) What is the kinetic energy 𝐾! of the muon? [15 points]
(c) The muon decays, too. Let 𝜏! be the mean lifetime of the muon in its
rest frame. What is the mean distance traveled by the muon in the
rest frame of the 𝜋 meson? [10 points]
Problem 4: (40 points)
A beam of either positively or negatively charged muons is stopped in a
piece of Pb. Answer the following questions separately for both a positive
muon and a negative muon. Hint: consider atom formation and the energy
levels of a hydrogen-like atom.
(a) Where is the most likely place for the muon to be after it stops in the
Pb for 𝜇 ! and for 𝜇 ! ? [10 points]
(b) What type of particles are emitted after the muon stops for 𝜇 ! and for
𝜇 ! ? [10 points]
(c) What are the energy ranges of the particles emitted in item (b)? Give
the answer in terms of eV or MeV for 𝜇 ! and for 𝜇 ! . [10 points]
(d) What is the lifetime of the muon in its own frame compared to it
decaying in vacuum? Do not calculate a number but state if it is the
same, a shorter, or a longer time and why for 𝜇 ! and for 𝜇 ! .
[10 points]
masses: muon 105 MeV
proton 938 MeV
electron 0.5 MeV
neutron 940 MeV
pion 135 and 140 MeV
photon and neutrino 0
Problem 5: (40 points)
Classical Thermodynamics of a Two-state System: A (very small) discrete
system has only two states 1 and 2 with energies 𝐸! = −𝜀! and 𝐸! = 𝜀! ,
respectively. This could, for instance, be a spin 1 2 particle in an external
magnetic field. Since this system contains only one particle, the different
thermodynamic ensembles (of the systems described below) do not provide
equivalent descriptions of the physics. We want to explore this difference
for this simplest possible system.
(a) If the system is isolated from the environment which are the possible
values for the internal energy of the system? [5 points]
(b) In the following we assume that the system is not isolated any more
but instead interacting with a heat bath of temperature 𝑇. Using the
canonical distribution of classical thermodynamics, what are the
probabilities 𝑝! to find the system in each of the two states in this
case? [10 points]
(c) Find the internal energy as a function of the temperature of the heat
bath (express it in terms of a familiar hyperbolic function).
[15 points]
(d) Using the limiting values of the expression in (c), what are the
possible values for the internal energy of the system when coupled to
the heat bath? [10 points]
Ph. D. Qualifying Exam
Feb. 2012
M odern and Statistical Physics
Do
3
out of
5
problems
Problem 1: (40 points)
1. Phase diagram, the critical point: The equation of state of van der
a
Waals gas is §¨ P 2 · v b RT , where a and b are characteristic
©
v ¹̧
constants for a given gas, and P, v, T and R are pressure, specific
volume, temperature and gas constant, respectively. When we regard the
above van der Waals equation as P = P(v, T)
(a) what are the three conditions at the critical point in the critical
isotherm (T = TC) on a P-v diagram? [15 points]
(b) Express the critical specific volume, vC, the critical temperature, TC,
and the critical pressure, PC, at the critical point, in terms of the
constants a and b. [25 points]
Problem 2: (40 points)
A metal ring is dipped into a soapy solution (index of refraction ݊௦ ) and
held in a vertical plane so that a wedge-shaped film formed under the
influence of gravity. At near-normal illumination with blue-green light
(wavelength ߣ௕௚ ) from an argon laser, one can see ࣨ fringes per cm.
Determine the wedge angle of the soap film. (Note: assume that the wedge
angle is very small).
Problem 3: (40 points)
2. A pi-mu atom consists of a pion and a muon bound in a Hydrogen-like
atom.
(a) What are the energy levels for such an atom compared to those for
Hydrogen expressed in terms of the electron, pion, and muon
masses? [20 points]
(b) The pi-mu atoms are produced in ‫ܭ‬௅ decays
‫ܭ‬௅ ՜ pi-mu atom ൅ neutrino.
If the ‫ܭ‬௅ has ߚ ൌ ͲǤͺ, what are the minimum and maximum energies
of the pi-mu atom in the moving frame of the ‫ܭ‬௅ ? (express these
energies in terms of the particle masses) [20 points]
Problem 4: (40 points)
Briefly explain or describe 4 of the following 6 phenomena (in no more
than 200 words for each phenomena): [10 points each]
(a) Electromagnetic structures of the neutron and proton
(b) The laser
(c) C, P, T, and CP symmetry (and any possible well known violations)
(d) The transistor
(e) 7KH-ȥSDUWLFOH
(f) Superconductivity
Problem 5: (40 points)
A system consisting of N (a very large number) identical weakly interacting
particles is in equilibrium with a heat bath. The total number of individual
states available to each particle is 2N. Of these, N states are degenerate with
energy 0, and N states are degenerate with energy ‫ܭ‬.
‫ܭ‬
N states
0
N states
ଵ
It is found by observation that the total energy of the system is ܰᖡ. Find
ଷ
the temperature of the heat bath under three different assumptions:
(a) That the particles are bosons. [10 points]
(b) That the particles are fermions. [10 points]
(c) That the particles obey the (unphysical) Boltzmann distribution.
[10 points]
(d) You should find from Parts (a), (b), and (c) that
ܶሺܾ‫݊݋ݏ݋‬ሻ ൐ ܶሺ‫݊݊ܽ݉ݖݐ݈݋ܤ‬ሻ ൐ ܶሺ݂݁‫݊݋݅݉ݎ‬ሻ
Explain why this is so. [10 points]
Modern and Statistical Physics Fall 2011
9/24/2011
Do 3 out of 5 problems
1. Classical Thermodynamics Ideal Gas [40 points]
Assume that the earth’s atmosphere is an isothermal ideal gas in a gravitational field.
Consider a thin layer of the ideal gas at height z and of thickness dz. There is a difference of
pressure across this thin layer, dP, which must just balance the gravitational force on the
mass. If the pressure at z = 0 is P0 , determine the pressure as a function of height z.
Fig.1
2. The Kinetic theory of Gas [40 points] The ditribution of particle speeds of a certain
hypothetical gas is given by
N ( v ) dv = Ave− v v0 dv ,
wherer A and v0 are constants.
a. Determine A so that f ( v ) ≡ N ( v ) N is a true probability density function; i.e.,
∞
∫ f ( v ) dv = 1 .
[10 points]
0
b. Find the mean speed, v , and the root mean square speed, vrms , in terms of v0. [10 points]
c. Find the most probable speed vm.
[10 points]
d. What is the standard deviation of the speeds from the mean, σ, in this case? [10 points]
3. Solid state physics, thermodynamics, statistical mechanics [40 points] There is a

semiconductor with two bands: The lower band is described by E k , the upper band by

E0 − E k . The two bands are separated by an energy gap of 2Δ. (Hint: Use electron-hole
( )
( )
symmetry. This is also evident from the way the upper band is defined.)
a. Calculate the chemical potential as a function of the temperature. [20 points]
b. Examine whether the system can be treated using classical statics at low temperatures.
[20 points]
4. High energy physics [40 points]
a. A particle of mass m is produced with energy E and decays after traveling a distance l.
How long did the particle live in its own rest frame? [8 points]
b. Use the result of part (a) to determine the spatial separation between the production and
decay vertices (a.k.a. “decay length”) of a B0 meson carrying an energy of 65 GeV, if it
lived 1.5 ps in its own rest frame. The mass of the B0 meson is 5.3 GeV. [8 points]
c. Determine the maximum energy that can be carried off by any one of the decay particles,
when a particle of mass m0 at rest decays into three particles with masses m1, m2, and m3.
[24 points]
5. Diffraction grating (Multiple slit diffraction) [40 points] Consider diffraction of an array of
multiple slits (N equal slits of width d and common spacing of a, numbered from 0 to N - 1) ,
or diffraction grating, illuminated by monochromatic plane wave normal to the array (Fig 2).
We assume that the length of the slit is long enough, and the obervation point P is far enough
from the slit.
a. Obtain the expression for the intensity of the multiple slit diffraction pattern at a point P.
[20 points]
b. Now using the above diffraction grating, we perform an experiment to determine the
wavelength of light. When the wavelength of the light λ changes to λ + dλ, the
diffraction pattern shifts. If this shifts is smaller than the width of a bright band of the
diffraction pattern, the wavelength λ and λ + dλ cannot be distinguished one another.
Determine the accuracy of the measurement of the wavelength by this experimental
method. [20 points]
Fig.2
Modern and Statistical Physics Spring 2011
2/26/2011
Do 3 out of 5 problems
1. High energy physics [40 points]
Consider the reaction p + gamma --> neutron + pi+.
a. Assume that the gammas are the cosmic blackbody radiation with average kT = 3 × 10 −4
eV. What is the threshold energy that the proton must have for the reaction to occur? [20
points]
mp ~ mn ~ 1000 MeV, mπ = 140 MeV, mµ = 105 MeV, mν = 0 eV.
b. The pion then decays to µ + ν. In the pion rest frame, what is the neutrino energy? What
is the neutrino energy in the "lab" frame? This is the source of the highest energy
neutrinos in the universe. Give answers in eV units. [20 points]
2. Relativity C-O white dwarf [40 points]
a. A C-O white dwarf has radius R and mass M. Assuming the electrons are degenerate,
what is the average energy of the electrons in terms of M, R and fundamental constants
assuming the electrons are non-relativistic? [15 points]
b. Repeat assuming the electrons are relativistic. [15 points]
c. Now assume that the radius shrinks by a factor of 2 so R' = R/2 while the mass remains
the same. What is the relative change in the gravitational and electron energies
(assuming both relativistic and non-relativistic) between R and R'? [5 points]
d. Is the star more stable (less likely to collapse) if the electrons are relativistic or nonrelativistic? Why? [5 points]
3. 2D harmonic oscillator [40 points]
Give a 2D harmonic oscillator with Hamiltonian, H =
p̂ 2x
2m
+
p̂ 2y
2m
+
(
)
1
mω 2 x 2 + y 2 + kmxy .
2
a. For k=0, what are the energies of the ground state and first and second excited states?
What are the degeneracies of each state? [20 points]
b. For k>0, using first order perturbation theory, what are the energy shifts of the ground
state and the first excited states? [20 points]
4
Classical mechanics Van der Waals Gas [40 points]
The Van der Waals equation of state for one mole of a non-ideal gas reads
a ⎞
⎛
⎜⎝ p + 2 ⎟⎠ (V − b ) = RT .
V
[Note: part (d) of this problem can be done independently of part (a) to (c).]
a. Sketch four isotherms of the Van derWaals gas in the p-V plane (V along the horizontal
axis, p along the vertical axis). Identify the critical point. [10 points]
b. Evaluate the dimensionless ratio pV/RT at the critical point. [10 points]
c. In a portion of the p-V plane below the critical point, the liquid and gas phases can
coexist. In this region the isotherms given by the Van der Waals equation are unphysical
and must be modified. The physically correct isotherms in this region are lines of
constant pressure, p0(T). Maxwell proposed that p0(T) should be chosen so that the area
under the modified isotherm should equal the area under the original Van der Waals
isotherm. Draw a modified isotherm and explain the idea behind Maxwell's construction.
[10 points]
d. Show that the heat capacity at constant volume of a Van der Waals gas is a function of
temperature alone (i.e., independent of V). [10 points]
5. Statistical mechanics [40 points]
There is a system of two identical particles, which may occupy any of the three energy levels
ε 0 = 0, ε1 = ε , ε 2 = 3ε .
The system is in thermal equilibrium at Temperature T, which means that the system is a
canonical ensemble. For each of the following cases, determine the partition function and the
energy and carefully enumerate the configurations.
a. The particles are Fermions. [10 points]
b. The particles are Bosons. [10 points]
c. The particles obey Boltzmann statistics and now they are distinguishable. [10 points]
d. Discuss the conditions under which Fermions or Bosons may be treated as Boltzmann
particles. [10 points]
Modern and Statistical Physics. September/October 2010
Do 3 of 5 problems
1. A top quark decays to a bottom quark and a W boson. The W boson then decays to a positron and a
neutrino. Give your answers below in terms of the masses mt , mb , mW of the top quark, the bottom quark,
and the W boson respectively. You may treat the positron and the neutrino as massless. You are
encouraged to set c=1.
(a) (14 points) What is the momentum of the bottom quark, in the rest frame of the top quark?
(b) (14 points) What is the maximum possible momentum of the positron, in the rest frame of the top
quark? (c) (12 points) Suppose the top quark has speed 0.5c in the lab frame, before it decays. What is the
maximum possible momentum of the positron in the lab frame?
2. A simple model of a rubber band is a one­dimensional (horizontal) chain consisting of N (N >> 1)
linked segments, as shown schematically in the diagram. Each segment has two possible states: horizontal
(H) with length a, or vertical (V), with length zero; with the figure indicating a state of HHHH­VHVH­
VHHH­HVHV­H. The segments are linked such that they cannot come apart. The chain is in thermal
contact with a reservoir at temperature T.
(a) (10 points) If there is no energy difference between the two states, what is the average length of the
chain?
(b) (25 points) Fix the chain at one end and hang weight from the other end, supplying a force F as shown.
Determine the average length of the chain at any temperature T and show that it is equivalent to
〈 L  T 〉=
.
aNe Fa / 2 kT
e−Fa /2 kT e Fa /2 kT
(c) (5 points) In which temperature limit is the extension proportional to F (Hooke's law)? Calculate the
constant of proportionality (that is the “k” from Hooke’s law).
3. Artificial materials (or metamaterials) can be engineered to provide a negative index of refraction n<0.
In this problem we explore the amazing properties of two arrangements of metamaterials described in the
Figure below. 1. (6 points) Consider an incoming optical ray at the interface from vacuum to a metamaterial. Write
down the expression for the refracted angle and draw the incoming and refracted ray for n=­1.
Indicate the difference(s) with the result obtained for conventional materials.
2. (17 points) We now consider a slab of metamaterial with thickness t as drawn in the Figure (a)
with n=­1. a. (6 points) Let a point­like source object be at a distance d=t upstream of the slab, show
that there is an image and give the location of this image. b. (6 points) Same question if the object is at a distance d=1/2t. c. (5 points) Comment on the properties of a simple slab with negative index of refraction
compared to a standard (n>0) slab. 3. (17 points) Consider the configuration shown in Figure (b). a. (6 points) Via geometrical construction, show that the source is imaged on the lower­
right quadrant. b. (6 points) Show that rays emitted by the source are also imaged on the source itself. c. (5 points) Discuss possible applications of such imaging/recirculating configurations. 4.
In the Big Bang theory of the universe, the radiation energy initially confined in a small region
adiabatically expands in a spherically symmetric manner. The radiation cools down as it expands.
a.
Derive a relation between the temperature T and the radius R of the spherical volume of
radiation, based purely on thermodynamic considerations. Here the radiation pressure is
expressed as p = U 3V , and the black body radiation energy density is
u = U V = aT 4 . [20 points]
b.
Find the total entropy of a phonon gas as a function of its temperature T, volume V and
the constants k,h, c . [20 points]
5. A flashlight is collimated so that its beam goes off in a 90 degree cone in its own rest frame.
a) (20 points) Calculate the opening angle of the flashlight cone when the flashlight
is moving forward at velocity v.
b) (20 points) Assuming that γ = 1/ 1 − v 2 / c 2 >> 1 show that the opening angle of
the cone is about 1/ γ
Modem and Statistical Physics. February 2010
Do 3 of 5 problems
*p)
e+ + n.
a. What is the threshold energy for the antineutrino
1. Consider the reaction nubar
if the reaction occurs off Hydrogen? Give answer
inMeV.
b. If the reaction occurs off Carbon 12 (with proton density of 0.1/F^3), the reaction will be suppressed
at low antineutrino energies. Explain why. (hint, note the two lowest spectroscopic states in nuclei.)
c. Assuming a Fermi gas model, what is the Fermi energy for protons nC 12? What is the average
proton kinetic energy? Give in MeV.
d. Up to what antineutrino energy will suppression in C 12 still be a factor?
:
g3g.6MeYlc2 hbar*c
mass neutron
mass proton: 938.3 MeV/c2
mass electron: 0.5 MeV/c2
2.
:
197
MeV*F
hc
:
1240 MeV*F
Forthis problem, do not calculate any exponentials or roots but put in all the other numbers for the final
answer
For a H atom, what is the ratio of the probability to be in the n:2Ievel compared to the n:l level at
3000 degrees K? E(lS) -- -13.6 eV and kT: .025 eV at T : 300 degrees K
b. For a metal with a Fermi energy equal to 4 eV and at T : 300 degrees K determine the following
assuming a Fermi gas model:
i. Wfrat is the density for conduction electrons? hc: 1240 eV*nm mass electron : 0.5 MeV/c2
ii. What is the ratio of the density of states between 4.4 eY and 4.0 eV?
iii. What is the ratio of the probability to have the energy be 4.4 eY compared to 4.0 eV?
c. For a photon in a Bosonic gas at T : 300 degrees K:
i. What is the ratio of the density of states between 4.4 eY and 4.0 eV?
ii. What is the ratio of the probability to have the energy be 4.4 eV compared to 4.0 eV?
T:
a.
3. A rocketship is traveling towards the Earth at 0.6c. It hres a projectile of mass M with a velocity of 0.8c
in the rocketship's frame. What is the energy of the projectile in the Earth's frame? If the rocket ship is
traveling perpendicular to a line from the ship to the Earth and again fires a projectile at 0.8c in the ship's
frame and uf 90 d.gr""s from the direction of the ship's motion (in the rocketship's frame), what is the
energy of the projectile in the Earth's frame? What is the tangent of the angle of the projectile relative
to the ship's motion (don't calculate but give as the ratio of two components)?
4. Optical mirage:
We explore the propagation of an optical ray in the (Oxz)
plane; see Figure. We assume the refractive index n obeys
the relation
a.
z = an2 * 6 where a andb are constants.
(10 points) Show that at a given height z we have
n(z)cosd:floaosd,o=A
b.
A is
a
constant.
(10 points) Deduce the differential equation for the
ray trajectory
c.
where
(#)' =(1)'-
l and show
that
the resulting ray trajectory is a pardbola.
(20 points) We now wish that the model discussed
in (f and (2) canbe used to explain the formation of optical mirages. We consider the (Oxz) plane
to be filled with an ideal gas. We assume the gas to be in mechanical equilibrium (so that the
change of pressure dP depends on the height variation dz as dP=-pgdz where g is the
gravitational constant and pthe gas density). We consider a temperature gradient along z of the
form T(z) : T0(l - kz)where k is a constant and firther assume the refractive index and the gas
density satisff the relation trn'z
-tIt p = const
.
(15 points) Show that the refractive index is related to
z via z : an2 -l b
and explicitly
determine the constants a and b.
(5 points) Qualitatively describe the formation of a "warm" mirage due to a hot floor.
5. Partition function of
an Einsteinsolid is
e "'
t7_t-'-en'r
where
ft
'
is the Einstein temperahtre. Treat the crystalline lattice as an assembly
a.
b.
c.
of 3N distinguishable oscillators.
Calculate the Helmholtz function F. [10 points]
Calculate the Enhopy,S. [10 points]
Show that the entropy approaches zero as the temperature goes to absolute zero. [10
d.
pointsl
Find the entropy at high temperatures. [10 points]
Modern and Statistical Physics. September 2009
Do 3 of 5 problems
I . The face-centered cubic (fcc) is the most dense and the simple cubic (sc) is the least dense of the three
cubic Bravais lattices. Suppose identical solid sphere are distributed through space in such a way that their
centers lie on the points ofeach ofthese three structures, and spheres on neighboring pointsjust touch,
without overlapping. (Such an arrangement of sphere is called a close-packing arrangement)
a). Assuming that the spheres have unit density, show that the density ofa set ofclose-packed spheres on
eachofthetbreestructures(the"packingfraction") ,"
centered cubic (bcc), and n l6 p 0.52 for sc.
tlinl6x0.l4
forfcc,
",linl8r0.68
forbody-
b). Show that for a fcc Bravais lattice the free electron Fermi sphere for valence 3 extends beyond point W
of the first Brillouin zone (see figure), so that the first Brillouin zone is completely frlled fHint: prove
u
kF
W = (1296 ll25 o')t' = 1.0081
lf
2. Considerthe reactiony+ e ) r| +n- +ewiththe electroninitiallyfree andatrest (withthis frame
designated as the lab frame).
a) What is the minimum photon energy for this reaction to proceed?
b) Assuming that the photon energy is the minimum determined in a), what is the velocity of either
the pions after the reaction?
of
c)Assumethepionthendecaystoamuonandneutrinon)V+v.Inthelabframe,whatisthe
maximum energy of the muon produced in the pion decay assuming the photon energy determined in a).
Give your answers in terms of the pion, muon, and electron masses while setting the photon and neutrino
masses to 0.
3. A 55 year old man can focus objects clearly from 100 cm to 300 cm. Representing the eye as a simple
lens 2 cm from the retina,
a) what is the focal length of the lens at the far point (focused at 300 cm)?
b) what is the focal length of the lens at the ner point (focused at 100 cm)?
c) what strength lens (focal length) must he wear in the lower part of his bifocal eyeglasses to focus at 25
cm?
4. A smooth vertical tube having two different sections is open from both ends and equipped with two
pistons of different areas. Each piston slides within its respective tube section. One mole of ideal gas is
inclosed between the pistons. The pistons are connected by a non-stretchable rod. The outside air pressure
is 1 atm. The total mass of the pistons is M. The cross sectional area of the larger upper piston A1, and the
lower piston A2 are related by 4 : A" + AA. Find the rise in the temperature of the gas between the
pistons required to lift the piston assembly by a distance L?
5. Assume that the neutron density in a neutron star is 0. l/frn3 lthat is 0. I neutron per
cubic Fermi). Assuming T:0, ignoringany gravitational forces, and using a Fermi gas model with uniform
density determine
a)
b)
c)
d)
the average energy ofthe neutrons
the average energy ofelectrons ifthe electron density is 1% ofthe neutron density
show two reactions, one which can convert a neutron to a proton and one which can convert a proton
to a neutron
determine the neutron to proton ratio. Hint, consider the Fermi energies of the neutrons, protons and
electrons at equilibrium
Give answers in i) and b) in MeV using hc : 1240 MeVfrn, the mass of the neutron: 1000 MeV/c2, and
the mass of the electron : 0.5 MeV/c2. You will need to decide if the particles are relativistic or nonrelativistic. The answer for d) can be given in terms of the particle masses.
February 2009. Stat/Modern/Thermo PhD. Qualifier. Pick 3 questions out of 5.
1.
a. (20) Show using conservation of energy and momentum that it is not possible for a free
electron moving through vacuum to emit a photon.
b. (20) Now consider the related problem of an electron moving through superfluid
helium. Show that in this case the electron can emit a phonon as long as it moves with a
velocity exceeding a critical velocity vc and find vc . The excitation spectrum of the
phonons in superfluid helium is given by E = up where u is the velocity of sound and p
is the momentum of the phonon. You may do this part in the non‐relativistic limit.
2.
a. (10) A C‐O white dwarf has a radius = R, mass = M and is assumed to have constant
density. Assuming the electrons are degenerate and non‐relativistic find the average
energy of the electrons in terms of M, R and fundamental constants.
b. (10) Repeat part a) assuming the electrons are relativistic.
c. (10) Assume the radius shrinks by a factor of 2 so R′ = R / 2 while the mass remains the
same. Find the relative change in the gravitational and electron energies (assuming
both relativistic and non‐relativistic) between R and R’.
d. (10) Discuss whether the star is more stable (less likely to collapse) if the electrons are
relativistic or if they are non‐relativistic.
3. When a large number of atoms come together to produce a solid each atomic level broadens
into a band. Draw a picture of simplified band structure, define the valence and conduction
bands, and describe the temperature‐dependent behavior of conductivity for:
a. (10) Metals (define the term Fermi energy)
b. (10) Insulators (define the term band gap)
c. (10) Intrinsic semiconductors
d. (10) The n‐type and p‐type semiconductors
4.
One mole of a monatomic ideal gas initially at temperature T0 expands from volume V0
to 2V0 . Calculate the work of expansion and the heat absorbed by the gas for the case
of expansion at:
a. (20) Constant temperature
b. (20) Constant pressure
5.
a. (20) Consider a large number of N localized particles in an external magnetic
field H (directed along the z direction). Each particle has a spin s = 1/2. Find the
number of states accessible to the system as a function of M s , the z component
of the total spin of the system. Determine the value of M s for which the number
of states is maximum.
b. (20) Define the absolute zero of the thermodynamic temperature. Explain the
meaning of negative absolute temperature, and give a concrete example to show
how the negative absolute temperature can be reached.
September 2008. Stat/Modern/Thermo PhD. Qualifier. Pick 3 questions out of 5.
1. The electronic structure of the atom is described by four quantum numbers.
a. Define these quantum numbers and describe the values they can acquire.
b. Write down the electronic configurations for elements with atomic numbers: 8
(Oxygen), 19 (Potassium), 29 (Copper).
c. Define the term atomic valence and find the valence for the elements O, K, and
Cu.
d. List four main types of bonding found in materials and give examples of the
elements and/or compounds for each type of bonding.
2. Consider the reaction ν + p → e + + n .
a. Find the threshold energy for the ν if the reaction occurs off Hydrogen.
b. Explain why the reaction will be suppressed due to Pauli Exclusion if it occurs
off 12 C (with proton density of 0.1/F3).
c. Assuming a Fermi gas model, find the approximate suppression as a function of
neutrino energy. ( mn = 940MeV/c 2 , =c = 197 MEV F, hc = 1240 MeV F)
3. A parallel beam of electrons is directed through a narrow slit of width a and a screen is
placed at a distance d from the slit. The first diffraction minimum is observed at a
distance y from the central maximum.
a. Find the velocity of the electrons in terms of a , d , y , and me assuming the
electrons are non-relativistic.
b. Find the velocity assuming the electrons are relativistic.
c. Estimate the spread of the electron beam through the slit using the Heisenberg
uncertainty principle and show that this is consistent with the spread estimated
based on the width of the central maximum.
4. We have an ensemble of N spins that have a Zeeman splitting described by the
Hamiltonian
N
H = −∑µ B Bσ .
(1)
i =1
Here B is the magnetic field, µ B is the Bohr magneton, and σ = ±1 is the direction of the
spin.
a. Show that the partition function is given by
µ B
Z = (2 cosh B ) N . (2)
k BT
b. Show that the energy in terms of the partition function is given by
∂ ln Z
(3)
E = k BT 2
∂T
and the magnetization by
∂ ln Z
M = k BT
(4)
∂B
c. Determine E and M for the partition function in Eqn. (2) and sketch them as a
function of temperature.
5. Chemical potential of an ideal gas.
a. Starting with TdS = dU + PdV show that the chemical potential of an ideal gas
can be written in terms of the temperature T and the volume V as :
µ = cPT − cV T ln(T ) − RT ln(V ) − S0T + constant .
Here, cP is the specific heat constant pressure, cv is the specific heat at constant
volume and S0 is a constant.
b. Starting from TdS = dH − VdP find a similar expression for the chemical potential
but now as a function of T and P.
c. Show that the chemical potential at the fixed temperature T varies with pressure:
P
µ = µ0 + RT ln  .
 P0 
Here, µ0 is the value of µ at the reference point (P0, T)
Stat/Modern/Thermo PhD. candidacy examination. March 2008
Pick 3 questions out of 5.
1) Consider a string of length L , mass density µ and string tension τ. The string is
maintained at finite temperature T .
a. Find the average amplitude of a mode of the string of wavelength λ = 2L n in the
limit of kB T >> hf , with f the frequency of the mode.
b. Explain why it is necessary to state that kB T >> hf .
€
2) Assume that the crystal lattice structure of solid comprising N atoms can be treated as an
€ of 3N distinguishable one-dimensional oscillators (Einstein solid).
assembly
€
a. What is the partition function Z? Use the Einstein temperature θE (≡ hν/k, where ν
is natural frequency).
b. Calculate the Helmholtz function F.
c. Calculate the entropy S.
d. Show that the entropy approaches zero as the temperature goes to absolute zero.
3)
The variation of the internal energy U as a function of entropy S is predicted by classical
n
equilibrium thermodynamics to have a functional form of U − U 0 = α ( S − S0 ) where α is
a constant for a fixed value of volume.
a. Show that the temperature as a function of entropy in the case of constant volume
is given by T = αn ( S − S0 )
n−1
€
.
1 n−1
b. Show that Cv =
€
S − S0
1 T 
=
 
n −1 n −1  αn 
.
 ∂T   ∂ 2U 
At some point it may be useful to derive the relationship   =  2  .
 ∂S V  ∂S V
€
4) Silicon has Z=14.
a. Relative to a hydrogen atom, what are the energy and radius of an electron in the
€
1s shell?
b. In the 3p shell?
c. Assume that there are 2 electrons in the 3p shell. What are the allowed
spectroscopic states for this 2-electron system? (Give S, L, J).
d. What is the energy ordering and why?
5) A charged kaon (at rest) decays by K + → π + + π 0 and then π 0 → γ + γ .
a. In terms of the masses of the particles what is the π 0 ’s energy?
b. What is the maximum photon energy?
€
€
Modern/Optics/Thermodynamics September 2007
Pick 4 out of 6 problems.
1 In inertial frame O a rod of length l is oriented along the x-axis and moving with
velocity u in the positive y direction. This rod is then viewed from an inertial reference
frame O’ moving with velocity v in the positive x direction.
a) What is the length of the rod in O’? [10 points]
b) What angle does the rod make with respect to the x’ axis? [30 points]
2 A cubic box (with sides of length L) holds diatomic H2 gas at temperature T. Each H2
molecule consists of two hydrogen atoms with mass of m each separated by distance d.
Assume that the gas behaves like an ideal gas. Ignore the vibrational degree of freedom.
a. What is the average velocity of the molecules? [10 points]
b. What is the average velocity of rotation of the molecules around an axis which is
the perpendicular bisector of the line joining the two atoms (assuming each atom
as a point mass)? [10 points]
c. Derive the expressions expected for the molar heat capacities Cp and Cv for such a
gas. [10 x 2 points]
3.
a. What is the threshold kinetic energy for the proton for p+ n  p+ p+ π− assuming
the neutron is at rest? [20 points]
b. If now the neutron is in a Carbon nuclei of size 60 F3 (with 6 protons and 6
neutrons), what is the Fermi energy of the neutron, and what is the threshold
kinetic energy in this case? [20 points]
mn = mp = 1000 MeV, mπ = 140 MeV, hc = 200MeVF
4. A smooth vertical tube having two different sections is open from both ends and
equipped with two pistons of different areas. Each piston slides within its respective tube
€
section. One mole of ideal gas is enclosed between the pistons. The pistons are connected
by a non-stretchable rod. The outside air pressure is 1 atm.
The total mass of the pistons is M. The cross sectional area of the larger upper piston A1,
and the lower piston A2 are related by A1= A2+ΔA.
How much (in Kelvin) must the inner gas (between the pistons) be heated in order to lift
the piston assembly by L=5cm?
p0
one mole
ideal gas
M=5kg
p0=1atm
A1-A2=ΔA=10cm2
CONSTANTS:
p0
g=9.8 m/s2
R=8.3J/(oK⋅mole)
kboltz=1.38x10−23J/oK
5. Eight non-interacting neutrons are confined to a 3D square well of size D= 5 F (10-15
m) such that V = -50 MeV for
0 < x < D,
0 < y < D,
0<z<D
and V = 0 everywhere else.
a. How many energy levels are there in this well? [10 points]
b. What is the degeneracy of each energy level? [10 points]
c. What is the approximate Fermi energy for this system? [10 points]
d. What is the relative probability to be in the lowest energy state to the fourth lowest
energy state at kT= 10 MeV? Just write down the ratio (don't calculate the value).
[10 points]
mn = 1000 MeV, hc = 200MeVF
6. Interference by a biprism. A plane wave (wavelength
λ) enters perpendicular to the biprism (the prism angle
€
α, refractive index n) as shown in the figure. The wave
transmitted through both sides of the biprism is bent
(refracted) and overlap at the viewing screen S (parallel
to the biprism) where an interference pattern can be
observed.
a. Find the refracted angle, β. [20 points]
b. Find the interval of adjacent interference lines.
[20 points]
β
biprism
Statistical and Modern. Spring 2007. Pick 4 out of 6.
1) a) Starting with the first law of thermodynamics and the definition of c p and cv , show
that

 ∂U    ∂V 
c p − cv =  p + 
 
 .
 ∂V T   ∂T  P

Here c p and cv are the specific heat capacities per mole at constant pressure and volume
respectively, and U and V are the energy and volume of one mole.
b) Use the above result plus the expression
 ∂U 
 ∂p 
p +
 = T  ∂T 
∂
V

T

V
to find c p − cv for a van der Waals gas with equation of state
a 

 p + 2  (V − b ) = RT .
V 

Here a and b are constants.
c) Use this result to show that as V → ∞ at constant p , you obtain the ideal gas result for
c p − cv .
2) The rotational motion of a diatomic molecule is specified by two angular variablesθ
and φ and the corresponding canonical conjugate momenta, pθ , pφ . Assuming the form of
the kinetic energy of the rotational motion to be
ε rot =
1 2
1
pθ +
pφ 2
2
2I
2 I sin (θ )
a) Derive the classical formula for the rotational partition function , r(T ),
r (T ) =
2 IkT
h2
b) Calculate the Helmholtz free energy Frot .
c) Calculate the corresponding entropy and specific heat.
The following may be helpful
∞
∫e
− ax
−∞
∫ sin
2
dx =
π
a
dx
= − cot( ax ) / a
2
(ax )
3) Assume that the neutron density in a neutron star is 0.1/fm 3 (that is 0.1 neutron per
cubic Fermi). Assuming T=0 and ignoring any gravitational forces calculate the ratio of
neutrons to protons to electrons.
Hint: determine their Fermi energy. The electron, neutron and protons masses are
.511 MeV/c 2 , 939.6 MeV/c 2 and 938.3 MeV/c 2 . The constant hc = 1240MeV fm . You
should be able to work out "by hand" an approximate value.
4) A π − µ atom consists of a pion and a muon bound in a Hydrogen-like atom.
a) What are the energy levels for such an atom compared to those for Hydrogen?
b) π − µ atoms are produced in KL decays ( K L → π − µ +ν ). If the KL has β = 0.8 what
are the minimum and maximum energies of the π − µ atom expressed in terms of the K ,
π and µ masses with mν = 0 ?
c) Approximately what fraction of KL decays will produce a π − µ atom (hint: use the
Heisenberg uncertainty principle)?
5) a) You are familiar with the quarter-wave thin film coating that acts as a “reflection–reducer”.
For the moment, let us look at a simpler thin film—the air gap between two pieces of glass such
as you would find in a Newton’s rings experiment. Why do we get constructive interference in
the reflected when the thickness is one-fourth of the wavelength of light or some odd multiple of
a quarter wavelength? Why isn’t it constructive at one-half wavelength of the light? For
assistance, I present two of the Fresnel equations (in two forms) for reflected light.
rP =
nt cos θi − ni cos θ t tan(θi − θt )
=
nt cos θi + ni cos θ t tan(θ i + θt )
r⊥ =
ni cos θ i − nt cos θt
sin(θ i − θ t )
=−
ni cos θ i + nt cos θt
sin(θ i + θ t )
Where the parallel and perpendicular symbols refer to the plane of incidence, and i,t refer to
incident and transmitted media, θ ’s are angles of incidence and transmission, and n’s are indices
of refraction.
b) In light of the previous, to get destructive reflection in a thin film-i.e.-a quarter-wave film, such
as the one illustrated below, what condition must prevail among the indices of refraction for the
three media (n0 may be taken as = 1.0 for air.)
c) The destructive interference described in part b) will generally not be complete. Find the value
n0
of n1 as a function of n2 which gives completely destructive interference at normal incidence.
n1
n2
6) In a big-bang theory of the universe, the radiation energy initially confined in a small
region adiabatically expands in a spherically symmetric manner. Here the radiation
(photon) pressure is expressed as p = U 3V , and the black body radiation energy density
is u = U V = aT 4 The radiation cools down as it expands.
a) Derive a relation between the temperature T and the radius R of the spherical volume
of radiation, based purely on thermodynamic considerations.
b) For the above problem, show the total entropy of a photon gas is expressed as
S=
4 3
aT V .
3
Modern Physics, Optics, Statistical Mechanics and Thermodynamics
September, 2006
Pick 4 out of 6 problems
1) A flashlight, in its own rest frame, is directed at an angle θ of 45 degrees to the x-axis
in the xy plane. What angle will the light beam appear to make to an observer moving
towards the flashlight along the x-axis at velocity β ?
a. Now consider a relativistic electron traveling in a circular orbit. Explain why the
radiation from the electron will be confined to a narrow cone, and calculate the
opening angle of the cone.
b.
2) A simple model of a rubber band is a one-dimensional (horizontal) chain consisting
of N ( N 1 ) linked segments, as shown schematically in the diagram.
Each segment has two possible states: horizontal with length a, or vertical, contributing
nothing to the length. The segments are linked such that they cannot come apart. The
chain is in thermal contact with a reservoir at temperature T.
a. If there is no energy difference between the two states, what is the average
length of the chain?
b. Fix chain at one end and hang weight from the other end, supplying a force F as
shown. Determine the average length of the chain at any temperature T . Find the
length in the limits T → 0 and T → ∞ .
c. In which temperature limit is the extension proportional to F (Hooke’law)?
Calculate the constant of proportionality.
3) The complete formula for multislit diffraction pattern in Fraunhofer diffraction is given by the
expression
2
 sin β   sin Nα 
I (θ ) = I 0 

 
 β   sin α 
where
2
I 0 is the flux density in the θ = 0 direction for one slit,
kb
sin θ , and b is the slit width,
2
ka
α = sin θ , and a is the slit separation, and
2
2π
is the propagation number.
k=
β=
λ
a. Show that I (0) = N 2 I 0 , where N is the number of slits.
b. For N ≥ 3 there are secondary maxima between the principal maxima. Deduce the
rule for the number of minima between principal maxima, and, then the number of
secondary maxima. Actually deduce this, don’t just recite it if you happen to remember it!
c. The result in a indicates that the flux density at θ = 0 varies as N 2 . Explain how that
can be true. After all, e.g., if you have three slits, then there is three times a much light.
How can the flux density be nine times as great?
4) Consider a system of two types of charge carriers in the Drude model. The two carriers
have different densities (n1 and n2) and opposite charge (e and –e), and their masses and
relaxation times m1, m2 and τ1, τ2, respectively. Determine the Hall coefficient RH of this
system.
5) Assume that the cross section ratio for the two body reactions given below depends only on
phase space of the final state particlesLarine o . Calculate the ratio.
ν µ + e →ν µ + e
ν µ + p →ν µ + p
6). Consider an infinitely long cylinder that has been cut in half, with the two halves
thermally insulated from each other (see figure). The top half is held at temperature
T=30ºC, and the bottom half at T=10ºC. Find the temperature T(x,y) inside the cylinder.
Hint: Use the conformal mapping
 z +1 
w = ln 

 z −1 
z = x + iy
Modern Physics, Optics, Statistical Mechanics and Thermodynamics
February, 2006
Pick 4 out of 6 problems
1. (Bohn) Consider an electron beam of circular cross section moving down the z-axis
and passing through a system of focusing magnets. Suppose the beam density
remains uniform, and its initial radius is R0 . Suppose further that the transverse
components of the electrons’ momenta stay uniformly distributed over a circle in the
momentum space , and the initial radius of this circle is P0 .
a. If the focusing system reduces the beam radius from R0 to R1 , how does the
distribution of transverse momenta change?
b. Part (a) is highly idealized. Suppose, instead, that the electron beam is bunched,
and each electron bunch moves non- relativistically through the focusing system.
Suppose further that Coulomb self-forces affect the beam, and that the focusing
system imparts an external magnetic field Bext. Write down the Vlasov-Poisson
kinetic equation for the distribution function of electrons in the six-dimensional
phase space of a single electron.
c.
The Vlasov-Poisson equations are, in general, notoriously difficult to solve.
Why?
d. Suppose, now, that the Hamiltonian for the beam bunch is independent of time
and that the distribution function is a function only of the Hamiltonian (again,
with regards to the six-dimensional phase space of a single electron). Is the
beam in equilibrium? Explain your answer.
 ∂T 
1
Tβ 
2. (Ito) The Joule coefficient may be written η ≡   =  P −
 , and the Joule ∂v  u c v 
κ 
 ∂T 
v
Thomson coefficient may be written µ ≡   = (Tβ −1). Here u is the internal
 ∂P  h c P
energy, β the expansivity, and κ the isothermal compressibility. Using these two
coefficients,
a. Find η and µ for a van der Waals gas and
b. Show that both are zero for an ideal gas.
3. (Ito) Consider a two-level system with an energy 2ε separating upper and lower
states. Assume that the energy splitting is the result of an external magnetic field B.
 ε 
Given that the total magnetic energy is U B = −Nεtanh  , show that the associated
 kT 
2
2ε kT
 2ε 
e
heat capacity is CB = Nk 
2.
2ε
 kT  (e kT + 1)
4. (Benbow) Suppose you have large, long-focal length achromatic lens to use as the
objective of an astronomical telescope. We are told in geometrical optics that if
parallel rays are incident on a lens, the rays will converge to a point at the focal length
of the lens.
If that is so, how can we expect to obtain an image of the moon (certainly not a point
source) if we place a screen at the focal point of the lens? Explain, using words, not
equations.
5. (Brown) Although a photon has no rest mass, it nevertheless interacts with electrons
as though it has the inertial mass
p p
m= =
υ c
where the velocity of the photon is υ = c .
°
a. Compute the photon mass (in units of eV) for photons of wavelengths 5000 Α
° (x-ray region). Compare to the mass of an electron
(visible region) and 1.0 Α
(which is 0.511 MeV). (Note: h = 4136
.
× 10−15 eV ⋅ sec ).
b. When we drop a stone of mass m from a height H near the earth’s surface, the
gravitational pull of the earth accelerates it as it falls and the stone gains the
energy mgH on the way to the ground. A photon of frequency ν that falls in a
gravitational field gains energy, just as a stone does. Using the photon mass
relation, determine the new frequency, ν ′ , of the fallen photon, and thus the
frequency shift, ν − ν ′ , suffered by the photon.
c. A passenger in an airplane flying at 20,000 feet aims a laser beam of red light
° ) towards the ground. What is the shift, λ − λ′ , in the laser beam
( λ = 7000 Α
wavelength an observer on the ground would measure?
6. (Hedin) Consider the two reactions of an elastic scatter of a neutrino off an electron at
rest for muon-type and electron-type neutrinos:
a) ν µ + e à ν µ + e
b) ν e + e à ν e + e
a. Which will have the larger cross section for Eν = 1 GeV?
b. Explain (best if you show the first order Feynman diagrams).
c. For Eν =1 GeV, what is the maximum angle a scattered electron will have with
respect to the incoming neutrino?
Modern Physics: Pick 4 of 6
1) The potential energy of gas molecules in a certain central field depends on the distance
r from the field’s center as U ( r ) = α r 2 where α is a positive constant. The gas
temperature is T, the concentration of molecules at the center of the filed is n0.
a) Find the number of molecules dN, located at the distances between r and r+dr
from the center of the field.
b) Find the most probable distance separating the molecules from the center of the
field.
c) How many times will the concentration of molecules in the center of the field will
change if the temperature decreases by η (Tnew = η × T). (Hint, how does dN/N,
that is, the fraction of molecules located in spherical layer between r and r+dr,
behave at the center)
d) Find the number of molecules dN, whose potential energy lies within the interval
from U to U+dU.
FORMULA SHEET for PROBLEM 1
 π ,n=0
 1, n = 0

2


1
∞
∞
 π ,n= 1
 2, n =1
n − x2
n −x
2
2
=
x
e
dx
x
e
dx
=


∫0
∫
0
 π ,n=2
 1, n = 1
4


 2, n = 2
 1 ,n=3
 2
2) Consider a particle of mass m undergoing Brownian motion in one dimension. The
particle is under the influence of a viscous friction force –mβv, an oscillatory driving
force –masin(ωt) and a random force mA(t). Its equations of motion are
x = v, v + β v = − a sin (ωt ) + A ( t ) .
a) Suppose A(t ) = 0 and A(t ) A(τ ) = αδ ( t − τ ) , where α is a constant and δ(t) is the
Dirac delta function. Suppose the initial conditions are q(0) =q0, v(0) = α/ω. Find the
mean speed v(t ) . Note:
t
βτ
∫ dτ e sin(ωτ ) =
0
e β t ( β sin(ωt ) − ω cos(ωt ) ) + ω
β 2 + ω2
b) Evaluate (by whatever means you chose) v(t ) in the limit β Æ 0, and provide a
physical interpretation of your result.
3) The density of states for a free particle of momentum p confined within a box of
volume Ω is given by
dn Ωp 2
=
dp 2π =3
a) Calculate the density of states per unit energy for a non-relativistic electron and for a
massless neutrino. Assume each is confined within a box of volume Ω but otherwise free
of interactions.
b) Assume that the transition probability for beta decay is dominated by the density of
states term. Take the electron to be non-relativistic and the neutrino massless. In terms
of the total decay energy, Etot, calculate the most likely energy for the emitted beta
particle?
4) An energetic proton strikes a proton at rest. A K+ is produced. Write down a reaction
for producing a K+ showing all the final state particles. What is the minimum kinetic
energy for the incoming proton to produce this final state (express in terms of the masses
of the particles)?
5) An interstellar proton interacts with the 3 degree cosmic microwave background
(CMB) and produces an electron-positron pair via p+photon -> p+electron+positron.
Assume the CMB is monoenergtic with E=kT= .002 eV. What is the minimum proton
energy to produce this reaction? What is the electron energy at this threshold proton
energy? The proton mass is 938 MeV/c2 and the electron mass is 0.5 MeV/c2.
6)
Mirror
L
L+d
Mirror
Half Silvered Mirror
Polychromatic
Light Source
Detector
A Michaelson interferometer has two arms, the first of length L and the second of length
L+d. The interferometer is illuminated by a polychromatic source of light with a
frequency distribution given by
2
I (ν ) = I 0 / 2πσ 2 exp  − (ν −ν 0 ) / 2σ 2 


a) Describe qualitatively how you expect the interference pattern for the case with
polydisperse light to differ from the case where the interferometer is illuminated with a
single frequency of light.
)
(
b) Calculate the intensity of light observed as a function of the arm offset d. Does this
result confirm your prediction from part a?
You may want to use the following integral.
∞
∫e
−∞
−
( x − y )2
2 a2
πσ 2 
cos ( kx ) dx =
1 + cos ( 2ky ) e−2 k σ 


2
2
2
2
Statistical Mechanics: Pick 2 of
3.
February,2005
1. Consider a large, spherical, self-gravitating system of identical stars for which the
distribution function in the phase space of a single star is
f (r,v) = CIH o - H (', O)]*'''
an-3t2t-
:c.lv?)-I,,
ro.vr!,,
2
L \/ 2 |
I
=o
for Y
<!r'
2
Here Y = V(t) = Ho-Q(r) denotes a relative potential, with O(r)being the actual
potential, and ydenotes speed- Using Poisson's equation V'?@(r) = 4trGp(r) , find:
a) The density distribution function
fr)
conesponding to the index n = 5.
b) Are the stellar orbits in this system chaotic? Why or why not? Clearly explain your
reasoning.
2. T\e equation for the distribution of free electrons in a metal in the vicinity of absolute
zero is
3/
. ,l2m/2
d"=:FJEat
.
Making use of this equation, find at T=0K
a) The velocity distribution of free electrons.
b) The ratio of the mean velocity of free electrons to their maximum velocityc) What is the functional dependence of the Ferrni Energy on the density of electrons?
3. The distribution functions for identical particles (indistinguishable or distinguishable)
can be represented by the equation:
Nr-
I
81
*o
Where a = 1 for Fermi-Dirac statistics, a= 'Gt-r)'tr
-1 for Bose-Einstein statistics and a=0 for
Maxwell-B oltzmann statistics.
g, us. (e, - a) t nf for all three distriburions.
b) For the Fermi-Dirac distribution, sketch N , I g, vs. e, for ?"= 0 and I slightly greater
a) Sketch and compare N , I
than zero.
c) Show that for a system of a large number, N, of bosons at very low temperature (such
that they are all in nondegenerate lowest energy state t= 0), the chemical potential varies
with temperature according to:
p-+-kTlNasI-+0.
Modern Physics: Pick 2 out of 4
1. Consider an assembly of identical two state atoms in equilibrium with a radiation
field. The two states of the atom are labeled 1 and2 and their energy difference is hvstate 2 has the higher enersy. Lnt
p(v)
=,yj\)denote
the energv densitv
of the radiation field per unit frequency interval. The probability for stimulated emission
from state 2 to I and from state 1 to 2 are given by
W|r= Br,rP(v)
Wrt.r-- Br,rP(v)
The probability of spontaneous emission from state 2 is given by, W),t-- A.
a) What is the
ratio
N|/ Nz
of the number of atoms in state 1, to the number of atoms in
state 2.
b) Show thatB2l = Br,z and use this to find the rutio A/B'
c) Explain briefly why it is necessary to have a non-thermal population distribution
between two states in order to achieve laser action.
2. A pion-muon atom (that is the two particles
decay via
KL
)
are bound together) can be forrned
in KL
pion-muon atom + neutrino.
a) What are the energy levels of the pion-muon atom relative to the hydrogen atom?
b) What is the lifetime of the pion-muon atom in terms of the pion and muon lifetimes?
c) If the KL is at rest when it decays, what are the momentum and kinetic energy of the
pi-mu atom?
Please state all answers in terms of the masses and lifetimes of the particles involved
(pion, muon, kaon, proton, or electron).
3. Find the energy Q, expressed in terms of the masses of the atoms Mp, M.' and the
electron m", liberated in B- and B+ decays, and in K-electron-capture if the masses of the
parent atom is Mn, the daughter atom M6, and an electron me are known. (Assume
electron binding energies are negligible)-
If two waves are to interfere, stipulations are that they must be traveling in the same
direction in the same region of space, and that their oscillatoryplanes be parallel. ff this
last condition is not met, something else will occur.
4.
a.
Suppose
Ex = Eoicos(kz- rn)
E, = Eoicos(kz-m)
What happens here? What do you see if the wave is traveling toward you?
b. Suppose
Et
:
Ev =
Eolcos(kz-
a)
Eolsrntkz- ax1
'
Whathappens now? What do you see if the wave is traveling toward you?
Ph.D. Qualifying Exam.
statistical Mechanics, Thermodynamics and Mo dem Physics.
September 2004
Part
L
Statistical Mechanics Thermodynamics. @ick 2 out of 3) If you answer all thrree
note that only the first two
will be graded.
1. A Carnot engine
is operated between two heat reseryoirs at temperatures of 400 K to 300 K.
(a) If the engine receives 1200 kilocalories from the reservoir at400K in each cycle,
how much heat does it reject to the reservoir at 300 K?
(b) If the engine is operated as a refrigerator (i.e., in reverse) and receives 1200
kilocalories from the reservoir at 300 K, how much heat does it deliver to the
reservoir at 400 K?
(c) How much work is done by the engine in each case?
(d) What is the efiiciency of the engine in (a) and the coefficient of perfonnance in (bX
2. An ideal monatomic gas consists
isothermally to
fill
a volume
ofN atoms in a volume V. The gas is allowed to expend
2v. showthatthe enhopy change is-Nkln2.
J. A mercury atom moves in a cubical box whose edge is I m\ong. Its kinetic energJ is equal
to the average kinetic energy of an atom of an ideal gas at 1000K. If the quantum numbers
n", r7y, and n" are all equal to z, calculafe z. Note that the atomic mass of mercury is 200.6
g/mol (hint calculate the mass of an individual atom).
Possible useful information
Boltzmannconstant,
1.38
k:
x
-23
JK:8.617 x 10-5 eV/K
Stefan-Boltzmann constan! o =A[o=Ta :5.670x l0{ JK4m-2s-r
'
l1h3ctPlanck's constant, h:6.62 x 10-3a Js
Speed of lighg c:2.99792458 x 108 ms-r
R: 8.34 x 103 J kmole-r K-r
10
dU: TdS - PdV
H:U+PV
F : U.TS
G: F+PV
Values of constan8 : h-6.63x10-a.r.s, ft =1.38x1
N e = 6.022x104 particles per mole.
0-23
J/K ,
Part
1..
II.
Modern Physics and Optics (Pick 2 out of 4)
Assume that the mu-neutrino vrandthe tau neutrino y.are composed of a mixture of two mass
eigenstates
vrandv, , with masses ml andm2.The mixing ratio is given by:
-sin(e)[v,\
f",) _fcos(o)
\",/ \sin(o) cos(o) /\v,/
In free space, the states v1 andv2 ovolve according to
(lv r(x, t)
r\
"
_ _*.,,
fl",(",t) ,)-
^("-'""'o
[v,
(o) t
\"-*""p,(o)
)
'/
a) Show tlrat the transition probability for a mu-neufino to turn into a tau neutrino is
given by:
PAt*
")=
sin2(29)sin'l(nr- nrlt nnf
b) Show that in the extreme relativistic limit this result becomes:
t
't)st1
\,, L o*
l
PQt-- t) = sin'(2o1"rr'l(*l
"
I is the distance taveled by the neutrino, and E is the total energy.
Herq
2. A proton is confined to a2D square well of size
F 5 F (10^-15 m) suchthat
for0<xcD
V:-50MeV
and
V:0
o<y<D
everywhere else.
Howmany energy levels are there inthis well?
b. What is the degeneracy of each energy level?
c. What is the approximate energy of the lowest energJ state? (and at least outline how
one can improve estimating the energy).
a-
mass
proton: 938
MeV/c^2
Hbar*c :197 MeV*F
3. An electron with p : 1 GeV/c strikes a proton at rest.
Their masses are 0.51 Meylc^2 and 93g MeV/c"2
a. What is the ma:rimtrm momenfum transverse to the incoming elecfion's direction that
the exiting proton can have?
b. what is the maximum total energy that the exiting proton can have?
c. What energy would the electron need to have to mate a rho meson, with mass of 770
MeV/C2? That is e+F>sFprrho.
4.
For the waveform faveling in a dense glass
E(r,t) =Xt(+i.3il(u/*)"os[o.oa+rr
x t06e
a^ Determine the ildex ofrefraction of the glass.
+t.rtlnx ldsr]
b. Detennine the wavelength ofthe light in air.
c. Resolve the" Eo" into an amplifude times a unitvector.
d. Determine which directionthe wave is propagating.
Ph.D. Qualifying Exam.
Statistical Mechanics, Thermodynamics and Modern Physics.
January 2044
Part
L statistical Mechanics Thermodynamics.
note that only the first two
(Pick 2 out of 3) If you answer all three
will be graded.
1)
Calculate the total electomagnetic energy inside an oven of volume 1 m3 heated to a
temperature of 600 degrees Fahrenheit.
2)
Anideal monatomic gas undergoes
specific volume vz.
(a)
(b)
(c)
3)
a reversible expansion
from specific volume vr to
Calculate the change in specific entropy As if the expansion is isobaric.
Calculate As if the process is isothermal.
Whioh is larger? By how much?
For N distinguishable coins the thermodynamic probability
i, ,
=ffi,
Nr is the number of heads andN-Nr ttre number of tails.
(a) Assume thatN is large enough that Stirling's approximation (lnn!-nlnn
valid. Show that lno is ma:rimum forNr : N/2.
(b)
Show that @^o-eNro2.
Possible useful information
-23
1.38 x 10
Boltzmann constant,
J/K: 8.617 x 10-5 eV/K
Stefan-Boltzmann constan! o:5.670 x 104 JK4m-2s-1
Planck's constanL h:6.62x 10-34 Js
Speed of l^ight, c:2.99792458 x 108 ms-l
R: 8-.34 x 103J knole-r K-l
dU: TdS - PdV
k:
H:U+PV
F : U.TS
G:
F+PV
where
-n) is
Part
II. Modern Physics and Optics
1) A proton is confined
V:
-50 MeV
(Pick 2 out of 4)
to a 2D square well of size D: 5 F (10"t m) such that
for0<x(D,0<y<D,
and
v:0
everywhere else.
a- How many enerry levels are there in this well?
b. What is the degeneracy of each energy level?
c. What is the approximate energy of the lowest energy state? Give both the zeroth
and first order approximation.
Mass proton:938
MeV/c2 h*c:197 MeV*F
2)
Consider a neufon star of mass M and radius R. Assume T: 0 K. Ignoring
gravitational energy and assuming a simple Fermi gas model, what is the ma:<imum
energy a neutron can have? What is the average energy of the neutrons? Do not
calculate actual values but leave in the form using the mass of neufion, mo, and other
constants.
3)
A photon from the cosmic microwave backgfound with energy 1.2x10-3 eV
(corresponding to a wavelength of I mm) collides head-on with a relativistic electron
having kinetiCenerry 1 Ge\l The photon is backscattered i.e., scattered by 180o,
after which the electron continues to move in its initial direction. Find the energy of
the photon after tlre collision. What is the wavelength of the backscattered photon?
4)
Consider a plane electromagnetic wave incident in the normal direction on a system
of nnarrowslits each separated by a distance d. Derive a formula for the intensity of
the soattering as a function of angle and wavelength in the Fraunhoffer
approximation.
Graduate
O-*J?r"
Stat-Mech Pick2 of 3.
Problem 1:
Consider the Fokker-Planck equation for the distribution function f:
#*y, +q.ff = Q.(40*O(D.A),
urhere 0 denotes a gradient with respect to velocity
dynamical friction and dimlsion:
anA
ab.
1is
v.
Take a simple model
of
{ = fy.and D = Dfr,, where F alrtdD ate constants
the unit vector in the direction
of y.
Describe the overall effect of the right-hand side on the evolution of the systern
How is the coefFrcient Brelated to the force fluctuations a constituent particle
e4periences?
c.
Deduce from the Fokker-Planck equation
bw B afr D are related for a system in
thermal equilibriurn
Problem2.
Consider the simplest model for desorption of atoms of mass m from a surface: one that
ignored the tanslational degrees of freedonl and takes the possible energy states of the
atom as -e (when adsorbed on the surfrce) and 0 when in the 'vacuum' at terperature
T.
*,
Use the canonical ensemble for Natoms to obtain an eqrression for the number
atdms inthe vacuum.
Wbat is the total energy ofthe system as a function oftemperature?
of
b.
Problem 3.
Conduction electrons are confined in a metal at T= 300 K. The Fermi energy is 4 eV.
What is the relative density of states N(E) for energy of 4.4 eV compared to 3.6 eV?
What is the relative probability for anelectronto have enErgy of 4.4 eV compared to 3.6
eV?
F'- il 't c'6i
Modem/Optics Pick 2 of 4
Problem 4.
Derive Malus's law: Malus's law relates the intensityof light passing tbrotr$ttwo ideal
polarizers with their polarizing directions at an angle A with each other. Polarizers act
on the electric field vector (amplitude) of the light wave traversing thern An ideal
plafiznr passes 50% ofpurely unpolarized light incident on its surface, the remainder
being now polarized parallel to the plarizng direction
Now, armed with Malus's law, consider two ideal polarizers arranged so that a beam of
light passing through both finds the second one at an angle of 45'to the fust. Then a 3'd
ideal polarizer is inserted between them and rotated at angular frequency ar, Derive an
expression for the intensity ofthe light emitted from the system as a function oftime.
Problem 5.
Two neutrinos leave a supenrcva explosion at the same time. They arrive at earth a
distance L away separated in time by At. The first neutrino to arrive is found to bave
energy Et andthe second energy Ez. Findthe mass ofthe neutrino.
Problem 6.
The disintegration constant )" of aradioactive nucleus is defined as the fraction of nuclei
that decay per unit time. Let N(t) bethe number of nuclei present at time /.
Derive the decay law, N(t): N(0)exp(-tr t)
What is the relation between half-life Tn afr ,L2
In an excitation experiment, a parent nucleusp is produced at arate.R, and decays
with a disintegration constant 2 p,to a daughter nucleus 4 whose disintegration constants
is La. At H, the number ofparent nuclei isifi0):N6oand the number of daughter
nuclei is ,va(0):0. Find nrlO and ila(Q for t>0, and show that after a long time, an
equilibrium is attained.
a.
b.
c.
Problem 7.
An electron is in the spin state
a.
b.
Determine the normal
Find the expectation values
^rr"^'"1"!r!^i)o
of S*, Sr, and Sr.
NIU Ph.D./Master qualifier examination 2003 Spring
Statistical Physics and Modern Physics
Solve 4 problems. Choose 2 problems from I, II and
from V and VI, choose 1 problem from IV and VII.
I
III, and choose 1 problem
Consider a nonrelativistic free particle in a cubical container of edge length
-L
a
Each quantum state r of this particle has a corresponding kinetic energy
on Z What is t,(V)?
b
Find the contribution to the gas pressure p,
^c crdll(J
^ ^-)
vL
c
L
and volume
Z
r3
f/
v
e,'
which depends
: -(dqldV) of aparticle in this state in terms
.
Use this result to show that the mean pressure
p of any ideal
particles is always related to its mean total kinetic energy
E
gas
of weakly interacting
AV V
=1rlr,
irrespective
of whether the gas obeys classical, Fermi-Dirac, or Bose-trinstein statistics.
II
Consider a system of two atoms, each having only 3 quantum states of energies 0, e and 2e .
The system is n contact with aheatreservoir at temperature T. Write down the partition
function Z far the system if the particles obey
A.
B.
C.
D.
lll
Classical statistics and are distinguishable.
Classical statistics and are indistinguishable.
Fermi-Dirac statistics.
Bose-Einstein statistics.
oistate P(v - b) = RT" where r? is the universal gas
constant, andb is a constant suchthat 0 <b <Vfot allV.
A certain
gas obeys the equation
a
Derive an expression for the work obtained from a reversible isothermal expression
one mole of this gas from an initial volume Vito aftnalvohsne V7.
b
If the gas were ideal, would the
of
same process produce more or less work?
IVFindthethresholdenergyfortheprocess,y+p-uo*p,inwhichasingle'zrmesonis
produced when a photon strikes a proton at rest. The rest energy of the pion is 135 MeV and
the nroton 938 MeV.
.
V
Take an ideal monatomic gas (y: 5/3) around the Carnot cycle, wherc point I atthe
beginning of the adiabatic compression has pressure Pr : Pa (atmospheric presswe), volume
Vt:13liters, andtemperatureTt:300 K. Point 3 has pressureP::2Po andvolume V3:26
liters. Calculate the values of volume and pressure at all four points of the Carnot cycle.
W An ideal
gas initially at temperature, Ti, pressure, Pi and volume, V has its pressure reduced
to a final value Pi< Pi via one of the following processes: (1) isochoric; (2) isothermal; (3)
adiabatic.
a
b
c
d
Sketch each process schematically on a P - V diagram.
In which process is he work done on the gas zero?
In which process is the work done on the gas greatest?
Show that the ratio the absolute magnitudes of the heat transfers, Qi*t1,.r".-u1 to Qiro"1,o6" is
given by
*e l,,lg)l
lril=4Eilt'tp
lYisochnncl
ll
leu,,*^^l=
'ul*
e
tt
\'/ ll
In which process is the change in internal energy of the gas the greatest? Explain your
reasoning.
VII
The k-alpha radiation from copper tZ19) occurs via the following process- An incident
high energy electron removes an electron from the n:1 orbital and subsequently an electron
from the n:2 orbital falls down to the n:l orbital releasing an x-ray.
a
Ignoring electron-electron interactions calculate, in units of the Rydberg constant R, the
energy of the Cu k-alpha x-ray.
b
More realistically, the second 1S electron will shield the nuclear charge. Assuming
perfect shielding, calculate a better estimate of the Cu k-alpha energy.
c
Discuss, in qualitative terms, how inter-electron interactions might modifu the Cu k-alpha
energy, beyond just the perfect shielding approximation.
d
The energy of the Cu k-alpha x-ray is slightly modified by the electron's spin-orbit
interaction. This leads to a splitting of the k-alpha line into a k-alpa-l and k-alpha-2.
i
List all possible values of total (orbital + spin) angular momentum for the n:1 and
n:2
states.
ii
List all transitions from states with n:2 to states with
selection rules for atomic transitions.
iii
Explain the observ'ed ratio of 2:1 for the Cu k-alpha-l to k-alpha-2 fluorescence
yields.
n:l that are permitted
by the