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Name:______________________________ Exam Number:_______
University of Michigan Physics Department
Graduate Qualifying Examination
Part II: Modern Physics
Saturday, May 18, 2013 9:30 am – 2:30 pm
This is a closed book exam, but a number of useful quantities and formulas are provided
in the front of the exam. ( Note that this list is more extensive than in past years. ) If you
need to make an assumption or estimate, indicate it clearly. Show your work in an
organized manner to receive partial credit for it. Answer the questions directly in this
exam booklet. If you need more space than there is under the problem, continue on the
back of the page or on additional blank pages that the proctor will provide. Please clearly
indicate if you continue your answer on another page. Label additional blank pages with
your exam number, found at the upper right of this page (but not with your name). Also
clearly state the problem number and “page x of y” (if there is more than one additional
page for a given question).
You must answer the first 8 required questions and 2 of the 4 optional questions.
Indicate which of the latter you wish us to grade (e.g. by circling the question number). We
will only grade the indicated optional questions. Good luck!!
Some integrals and series expansions
2
2 2 3 2 3 4
3 5
2 4
2 3 4
2 3
exp( d
exp( d
exp( ) 1
sin( )
cos( ) 1
ln(1 )
(1 ) 1 (^ 1)^ (^ 1)(^ 2)
x x
x x x
x x x^ x^ x
x x x^ x
x x^ x
x x x
x x
x^ x x x
^
^ ^
Some Fundamental Constants 8 19 27 31 34 15 7
2.998 10 m/s
proton charge 1.602 10 C
proton mass 1.67 10 kg
electron mass 9.11 10 kg
Planck's constant 6.626 10 J·s 4.
speed of light
10 eV·s
Rydberg constant 1.097 10
p e
c
e
m
m
R
1 1 9 2 2 0 7 0
23 - A
B A
m
Coulomb constant (4 ) 8.988 10 N·m / C
vacuum permeability 4 10 T·m/A
universal gas constant 8.3 J / K·mol
Avogadro s number N =6.02 10 mol
Boltzmann s constant k =R/N =1.38 10 J/K=8.
k
R
8 2 4 8 sun 6 earth 30 sun 24 earth
7 10 eV/K
Stefan-Boltzmann constant 5.67 10 W / m K
radius of the sun 6.96 10 m
radius of the earth 6.37 10 m
mass of the sun 1.99 10 kg
mass of the earth 5.97 10 kg
gravitati
R
R
M
M
onal constant G 6.67 10 ^11 m / (kg·s )^3
- ( Quant
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- ( Quantum Mechanics ) Consider a particle placed in a one‐dimensional infinite square well potential
such that V x ( ) 0 for 0 x a , but V x ( ) everywhere else.
a) Find the wavefunction and energy for both the ground state and the first excited state.
Now, consider two (non‐interacting) particles placed in this potential. What are the wavefunctions and energies of the ground and first excited states of this two particle system in the cases where
b) the two particles are distinguishable (i.e., different species of particles altogether), c) the two particles are identical bosons, and, d) the two particles are identical fermions?
- ( Statistical Mechanics ) Consider a gas of non‐interacting particles in 3 dimensions. Assume that these particles follow the classical Boltzmann distribution and that the energy of a particle is
2 2 2
E p^ x^ p^ y^ pz ax bx c
m
where a , b , and c are positive constants and the particle’s position is ( x , y , z ). Find the average energy per particle at temperature T.
- ( Statistical Mechanics ) Consider a two‐dimensional crystal, as shown. Usually, atoms sit at the “normal” sites, indicated by circles, where they have the lowest energy. At non‐zero temperature, however, it is also possible for some atoms to be excited into the “interstitial” sites, indicated by X’s. An atom in an interstitial site has energy 0 0 , while one in a normal site has energy 0.
a) Suppose that the N atoms are divided between a total of N normal sites and N interstitial sites. If n of these atoms are in interstitial sites, what are the system’s internal energy U and entropy S? Assume the locations of the filled interstitial and empty normal sites are uncorrelated and
that you are in the thermodynamic limit N 1 and n 1. Hint : If there are n filled
interstitial sites, there must also be n empty normal sites, and there is entropy associated with both.
b) Find the fraction n / N of atoms that are excited into interstitial sites as a function of the
temperature T.
- ( Atomic Physics ) An atomic beam of initial velocity 100 m/s moves along the axis of a solenoid, starting in a region where the magnetic field is practically zero and moving into the center of the solenoid where the field is non‐zero. The atoms have a mass of 108 atomic mass units (amu), a total electron spin S=1/2 , a g‐factor of 2, and orbital angular momentum L=0. The atomic beam is unpolarized. The atoms propagate along the z ‐axis, and the magnetic field of the solenoid is parallel to
its axis and has the form B B ( z ) z ˆ
(a) List the relevant magnetic sublevels of the atoms’ ground state. What are the force vectors acting on
atoms in each of these states as a function of B ( z )?
(b) Determine the velocities of atoms in each magnetic sublevel at a point inside the solenoid where the magnetic field B ( z ) is 10 Tesla.
Note : 1 amu = 1.66 10 ^27 kg, and 1 Bohr magneton = 1 9.27 10 2 4 A·m^2
B 2 e
e
m
Optional (do 2 of 4)
9. ( Nuclear Physics ): The half‐life of Uranium‐ 238 is about = 4.47 x 10 9 yr. Estimate the fraction fU of
the Earth (by mass) that would have to be Uranium‐ 238 at the present time in order for the radioactive heating from its decays to equal the energy input from the Sun. (A single Uranium‐ 238 decay releases 4.3 MeV, and the solar luminosity is 4 x 10 26 J/sec, leading to an average surface temperature on the Earth of ~300 K. You may find other useful numbers at the front of the exam.)
- ( Particle Physics ) Answer the following:
a) The cross section to make a new particle at the LHC is 200 pb. This particle has three decay channels with branching ratios of 10%, 30%, and 60%. These decay channels have detection efficiencies of 2%, 0.3%, and 0.1%, respectively. How many of these new particles will be produced with an integrated luminosity of 100 fb‐^1? About how many will be detected? b) For a proton of rest mass 0.938 GeV/ c^2 and with momentum 7 TeV/ c , what is the speed of the proton? (These numbers are typical of the LHC.) c) Which of the following processes are allowed and which are forbidden: i. ݊ ି݁ → ̅ߥ ii. ିߤ ି݁→ ̅ߥ ߥఓ iii. ߤ ା^ ିߤ ߛߛ → iv. ି݁ ߛߛ → v. ̅ →
- ( Condensed Matter ) Consider a ܦ2 material. The elastic deformation in this material can be described by a ܦ2 vector field ݑሬԦሺݔ, ݕሻ. Here, ݔ and ݕ are the 2D coordinates and the ݔ and ݕ components of the vector field ݑሬԦሺݔ, ݕሻ are functions of ݔ and ݕ: ݑ௫ ሻݕ ,ݔሺ and ݑ௬ ሻݕ ,ݔሺ. Assume that the energy cost for a deformation ݑሬԦሺݔ, ݕሻ in this ܦ2 material is
ܥሾ ݕ ݀ ݔ ݀∬ ൌ ܧ (^) ଵ ߲൫ (^) ௫ ݑ௫ ߲െ (^) ௬ ݑ௬ ൯ ଶ^ ܥ ଶ ߲൫ (^) ௫ ݑ௬ ߲ (^) ௬ ݑ௫ ൯ଶ^ ሿ.
Here, ܥଵ and ܥଶ are positive constants.
(1) Distortions that cost no energy ( ܧൌ 0) are known as zero‐energy distortions. Prove that for any zero‐energy distortion in this system, ݑ (^) ௫ ሻݕ ,ݔሺ and ݑ௬ ሻݕ ,ݔሺ must be harmonic functions. [Harmonic functions are functions that satisfy the Laplace equation ݂ଶ^ ൌ 0 ].
(2) In ܦ2, harmonic functions can be written in terms of ݂ ݁ ܣ ൌ ሻݕ ,ݔሺ ೣ^ ௫^ sinሺ݇ (^) ௬ ሻ ߮ ݕ , where ܣ, ߮ , ݇ ௫ and ݇ ௬ are constants. The physical meaning of this function ݂ ሻݕ ,ݔሺ is that these zero‐energy distortions are sine waves along the ݕ direction but the amplitudes of these waves decay exponentially along the perpendicular direction, which is the ݔ direction here. Find the relation between the wave vector ݇ ௬ and the decay rate ݇ ௫.
