Name:______________________________  

Exam  Number:_______  

University  of  Michigan  Physics  Department  Graduate  Qualifying  Examination  

Part  II:    Modern  Physics  Saturday  17  Jan  2015     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.  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  

 

exp−∞

∞

∫ (−αx2 ) dx = πα

x2−∞

∞

∫ exp(−αx2 ) dx = 12

πα 3

exp(x) =1+ x + x2

2+x3

3!+x4

4!+

sin(x) = x − x3

3!+x5

5!+

cos(x) =1− x2

2+x4

4!+

ln(1+ x) = x − x2

2+x3

3−x4

4+

(1+ x)α =1+αx +α(α −1)2

x2 +α(α −1)(α − 2)3!

x3 +

   

Some  Fundamental  Constants  

 

8

19

34 15

7 1

1 9 2 20

2.998 10 m/sproton charge 1.602 10 C

Planck's constant 6.626 10 J·s 4.136 10 eV·sRydberg constant 1.097 10 m

Coulomb cons

s

tant (4 ) 8.988 10 N·m / C

vacuu

peed of light

m p r

e

ce

Rk π

−

− −

−∞

−

= ×

= ×

= × = ×

= ×

= = ×

h

Ú7

0

23 -1A

-23 -5B A

8 2 4

meability 4 10 T·m/Auniversal gas constant 8.3 J / K·molAvogadro s number N =6.02 10 mol

Boltzmann s constant k =R/N =1.38 10 J/K=8.617 10 eV/K

Stefan-Boltzmann constant 5.67 10 W / m Kradiu

Rµ π

σ

−

−

= ×

=

ʹ′ ×

ʹ′ × ×

= ×8

sun

6earth

6moon

11 3 2

s of the sun 6.96 10 m

radius of the earth 6.37 10 m

radius of the moon 1.74 10 m

gravitational constant 6.67 10 m / (kg·s )

R

RRG −

= ×

= ×

= ×

= ×    

 

1. Quantum Mechanics:Consider a permanent planar quantum mechanical dipole rotor p that isconfined to the x-y plane. For a free rotor, the Hamiltonian is given by

H0 = − h2

2I

∂2

∂φ2,

where I is the moment of inertia and where φ is the angle measured withrespect to the x-axis. We will also consider a small perturbation H1 to theHamiltonian due to an electric field D along the y-axis, where

H1 = −p ·D .

a) Find all of the energy eigenvalues and eigenstates of H0. For the sake ofdefiniteness, arrange for the states to be eigenstates of the angular momen-tum operator Lz = −ih(∂/∂φ).

b) Find all of the matrix elements (diagonal and off-diagonal) of the per-turbed energy operator H1 between the original eigenstates (of H0) found inpart (a).

c) Find the first order corrections to the energy levels.

1

2. Quantum Mechanics:Consider a free particle having mass m with initial wave function ψ(x, 0) ofthe form

ψ(x, 0) =1

2√σ

1 |x+ a| ≤ σ1 |x− a| ≤ σ0 otherwise

.

where σ � a and both σ and a are real and positive.

a) Sketch |ψ(x, 0)|2 and prove that ψ(x, 0) is normalized correctly.

b) Using the uncertainty principle, estimate the earliest time at which inter-ference between the two parts of the wave packet become important.

c) Calculate the k−space amplitude a(k) and calculate the k−space dis-tribution function |a(k)|2. Show that the distribution function in k spaceresembles that of the spatial distribution function for two slit diffraction.

[Note:∫ bae−ikxdx =

(e−ikb−e−ika

−ik

)]

d) Write an integral expression for ψ(x, t).

2

3. Statistical Mechanics:In this problem you will examine the properties of blackbody radiation in onedimension. Consider photons in a one dimensional cavity of length L. TheHamiltonian is H =

∑i c|pi|, where pi is the momentum of the ith photon.

a) Calculate the density of states g(ε), where ε is the energy of the state.

b) Calculate the average internal energy and specific heat CV as functions of Land the temperature T . You may need the following integral

∫∞0

x dx(ex−1)

= π2

6.

c) Calculate the entropy S, Helmholtz free energy, and pressure p.

3

4. Particle Physics:A top quark with mass mt = 172 GeV/c2 decays to a b quark with mb =5 GeV/c2 and a W boson with mW = 80 GeV/c2.

a) What force mediates this interaction?

b) In the rest frame of the decaying top quark, what is the energy of the Wboson in GeV?

4

5. Quantum Mechanics:a) Assume |ψ〉 simultaneously obeys the following equations:

Jx |ψ〉 = ax |ψ〉Jy |ψ〉 = ay |ψ〉Jz |ψ〉 = az |ψ〉 ,

where Ji represent the components of the angular momentum operator. Findthe values of ax, ay, az. (Hint: use the commutation relation).

b) If we add four spin-1/2 operators, i.e. ~S = ~S1 + ~S2 + ~S3 + ~S4, what are the

possible eigenvalues of the operator ~S2? How many orthogonal states |Ψ〉 of

the four spins have the property ~S2 |Ψ〉 = 0.

5

6. Statistical Mechanics:Consider a system of spin-1/2 particles with magnetic moment µ in a uni-form magnetic field of strength B. The system thus has two energy states±E corresponding to spins that are aligned and anti-aligned with the back-ground field. Assume that the system is in contact with a thermal reservoirof temperature T . Electromagnetic radiation impinging on the system canbe absorbed if the frequency of the radiation satisfies the Bohr condition.

a) Write down the constraint on the frequency required for the incoming ra-diation to be absorbed.

b) The power absorbed from the radiation field Pab is proportional to thedifference between the number of particles in the two energy states. Find thetemperature dependence of the absorbed power Pab. Evaluate the result inthe high temperature limit kT � E.

6

7. Statistical Mechanics:Consider a volume of size 2V split in two equal parts. On one side of thepartition, there are N molecules of oxygen gas held at temperature T . Onthe other, N molecules of nitrogen are held at the same temperature. Theenergy of a molecule can be written as:

Eik =

~p2

2m+ εik

where εk denotes the energy levels corresponding to the internal states of thegas, and the i index reminds us that the internal energy levels may differ forthe different types of molecule.

a) Evaluate the free energy F for the oxygen gas as it resides separately on itsside of the partition. You should display explicitly any dependence on V , butif you encounter a function solely of the temperature you should parametrizeit as f(T ). The following integral may be useful:∫ ∞

−∞e−y

2/a =√πa

b) Calculate the entropy of the oxygen gas. You answer should be in termsof N , V , and (the derivative of) your unknown function f(T ).

c) The gases are now allowed to mix, but no heat is exchanged with thesurroundings. Neglect interactions between the molecules. Does the temper-ature change? If so, by how much? Does the total entropy of the systemchange? If so, by how much?

7

8. Quantum Mechanics:Use the WKB approximation to determine how the energy levels of a poten-tial V (x) = a |x|µ (µ > 0) vary with quantum number n for large n. Note, itis not necessary to explicitly evaluate any integral in this problem. What arethe only values of µ for which the the energy does not depend on fractionalpowers of n? Physically to what potentials do these values of µ correspond?

8

9. Atomic Physics (Optional) :For the ground state of the hydrogen atom, the only nonvanishing contribu-tion to the hyperfine structure comes from the component of the magneticfield of the proton given by Bcontact = 2µ0

3µpδ(r), where µp is the magnetic

moment of the proton,

µp =egpSp2mp

(1)

and Sp is the spin of the proton. This field gives rise to an interaction energywith the electron given by

H ′hf = −µe ·Bcontact. (2)

Using the fact that the g factor of the proton is 5.59 and that the proton hasspin 1/2, calculate the hyperfine splitting in the ground state of hydrogenin frequency and wavelength units. You can use the fact that ψ100(r) =(1/πa3

0)1/2e−r/a0 and δ(r) =δ(x)δ(y)δ(z).

9

10. Condensed Matter (Optional):Consider a dimerized linear chain where all the atoms are identical, butk1 6= k2. M is the mass of the atoms and k1, k2 are the spring constantsconnecting two neighboring atoms.

a) Find and plot the phonon dispersion (ω vs. q) for longitudinal modespropagating along the chain.

b) repeat for k1 = k2.

page 5 of 7

CONDENSED MATTER PHYSICS 1

Consider!a!dimerized!linear!chain!where!all!the!atoms!are!identical,!but!k1#≠!k2.#M#is!the!mass! of! the! atoms! and! k1,! k2! are! the! spring! constants! connecting! two! neighboring!atoms.! (a)! Find! and! plot! the! phonon! dispersion! ( vs. qω )! for! longitudinal! modes!propagating!along!the!chain!and!(b)!repeat!for!k1#=!k2.!

SOLUTION

(a) The equations to solve are

1, 1 1, 2, 2 1, 2, 1

2, 1 2, 1, 2 2, 1, 1

( ) ( )

( ) ( )l l l l l

l l l l l

MU k U U k U U

MU k U U k U U−

+

= − − − −

= − − − −

&&&&

Replacing

( )1, 1

( )2, 2

i ql tl

i ql tl

U Q e

U Q e

−ω

−ω

=

=

we get 2

1 1 1 2 2 1 22

2 1 2 1 2 2 1

( ) ( )

( ) ( )

iq

iq

M Q k Q Q k Q Q eM Q k Q Q k Q Qe

−− ω = − − − −

− ω = − − − −

Setting the determinant equal to zero

21 2 1 2

21 2 1 2

( )0

( )

iq

iq

M k k k k ek k e M k k

−

+

ω − + +=

+ ω − +

The result is

2 4 2 21 2 1 22 ( ) 4 sin ( / 2) 0M M k k k k qω − + ω + =

(b) For k1 = k2, we have 2 4 2 2 24 4 sin ( / 2) 0M Mk k qω − ω + = . The gap at q = π closes and

4 sin( / 4) : 0 2k q qM

ω= → π

10

11. Condensed Matter (Optional):Consider a powder, that is, a random distribution of crystallites of a materialthat cystallizes in the face-centered cubic (fcc) structure with lattice spacinga = 3.55 A. Calculate the positions of the first three x-ray Bragg diffractionpeaks in terms of the quantity K = 4π sin θ/λ, where λ is the wavelengthof the x-ray radiation and 2θ is the scattering angle (the angle between theincident and scattered x-ray beams).

11

12. Nuclear Physics (Optional):A neutrino detector was composed of a large (615 metric tons = 6.15 ×105

kg) tank of tetrachloroethylene (C2 Cl4) located in a mine. Solar neutrinosincident on Chlorine atoms converted these atoms to Argon atoms.

a) Write down an equation summarizing the reaction. What else is producedbesides an Argon nucleus?

b) The mass of a neutral Cl37 atom is 34433.52 MeV/c2, and the mass ofa neutral Ar37 atom is 34434.33 MeV/c2. What is the threshold neutrinoenergy for this process to proceed?

c) Notably, there are neutrinos produced in sun via the beta decay of B8

with energies all the way up to 15 MeV with a flux at the Earth of ΦB8

ν =1.4 × 107 cm−2 sec−1. (these neutrinos are part of the pp-chain) The crosssection for these B8 neutrinos with the Cl atoms is σνCl = 1.14× 10−42 cm2.The atomic mass of carbon is 12.01, and the atomic mass of Chlorine is35.45. Chlorine-37 makes up 24% of naturally occurring Chlorine. What isthe average number of Argon atoms that can be expected to be produced bythese B8 neutrinos in a year?

d) These Argon atoms are then chemically separated from the detector, andcounted using the fact that the created Argon isotope is unstable with half-life = 35 days. Given a sample of 10 radioactive Argon atoms, how longshould the sample be observed such that you would have at least a 99%chance that the Ar atoms would have all decayed?

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