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Canadian Association of PhysicistsSUPPORTING PHYSICS RESEARCH AND EDUCATION IN CANADA

2013 CAP Prize Examination

Compiled by the Department of Physics & Engineering Physics, University of Saskatchewan

Tuesday, February 5, 2013 Duration: 3 hours.

Last name: First name:

Institution:

Instructions:

1. You have 3 hours to complete this exam.

2. A scientific calculator is permitted but textbooks or reference materials are not.

3. There are 10 questions, each weighted equally. It is unlikely that you will be able toanswer each question, therefore you should budget your time wisely.

4. Write your solutions on the pages provided. Use the back of the pages if more space isneeded.

5. This exam consists of twenty three (23) pages.

Fundamental Constants:

Atomic mass unit: u= 1.661 1027 kg = 931.5 MeV/c2

Boltzmanns constant: kB = 1.381 1023 J/K = 8.617 105 eV/K

Coulombs Constant: k= 1/40= 8.988 109 Nm2/C2

Electron Mass: 9.109 1031 kg = 5.110 101 MeV/c2

Proton Mass: 1.673 1027 kg = 938.2 MeV/c2

Electron Compton Wavelength hmc = 2.426 1012 m

Electron Volt: 1 eV = 1.602 1019 J

Elementary Charge: e= 1.602 1019 C

Permittivity of Vacuum: 0 = 8.854 1012 C2N1m2

Permeability of Vacuum: 0 = 4 107 Hm1

Plancks Constant: = h2

= 1.055 1034 J s = 6.586 1016 eV sh= 6.626 1034 J s = 4.136 1015 eV shc= 1.986 1016 J nm = 1.240 keV nm

Gravitational constant: G= 6.672 1011 Nm2kg2

Speed of Light: c= 2.998 108 m/s

Radius of the Sun: R= 6.950 105 km

Minimal distance between Sun and Earth (perihelion distance): rp= 1.471 108 km

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Department of Physics & Engineering Physics, University of Saskatchewan page 2 of 23

Problem 1 : Mechanics/Gravitation

A rotating planet is an oblate spheroid whose equatorial radius a exceeds its polar radiusb.

The oblateness = 1 b/a is a measure of this difference. The planets oblateness has aneffect on the gravitational field of the planet. To find the effect, one may replace the spheroidof volumeVby a sphere of radius R with the same volumeV. The gravitational field of theplanet is then made up of two terms: one term represents the gravitational field of the sphereof radius R and the second term represents the gravitational field of a surface mass density(mass per unit area) that must be added to the gravitational field of the sphere to form thegravitational field of the actual spheroid. Find the required surface mass density in the firstorder of oblateness and express the final result in terms of equatorial radius a, oblateness, the uniform volume mass density of the planet 0 and the latitude on the planet.

Hint: Find the distance between any point on rotating planet and the center of planet in

terms of latitude.

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Problem 2 : Electromagnetism

A permanently polarized sphere of radius R has a uniform polarizationP. Find the electric

field inside and outside the sphere and express them in terms ofP, R, permittivity of freespace and appropriate coordinates.

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Problem 3 : Experimental Physics

A Compton scattering experiment is performed using a source of monoenergetic gamma rays

and an aluminum target. The energies of the gamma rays scattered by the target, E

, aremeasured for various scattering angles, , using equipment consisting of a scintillation crystal,photomultiplier tube, and multichannel analyzer.

detector

shielding

sourceAl target

(a)Derive the equation for the energy of the scattered gamma ray, E, in terms of the energyof the incident gamma ray, E, the rest energy of the electron, E0, and the scattering angle,.

(b) The following data were collected:

Scattering Angle, (degrees) Energy of Scattered Gammas,E (MeV)

41.4 0.50060.0 0.40275.5 0.33690.0 0.288

Using these data only, calculate the rest energy of the electron and the energy of theincident gamma rays.

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Problem 4 : Optics

Relative to the line from the sun and passing through an observer (the antisolar direction),

calculate the range of angles at which a primary rainbow appears. The index of refraction ofwater for red light is 1.330 and for violet light is 1.343. You may take the index of refractionof air as 1.000.

Hint: Consider a ray of light entering the upper-half of a spherical drop of water.

lightfrom sun

b t

top of

rainbow

bottom of

rainbow

antisolar

direction

observer

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Problem 5 : Quantum Mechanics

An electron on a surface experiences a two-dimensional potential

V(x, y) =12

m21x2 +1

2m22y

2 +Ax4, (1)

with2 > 1 >0, A >0. We also assume that the ratio1/2 is irrational, i.e., cannot bewritten as a ratio of integers.

(a)Find an approximationH0to the Hamiltonian for which you can write down exact boundstates and energy eigenvalues for this two-dimensional problem. Write down the unperturbedeigenstates and eigenvalues.If you cant recall exact results for H0 from your quantum mechanics course, just write inwords what you remember about those eigenstates and energy levels.

(b)How much does the x4 term in (??) shift the energy levels from (5a) in first order ofA?In this part and the following two parts: If you do not know how to solve the problem,describe how you would attack the problem.

(c) How much does the x4 term in (??) shift the eigenstates from (5a) in first order ofA?

(d) Now we expose the electron to a time-dependent electric field

E(t) =

E00

exp(|t|/)

in x-direction. E and are constant parameters. How large is the first order transitionprobability between the lowest two energy levels for the Hamiltonian H0 from (5a)?

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Problem 6 : Special Relativity

A photon is emitted from a source moving in a circle at a constant speed and the same

photon is received by a detector moving on the same circle at the same speed. Calculate theratio of the detected photon energy and the emitted energy as measured in the instantaneousrest-frame of the source.Hint: It may be useful to recall that four velocity is defined by u = (c; v), and thefour-momentum is p = (E/c; p). From these results demonstrate that in the conventionwhereu u= c2, the energy of a photon of four-momentum p measured by an observer withfour-velocityu is E= p u.

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Problem 7 : Statistical and Thermal Physics

Choose and answer only one part: (a)or (b)

(a) Consider a gas ofNnoninteracting electrons in a uniform magnetic field B = Bz in amacroscopic system. Due to the interaction between the magnetic moment associated withthe electron spin and the field, the single-particle energy is shifted by BB and BB forspin up and down, respectively, where B is the Bohr magneton. Assume that the fieldaffects only spin of the electron and not its orbital motion. Show that the spin magneticsusceptibility at arbitrary temperature is given by

= 22B

d

dD()

d f() , (2)

whereD() is the density of orbital states and f() is the Fermi-Dirac distribution function,

f() = 1

e()/ + 1, (3)

with fundamental temperature and chemical potential . Express in terms ofN and in the thermodynamic limit ( ), assuming that the lowest single-particle energy is zeroandD(0) = 0.

(b) Consider a gas ofN free, noninteracting electrons (mass m) confined in a cubic box ofside length L (volume V =L3) in the ultra-relativistic limit. The relativistic single-particleenergy can be approximated by =

(pc)2 + (mc2)2 pc. Momentump is quantised as

p=

L(nx, ny, nz) ; nx,y,z = 1, 2, 3, . (4)

Assume that the system is in the macroscopic limit, L , and consider zero temperature.The ultra-relativistic limit cannot be reached if the electron density n= N/V is too small.Express the conditionpFc mc

2 in terms ofn. Also express the internal energy U in termsofNand Fermi momentum pF, and the pressure of the gas in terms ofU and V.

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Problem 8 : Particle Physics/Quantum Mechanics

Neutrino oscillations can originate from a mismatch between the neutrino states created

by the weak interaction and the mass eigenstates. In a model with two neutrinos, the weakinteraction states |eand|are related to the mass eigenstates |1 and |2 by

|e|

=

cos sin sin cos

|1|2

(5)

where|1 and |2 respectively correspond to neutrino masses m1 and m2. If a weak inter-action produces an electron neutrino state |e of momentum p at t= 0, find an expressionfor the probability of detecting a tau neutrino state | at time t. You may assume thatthe neutrino masses are small compared with the momentum scale, i.e., mc2 |p|c. Basedon your results, what information on the neutrino mass spectrum could be inferred from

neutrino oscillations?

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Problem 9 : Condensed Matter Physics

For some metals, one can model the conduction electrons as a gas of nearly free electrons

(mass m) moving in a weak periodic potential V(r) created by the lattice of ions. One canwrite the single-electron wave function for a given wave number k as a Fourier expansion,

k(r) =K

CkK ei(kK)r , (6)

whereKs are reciprocal lattice vectors.

(a)Show thatk(r) above satisfies the Bloch theorem. Also expressV(r) in terms of Fouriertransforms.

(b) By substituting the above wave function into the Schrodinger equation, obtain the

Schrodinger equation in momentum space:E

2

2m(k K)2

CkK=

K

VKKCkK, (7)

whereVKK is the Fourier transform ofV(r) for wave vector K K.

(c) Consider a two-dimensional square lattice of ions with lattice spacing a. Define the firstBrillouin zone bykx= [0, 2/a] andky = [0, 2/a]. What is the degeneracy atk= (/a,/a)in the unperturbed system?

(d) The reciprocal lattice vectors are K00 (0, 0), K10 (1, 0), K01 (0, 1), and K11 (1, 1) in units of 2/a. Thus, the possible values ofK K are (0, 0), (1, 0), (0,1), and(1,1). Explain that there are only three distinct values ofVKKand denote them as V00,V10, andV11. Using the notations,

Eij =

2

2m(k Kij)

2 , Cij =CkKij; i, j= 0, 1, (8)

and assumingV(0,0) V00 = 0, rewrite Eq.(??) in matrix form.

(e) Find the energy eigenvalues at k= (/a, /a) for V10 = 0 and V11 = 0. What happensto the degeneracy at this k-point?

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cap 2013 exam