COMPREHENSIVE EXAMINATION 2013Classical Mechanics

1 A BEAD MOVING ON A HELIX

A bead moves without friction on a piece of wire which is twisted into theshape of a cylindrical helix (see �gure 1). The origin is a center of attractiveforce varying directly with the distance (constant k). Find the equation ofmotion for the bead sliding on the wire. The radius of the coil is b and the zcoordinate is proportional to the coordinate �; say z = a�:

2. A SYSTEM OF OSCILLATORS

Consider the vertical oscillations of the system of springs and masses (shownin Figure 2) with the spring constants k1 = 78; k2 = 15 and k3 = 6 dynes=cm:(a) Set up the force matrix and �nd the normal modes, normal coordinates andassociated angular frequencies. (b) If the 1g block is displaced by 1cm fromits equilibrium position with the 3g block held at equilibrium and both blocksreleased from rest, describe the subsequent motion of both blocks. (The role ofgravity in this problem is restricted to de�ne the equilibrium positions of themasses. It does not e¤ect the vertical oscillations.)

3. ANGULAR ACCELERATION ACCOMPANYING CONTRACTION

A particle of massM is attached to a string (Fig. 3) and constrained to movein a horizontal plane (the plane of the dashed line). (a) The particle rotateswith velocity v0 when the length of the string is r0. How much work is done inshortening the string to r? (Hint: The force on the particle due to the string isradial, so that the torque is zero as the string is shortened.)(b) How does this result compare to the case of a particle rotating on a string

that is freely winding up on a smooth �xed peg of �nite diameter?(c) The result of the preceding example has a bearing on the �at shape of

many galaxies. How could you use the preceding example to build a simplemodel of the formation of a galaxy from a large spherical cloud of gas endowedwith some angular momentum to start with?(Hint: It may be preferable to use energy considerations in working out this

problem.)

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4. LARMOR�S THEOREM

The force exerted upon a charge e moving with velocity ~v in an electromag-netic �eld ( ~E; ~B ) is given by

md2~r

dt2= e

�~E +

1

c~v � ~B

�:

Here the vector ~r indicates the position of the charged particle in an inertial sys-tem. Show that the e¤ect of a weak uniform magnetic �eld ~B on the motion of acharged particle in a central electric �eld E(r)r̂ can be removed by transformingto a coordinate system rotating with an angular frequency ~!L = � e

2mc~B: How

weak does the weak ~B� �eld have to be? To normalize notation let us de�ne~r = ~ainertial + ~rL; where ~ainertial is the instantaneous position of the origin ofthe central electric �eld in the inertial frame.

5. EULER�S ANGLES

A rigid body (e.g. an atomic nucleus - approximately) has its symmetry axesaskew to axes de�ned in the laboratory. In order to align its axes parallel to thelab system it has been found in a particular case that one needs just two Eulerrotations: (a) Find the matrix that represents the transformation obtained by(a) rotating about the x�axis by 45� counterclockwise, and then (b) rotatingabout the y0�axis by 30� clockwise. (b) What are the components of a unitvector along the original z�axis in the new (doubleprime) system?

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Comprehensive Examination EM (2013 summer) Department of Physics

Electricity and Magnetism

Answer any four problems. Do not turn in solutions for more than four problems.Each problem has the same weight.

1. Radiation loss: When an electron is accelerated, it is losing energy by radiatingelectromagnetic waves. Using the synchrotron radiation power, find a life time of theRutherford’s atomic model and explain why this model is not adequate for the atomicmodel in our life. Use the initial radius of the hydrogen atom r0 = 0.529× 10−8 [cm].The electron orbital speed is not relativistic, namely, you can use γ ∼ 1. Direction: 1)Find the total energy of a atom (one electron rounding around a proton), 2) Derivean equation of power loss when the electron’s orbital radius changes, 3) Use thesynchrotron power obtain the life time of the atom. The life time is the time theelectron’s orbital radius going down to zero.

Fig. Q1

2. Biot-Savart’s law: Consider a circular current with a radius R flowing on aconducting wire. (a) Obtain the magnetic fields on axis by Biot-Savarts law. Thewire width is negligible. (b) Place two identical circular coils symmetrically alonga common axis, and separated by a distance R, which is equal to the radius R ofthe current (Helmholtz coil). Obtain the magnetic fields on the axis. (c) Show themagnetic field on the axis is almost uniform around the center between two coils.

3. Gauss’s law: Two concentric conducting spheres of inner and outer radii a and b,respectively, carry charges ±Q. The empty space between the spheres is half-filled bya hemi-spherical shell of dielectric (of dielectric constant ε/ε0), as shown in the figure.(a) Find the electric field between the spheres by the Gauss’s law. (b) Calculate thesurface-charge distribution on the inner sphere.

Fig. Q3

1

Comprehensive Examination EM (2013 summer) Department of Physics

4. Boundary condition: A plane electromagnetic wave (E and H) is comingthrough a material (ε1 and µ1), and it is going into another material (ε2 and µ2).The wave is crossing the interface of these two material normally. a) Obtain thereflected wave (E1 and H1) and the transmitting wave (E2 and H2). b) Calculatethe reflection rate R and the transmission rate T of the incoming energy flux. c)Calculate the reflection rate and the transmission rate when the light is crossing theinterface of air/glass. The refractive index of the glass against the air is n = 1.5.

Fig. Q4

5. Lorentz transformation: A charge q is at rest in K-frame (laboratory) in astatic magnetic field B = (0, 0, B) but no electric field. In the K’-frame, which ismoving in x−direction with velocity v against K-frame, a person ‘b’ see the charge ismoving with −v in x−direction. How does the fields look in the moving K’-frame?Is the charge rotating by v ×B force in K’-frame?

Fig. Q5

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Comprehensive Examination EM (2013 summer) Department of Physics

Supplements of Electricity and Magnetism

Constant parameters

• Electric permitivity of free space ε0 = 8.854× 10−12 (mks) or 1/4π (cgs)

• Magnetic permeability of free space µ0 = 4π × 10−7 (mks) or 4π/c2 (cgs)

• Electron charge e = 1.6× 10−19 [C] or 4.8× 10−10 [esu]

• Electron mass m = 0.91× 10−30 [kg] or 0.91× 10−27 [g]

Maxwell equations

MKS cgs∇ ·D = ρ ∇ ·D = 4πρ (Coulomb′s law)

∇× E +∂B

∂t= 0 ∇× E +

1

c

∂B

∂t= 0 (Faraday′s law)

∇×H− ∂D

∂t= J ∇×H− 1

c

∂D

∂t=

4π

cJ (Ampere−Maxwell′s law)

∇ ·B = 0 ∇ ·B = 0 (Absence of free magnetic poles)

Synchrotron radiation

Power of the synchrotron radiation from a charge q with acceleration β̇ is givenby the following equation. Here β = v/c, γ = 1/

√1− β2.

MKS cgs

P =1

6πε0

q2β̇2γ4

cP =

2

3

q2β̇2γ4

c

Biot-Savart’s law

Magnetic field δB from a small current δI is

MKS cgs

δB =µ0

4π

δI× (x− x′)

|x− x′|3, δB =

1

c

δI× (x− x′)

|x− x′|3,

here x’ is the location of the current and x is the observation point.

3

Comprehensive Examination EM (2013 summer) Department of Physics

Taylor expansionGenerally the Taylor expansion is given by

f(x) =∞∑n=0

f (n)(a)

n!(x− a)n.

and the following Taylor expansion is useful for the Helmholz coli problem,

1

[(z ±R/2)2 +R2]3/2=

[8

5√

5R3∓ 48

25√

5R4z ± 256

125√

5R6z3 − 1152

625√

5R7z4 +O(z5)

]

A plane wave and the refractive indexA plane wave has the following relation between electric field and magnetic field,

H

E=

√ε

µ,

and the refractive index is given as n =√ε∗µ∗, here ε∗ (µ∗) is the ratio of the electric

permitivity (magnetic permeability).

Lorentz transformation

x′µ = Λµνxν

4-dimensional vectors and tensors

• Lorentz transformation matrix to the frame K ′ moving in X-direction withvelocity β and γ = 1/

√1− β2:

Λµν =

γ −iγβ 0 0iγβ γ 0 00 0 1 00 0 0 1

• Space and time: xµ = (ict, x, y, z)

• Charge and current: jµ = (icρ, jx, jy, jz)

• Potential: Aµ = (icφ, Ax, Ay, Az)

• Velocity wµ = (icγ, γvx, γvy, γvz)

• Lorentz transformation formula of E and B fields (transferred by Λµν).

E ′x = Ex B′x = Bx

E ′y = γ(Ey − βBz) B′y = γ(By + βEz)E ′z = γ(Ez + βBy) B′z = γ(Bz − βEy)

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Comprehensive Examination Physics Department

University of Nevada, Reno

Quantum Theory August 2013

Complete 4 out of 5 problems. Each problem has the same weight.

1.) a.) Suppose that A and B are two commuting observables. If |a1> and |a2> are two eigenvectors of

A with different eigenvalues, prove that the matrix element <a1|B|a2> =0.

b.) Now suppose the matrix

a

a

a

A

200

020

00

and

00

00

00

b

b

b

B , in the basis |1>,|2>,|3>

where a and b are both real constants. Show that A and B commute. Is it possible to find an orthonormal set of kets which are simultaneous eigenvectors of both A and B? If so, construct such a set, or if not, explain why this is not possible.

2.) The Zeeman Hamiltonian describes the interaction of an atom with an external magnetic field B0.

Assume this Hamiltonian can be written as )2(0 zzZ SLW where 00 2B

m

q

e

is the

Larmor frequency of the atom, L is the orbital angular momentum of the electron, S is the spin angular momentum of the electron and I is the nuclear spin. Let F be the total angular

momentum: .ISLF

a.) For a hydrogen atom in its ground state |n=0, l=0, m=0> in an external field B0 (neglecting

fine and hyperfine structure), determine the energy shifts of this level resulting from the magnetic field for each of the spin states |mS,mI> of the electron and nuclear spins. What is the degeneracy of each energy level?

b.) Now add a small perturbation SIA

where A is a positive real constant. Use perturbation theory at the lowest non-vanishing order to determine the energy shifts of the levels you

found in part a.) You may assume 02 A . What is the degeneracy of each level?

3.) A spinless particle is bound in a two-dimensional harmonic potential

)(2

1),( 222 yxmyxV . Let the kets yx nn ,| with nx = 0,1,2, etc. and ny = 0,1,2 etc.

describe the energy eigenstates of the system. Assume the particle is initially in the first excited state (with the excitation along the y-direction) i.e. 1,0|| . At time t=0, a perturbation of

the form /2),( teAxtxW is turned on. Using first-order time dependent perturbation theory,

calculate the probability that after a long time t , the system will occupy any final state.

4). Consider a system of three distinguishable spin-1/2 particles. Assume the initial state of the

system is given by |||3

1| . Let the total spin be 321 SSSS

.

a.) If zS3 is measured, what results are possible and with what probability?

b.) If instead zS is measured, what results are possible and with what probability?

c.) Suppose instead xS3 is measured, what results are possible and with what probability?

5.) A spinless particle of mass m and energy mkE 2/22 scatters from a spherical -function

shell potential )(2

)(2

Rrm

rV where is a positive real constant. Determine the

differential cross section for scattering ),( using the first order Born Approximation. Is the

result isotropic? If not, indicate the explicit dependence on and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Quantum Theory Supplement - August 2013

Schrodinger’s Equation:

H

ti

Hamiltonian:

Vm

H 22

2

Raising and lowering operators:

Angular Momentum:

1,|)1()1(,| mjmmjjmjJ

Scattering:

)'(]')'(exp['2

4

1)',( 3

2)1( xVxkkixd

mkkf

1|1|ˆ

1||ˆ

ˆˆ2

ˆ

ˆˆ2

ˆ

nnna

nnna

pm

ix

ma

pm

ix

ma

1

Comprehensive Examination Physics Department University of Nevada, Reno Statistical Mechanics August 2013 Complete any 4 out of 5 problems. Each problem has the same weight. 1. Partition function. Two-level system.

Consider a system composed of a very large number N of distinguishable and mutually non-interacting atoms, each of which has two (non-degenerate) energy levels: 1=0 and 2= > 0.

a) Using the microcanonical ensemble, find the partition function and the entropy (as a function of E, N, and ), and the average energy E (as a function of N, , and temperature T).

b) Using the canonical ensemble, find the partition function, the Helmholtz free energy F, and the average energy E (as a function of N, , and temperature T).

c) Compare the results for the average energy E from (a) and (b) and comment on it.

2. Formal thermodynamic manipulations.

From the fundamental thermodynamic relation show that

a) PSNS N

V

P ,, )()(

where ,,, PNV and S are volume, number of particles, pressure, chemical potentail, and entropy.

b) ,, )()( PT T

V

P

S

where T is temperature and the rest of the variables is specified in (a).

Hint: The fundamental thermodynamic relation is dNPdVTdSdU , so the

independent variables are SNV ,, .To prove the desired expressions you need to use a Maxwell relation approach and define the thermodynamic potentials that has independent variables of interest in (a) and (b).

3. Equilibrium of vapor and solid. A solid exists in equilibrium with its vapor, which is treated as a classical ideal gas. Here we choose the energy reference such that a particle in the gas has zero energy at zero velocity, so the solid has a binding energy per atom. (i.e. the total energy of the solid with N atoms is U0 = -N at T = 0).

2

a) Use the Einstein model to describe the excitations in the solid above the energy U0 = -N of the ground state. Find the partition function of one oscillator and Helmholtz free energy of the Einstein solid.

Hint: Einstein assumed that a solid comprising N atoms can be treated as an assembly of 3N distinguishable one-dimensional oscillators with the same frequency . Also you need to include the binding energy in the Helmholtz free energy of the solid.

b) Find the Helmholtz free energy of the vapor. c) What is the condition for the equilibrium between the vapor and the solid? d) Find the vapor pressure P(T).

4. Chain. Consider a one-dimensional chain consisting of N molecules which exist in two configurations α and , with corresponding energies α and , and lengths a and b. The chain is subject to a tensile force f (see Fig. 1).

a) Calculate the average length <L> as a function of f and the temperature.

Hint: you may consider using the following expression )ln

(f

ZL

, where Z is a

partition function.

b) Assume that α > and a > b. Estimate the average length <L> in the absence of the tensile force (f=0) as a function of temperature. What are the high- and low-temperature limits and what is characteristic temperature at which the changeover between the two limits occurs?

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c) Calculate the linear response function

.)( 0

ff

L

Produce a general argument to show that .0

5. Quantum gas.

The number of bosons Nexc in volume V and each of mass m in excited states are given by the following expression (under assumption that it is a sufficiently large number of particles in the ground state)

0

2/12/32/3

22 1)

2(

4 xexc e

dxxmVN

,

where the integral is evaluated using the Riemann function

.612.21

2)2/3(

0

2/1

xe

dxx

a) Find the expression for the Bose-Einstein condensation temperature C. b) Calculate the Bose-Einstein condensation temperature for a vapor of 87Rb atoms

(86.9 amu) with a density of 2.5x1012 cm-3. c) For experiment in (b), find the temperature at which half of the atoms are in the condensate. d) What are the values of the chemical potentials for Bose and Fermi gases at ?0

Compare and comment on your answer. e) Does boson condensation occur in two dimensions? Provide the detailed explanation.

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Supplemental expressions that may (or may not) be useful in this exam

Thermodynamic potentials

Stirling’s approximation for large N .)ln()!ln( NNNN

Multiplicity, entropy, and partition functions

ii

ii

iB PXXZkS ),/exp(,ln

/1)/(1 TkB ZTkFZ

E BNV ln]ln

[ ,

PTVTi

ii N

GNPT

N

FNVTN ,, )(),,()(),,()](exp[

Ideal gas

2/12

2/12

3 )2

()2

(;! m

h

Tmk

h

N

VZ

BN

N

Quantum gas

1

1)(

jen j

F

E

H

G

5

Fundamental Physical Constants

Name Symbol Value

Speed of light c

Planck constant h

Planck constant h

Planck hbar

Planck hbar

Gravitation constant G

Boltzmann constant k

Boltzmann constant k

Molar gas constant R

Avogadro's number NA 6.0221 x 1023 mol-1

Charge of electron e

Permeability of vacuum

Permittivity of vacuum

Coulomb constant

Faraday constant F

Mass of electron

Mass of electron

Mass of proton

Mass of proton

Mass of neutron

Mass of neutron

Atomic mass unit u

6

Atomic mass unit u

Avogadro's number

Stefan-Boltzmann constant

Rydberg constant

Bohr magneton

Bohr magneton

Flux quantum

Bohr radius

Standard atmosphere atm

Wien displacement constant b