QUALIFYING E)いM

Part IA

November 18.2016

8:30-11:30 AM

NAⅣIE

1.

TOTAL

INSTRUCTIONS: CLOSED BOOK. Formula sheet is provided. WORK ALL PROBLEMS.Use back of pages if necessary. Extra pages are available. If you use them, be sure to write yourname and make reference on each page to the problem being done.

2.

3.

4.

PUT YOUR NAl■IE ON ALL THE PAGES!

Physics Qualifying Exam I l/18 Part IA Name

l. In this problem, we will consider the phenomenon of gravitettional slingshot, which can occur

when a space probe passes by a massive astronomical object. We will consider the case of a

space probe passing by a dense asteroid, rvhich we rvill take to have velocity uo : 13 km/s in

the .i-direction, in the reference frame of the Sun. Assume that the probe approaches the

asteroid traveling with a velocity of u, : l2 km/s in the f -direction in the reference frame

ofthe Sur, where ff : -(cos 30"i - sin 30"!). (See figure below.)

Assume that the mass of the space probe is negligible compared to the mass of asteroid,

and that both the probe and asteroid can be treated as point objects. Ignore relativisticet-fects, as well as the gravitational effect of any other objects.

(a) What is the maximum speed which the probe can achieve, in the ref'erence frame

of the Sun, after this encounter?

(b) If the probe does emerge from the encounter with the maximum possible speed

(to be determined above), in what direction will it be traveling with respect to the

Sun?

asteroid

o v, = 13 km/s

o

space probe

ＶＬ

Physics Qualifying Exam I l/18 Part IA Name

2. Consider a mass m on the end of a spring of natural length t and spring constant lr.

Let y be the vertical coordinate of the mass as measured downward from the top of the

spring. Assume the mass can only move up and down in the vertical direction.

(a) Show that

L =:m,2_:た (y――′)2 _mθ y.

(b) Determine and solve the corresponding Euler-Lagrange equations of motion.

Consider the same spring as in parts (a) and (b) but now allow the mass to also swing

from side to side in two dimensions. This time use polarcoordinates (r,Q) centered on

the top of the spring. Let @ be the angle as measured from the downward vertical.(Consider the entire @ range fiom -z to n) .

(c) Find the Lagrangian that describes the system.

(d) Write down the corresponding Lagrange equations of motion.

(e) Determine the equilibria: configurations where all time derivatives vanish.

(0 For each equilibrium approximate the Lagrange equations near the equilibrium to

first order and then solve the resulting linear ODEs.

(g) Which equilibrium is stable, which is unstable?

(h) What are the frequencies of motion near the stable equilibrium?

Physics Qualifying Exam I l/18 Part IA Name

3. (l) A proton-proton collision in a high energy accelerator can result in the production

of negatively and positively charged kaons (mp:938 MeV, mv:493.7 MeV)

via the reaction,

pp ---- pp K- K*

(a) Calculate the minimum kinetic energy of the incident proton that willallow this reaction to occur for a fixed target.

(b) How does the result of (a) compare with the rest mass of the produced

kaons?

(c) Suppose instead that the two protons collide head-on with equal momenta.

What is the minimum kinetic energy needed in this case?

(ll) A rocket ship flies past Earth at 0.91c. An astronaut inside is lying down and

having his height measured while he is parallel to the direction of motion of the

rocket.

(a) If the doctor inside the ship measures his height to be 2.00 m, what would

an observer on Earth measure his height to be?

(b) If the astronaut now stands vertically upright after the examination, what

would the doctor find for his height'?

(c) What would an observer on Earth find for his height?

(d) Ifthe astronaut is confined to a bed tilted at an angle of 300 relative to the

direction of the space ship, what is his length according to the ship's

doctor and an observer on Earth?

4.

Physics Qualifying Exam I 1/18 Palt IA Name

On Earth a baseball player can hit a ball 120 m by giving it an initial angle of 45' to thehorizontal. Take the acceleration due to gravity as g: l0 m/s2 .

Suppose the batter repeats this exercise in a space 'habitat'that has the form of a circularcylinder of radius R = Llkm and has an angular velocity about the axis of the cylindersufficient to give an apparent gravity of g at radius R. The batter stands on the innersurfbce of the habitat (at radius R) and hits the ball in the same way as on Earth (i.e., at

45" to the surface), in a plane perpendicular to the axis of the cylinder (see Figurebelow).

What is the angular velocity of the cylinder?

What is the furthest distance the batter can hit the ball, as measured along thesurface of the habitat?

(a)

(b)

QUALIFYING EXAM

Part IB

Novcmber 18,2016

1:30-4:30 PM

NAME

TOTAL

INSTRUCTIONS: CLOSED BOOK. Formula sheet is provided. WORK ALL PROBLEMS.Use back of pages if necessary. Extra pages are available. If you use them, be sure to write yourname and make reference on each page to the problem being done.

つ４

3.

4.

PUT YOUR NAl■IE ON ALL THE PAGES!

Physics Qualifying Exam I l/18 Part IB Name

1. A large, flat slab of linear dielectric material with €,.: 3 is placed in a uniform electricfield Eo. The resulting electric fields E, and E, just above andjust below the surface ofthe dielectric are shown in the figure below, with E, : I Vim and 91 = 600

ａ

　

　

ｂ

(c)

Calculate d, in degrees, and the magnitude E, in V/m.

What are the sign and magnitude of the bound surface charge density, o6 , inClm2?

What is the magnitude of the original field Eo (in V/m), and the angle do (indegrees) that it makes with respect to the horizontal?

Physics Qualifying Exam I l/ I 8 Part IB Name

2. A relativistic electron with total energy ymcz travels along the *,? axis of a

free-electron laser (the "laboratory frame"), whose wiggler magnetic field is given byB = I By7 sin (kwx), rvhere kw = 2n/711y. Assume that B* is sufficiently weak thatthe transverse velocity of the electron is substantially smaller than the speed of light.Thus, in an inertial frame travelling in the forward direction with the same gamma factory (the "beam frame"), the electron appears to be at rest except for a small transverseoscillation. Let y = 80, and assume that the period of the wiggler magnet in thelaboratory frame is 761 - 2.3 cm and that its peak magnetic field is Bw :0.1T.

(a) In the beam frame, the oncoming wiggler field appears as an oncoming planeelectromagnetic rvave. What are the peak electric and magnetic fields Es and 86of this plane wave in the beam frame (in SI units)? Along which axes doE and B point?

(b) What are the wavelength and frequency of this plane wave in the beam frame,in SI units?

(c)

(d)

(e)

What are the maximum transverse acceleration and maximum transversedisplacement of the electron in the beam frame, in SI units'/

What is the maximum transverse displacement of the electron in the laboratoryframe, in SI units?

What is the wavelength of the Thomson-scattered radiation in the laboratoryframe (in pm) that propagates in the direction of the electron beam in thelaboratory frame? This is the wavelengh of the laser light generated by thefiee-electron laser.

HINT: The following Lorentz transtbrmations may be useful

EL=

Bl=

E,: EL

B*=Btr

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γ

切れ

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ら

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＜

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ろ

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み

み

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辱

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一　

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尾

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＜

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γ

一一　

〓

ち

ら朔μ

ろ

ら

＜

＜

γ

　

γ

一一　

〓

ら

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朔詞

り知

Physics Qualifying Exam I l/18 Part IB Name

3. A rectangular conducting loop rvith side lengths I and b is dropped into a magnetic field

do at the time f = 0. The coil has a self-inductance L and the mass m. The resistance R

is negligible. The current in the loop is zero at t : 0.

ι>0:

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↓Ｊ

Ｉ

Ｉ

Ｉ

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↓

↓偽θ一〓

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3=30

× XX× × X×

× X× × XXX

× × × × × × ×

× X× × X× X

× X× × × × X

× XXXX× ×

× XX× X× X

× × X× × X×

(a) Calculate the current /(f) and the velocity v(t) of the loop as function of time.Only consider the time frame when the upper edge of the loop is not in the

magnetic field.

How strong does the magnetic field have to be to prevent the loop fromcompletely entering the magnetic freld ? What kind of motion is the loopundergoing?

HINT: Treat the loop as a circuit.

(b)

Physics Qualifying Exam

4.

Part IB NameH/18

Figure l. The Sagnac interferometer.

A Sagnac interferometer is a device that uses the interference of counterpropagating lightbeams to measure a phase shift associated with rotation of the interferometer. In the figurea realization of this type of interferometer is shown using fiber light guides.

Assume the light guides provide ideal light propagation with an index of refraction of unity.A four-port beam splitter/combiner (rvhich works like a half-silvered mirror) splits theinitial beam entering port I equally into ports 3 and 4 of the circular-loop light guide. Theclockwise (CW) and counter-clockwise (CCW) beams then re-enter the beam combiner atports 3 and 4 and the superposed signal is combined at port 2 and detected by a lightdetector as shown in the figure. (You may ignore the returning combined light that alsoappears at port l, and assume that no light propagates directly between port I to port 2, orbetween ports 3 and 4.)

The entire interferometer is on a fixed rigid table rotating at angular frequency O aroundthe center of the loop axis. The loop has radius R and you can ignore the size of the beamsplitter/combiner compared to the size of the loop, and assume that the tangential velocityat the loop is v = RQ 11 c where c is the speed of light. Monochromatic light withfrequency a,l (in radians) enters the light guide from the source as shown in the Figure.

(a) Derive the time difference between the arrival times of the CW and CCWwavefronts when they retum to the beam combiner.

(b) Derive the phase between the two beams. If the axis of the loop is aligned withthe rotation axis of the Earth and placed on the North pole and light with awavelength of l3l0 nm is used, and the minimum measurable phase change is L',what is the minimum R needed to detect the rotation of the Earthi

Beam splittencornbiner

t L,ght _l

I source I

地

ア ｀

QUALIFYING EXAM

Part IIA

Novembcr 21,2016

8:30-11:30 AM

NAME

TOTAL

INSTRUCTIONS: CLOSED BOOK. Formula sheet is provided. WORK ALL PROBLEMS.Use back of pages if necessary. Extra pages are available. If you use them, be sure to write yourname and make reference on each page to the problem being done.

つ

４

3.

4.

PUT YOUR NAl■IE ON ALL THE PAGES!

Physics Qualifying Exam lll2l Part IIA Name

I . The states of a complex atom are denoted by la, j ,m) where 7 denotes the total angularmomentum quantum number and m the corresponding quantum number for thez-component of total angular momenfum. The set cr denotes the remaining quantumnumbers needed to specify the state. Assume that the transition amplitude from an initialstate lc;,7;,m;) to a final starclo.f ,if ,msl canbe written as:

(町′Jif′ 響f10ズ・rlα

`,.,mi)

Here O is a rotationally invariant operator made of dynamical variables such as

*, .i^, it .F*, and F, .F^. Here,i is the center-of-mass operator while the vector

E denotes an external electric field rvhich can, without loss of generality, be taken to be

along the z-direction.

Assume that the atoms are initially in the jt - 7 state and unpolarized

(a)

(b)

(c)

(d)

What are all possible values of iy that the atom can make transitions to?

For now, focus on transitions to the states with iS: 1 for which ffif = 0, + L

Can you evaluate the relative probabilities for transitions to the different m;states? [f yes, do so. [f not, state clearly what further information you need to be

able to do so.

Next consider transitions to the states with the other values of 4 that you foundin part (a). Can you evaluate the relative transition rates to these states? If yes,

do so. If not, state clearly what further information you need to be able to do so.

Evaluate the transition rate to the state

lot = qi; jf = 2, Tftf - -21

Physics Qualifying Exam lll2l Part HA Name

2. The time-dependent wave function of two spinless particles is given in spherical polar co-ordinates by,

Ψ('1′ F2,ι )=ⅣC~号~子

c~αjι

/え

〔yl,1(θ l,91)yl,_1(θ 2′ ψ2)~yl,_1(θ l′ 91)yl,1(θ 2′ 92)〕

(a)

(b)

(d)

(C)

(C)

where' N',' &', and' a' are constants and Ys.1y1 ?tE spherical harmonics.

What are the allowed values of total energy (or the range of allowed values oftotal energy) for this system?

Explain your answer:

What are the possible outcomes of a measurement of the z-component of totalangular momentum for this system, and rvhat are the corresponding probabilities?Is your answer dependent on when you make the measurement?

What are the possible outcomes of a measurement of L2 , (the square of the totalangular momentum) and rvhat are the corresponding probabilities? Does youranswer depend on when you make the measurement'?

What are all possible outcomes of a measurement of the distance of particle 2

from the origin'/ What is the most likely value of this measurement'? Does youranswer depend on when you make the measurement?

If you make many measurements of the distance of particle 2 from the origin, in aseries of independent experiments, what is the average value of thesemeasurements? Does your answer depend on when you make the measurements?

=

｀

Physics Qualifying Exam lll2l Part HA Name

3. A particle of mass m is confined to move in a l-dimensional potential, V(x) given by

V(x) =

where d(x) is the Dirac delta function, and g > 0 is a positive real number, as shown inthe figure.

α　α

＞

＜一

χ　χ

√

ノχ∞グ

ｒ

Ｊ

ｌ

ヽ

ａ　　　ｂ

Figure: Infinite square-well potential, with a d function at the origin

Write down the general form of the wavefunction in the regions x > 0 and x < 0.

What are the four boundary conditions that the wavefunction must satisfy in the

two regions, including those arising from the d(x) function at the origin?

Give an analytic expression for the energy of the odd eigenstates and give theirwavefunctions. Sketch the two odd wavefunctions with lowest energy.

Derive a transcendental equation whose solution yields the bound state energiesof the even wavefunctions. Sketch the functions you find to enter thetranscendental equation and graphically solve it to obtain the lowest energysolution for the cases g: 0 and g: oo.

Sketch the lowest energy even-wavefunction for g : 0 and g : .o. What is theenergy of this state for g : 0 and g : co? Explain this in terms of the solution to aninfinite square-well problem without the d' function.

(c)

(d)

(C)

Physics Qualifying Exam lll2l Part tlA Name

4. A tl■ree― state system has unperturbed Hamiltonian ρ(0)and is suttcct tO a pcrturbation

β(1).In the basis ofthe cigenstates of β(0),the corresponding matHccs are given below.

Calculate thc irst ordcr encrgy shifts and eigcnstate shits of an thrcc states:

(a)When glの′J`°)and E`°)are dl dfferent

(b)ヽVhCn g10)= F`0).

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ム」 [器 ]=[中 l=[器 l'

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ル1[甲 }=[欅 ]口 [甲 ]リ

QUALIFYING EXAM

Part IIB

Novcmbcr 21,2016

1:30‐ 4:30 PⅣI

NAME

TOTAL

INSTRUCTIONS: CLOSED BOOK. Formula sheet is provided. WORK ALL PROBLEMS.Use back of pages if necessary. Extra pages are available. If you use them, be sure to write yourname and make reference on each page to the problem being done.

つ

４

3.

4.

PUT YOUR NAl■IE ON ALL THE PAGES!

Physics Qualifying Exam lll2l Part IIB Name

' L Consider a system consisting of N bosons, where N >) l. Each particle may be in oneof two states: the ground state (with energy E: 0) or an excited state (with energyE : e > 0). The system is in thermal equilibrium rvith temperature L

(a) For this part, assume that the bosons are distinguishable. What is the maximumtemperature ?" such that the average fraction of particles in the ground state is atleast 90Yo?

(b) Now assume that the bosons are identical. What is the maximum temperature 7"

such that the average fraction of particles in the ground state is at least 90%?

(c) Explain qualitatively why the results of part (a) and part (b) are so different.

Physics Qualifying Exam lll2l Part IIB Name

2. A thin horizontal cylinder, closed on one end, rotates with constant angular speed co

around a vertical axis that passes through the other, open end of the cylinder. The outside

air pressure is Po, the air temperature is T and the molar mass is M.

Find the air pressure inside the cylinder as a function of distance r from the rotation axis.

Assume that air behaves as an ideal gas and that its molar mass is a constant, independent

of r. The system is in thermodynamic equilibrium.

↓ω一・・・・・　　一「一・

Physics Qualifying Exam lll2l Part IIB Name

3. A paramagnetic solid consists of Natoms in a volume V. Each atom has a total spin-land zero total orbital angular momentum. There is a strong internal magnetic anisotropy

in the z-direction of the crystalline solid. The paramagnetic solid is placed in a uniform

external magnetic held, ^8, in the z-direction. The magnetic Hamiltonian of each atomis:

In (l), l=-2pu(Sln) is the magnetic moment of the atom, p, is the Bohr magneton, .i

isthespinoftheatom, I isttreunitvectorinthe z-direction,and to>0 istheintemal

magnetic anisotropy energy.

「――――――ＩＪ

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リ

＾″

一名

ｒ

ｔ一

ｒｌｌｌｌｌｌｌＬ

ａ一一β一μ一〓〃

(a)

(b)

Calculate the paramagnetic partition function of the ly' -atom solid in thermal

equilibrium at absolute temperature, T .

Calculate the magnetization in the z-direction, M-, of the N-atom solid in

thermal equilibrium at absolute temperature, I. (Recall that magnetization is the

average total magnetic moment per unit volume.)

Calculate the linear paramagnetic susceptibility,T, defined by, M.= lB,, in the

high-temperature limit, (e, lkrf )<< l, where q = 2ltuB., and ku is Boltzmann's

constant. Assume that (erf e,)>> l, so that @olkrf ) is not small.

(C)

Physics Qualifying Exam lll2l Palt IIB Name

4. Consider the Sun as a sphere of mass Mo :2 x L030 kg and radius Ro : 106 km; itsatmosphere is called the corona and it is composed of ionized gas, which for the purpose

of this problem can be approximated as an ideal gas of fully ionized hydrogen atoms. Letn(r) be the number density of hydrogen in the corona, where r is the distance from the

center of the Sun, and T(r) :r, (*)" its temperature, with a : -217.

(a) If the solar corona is in static equilibrium, its pressure P(r) satisfies thehydrostatic equilibrium relation

dP(r)

dr

where G is the gravitational constant and m the mass of a hydrogen atom. Write

P(r) as a function of the density and the temperature (hint: since the gas is fullyionized, both electrons and protons contribute equally to the gas pressure), and

then solve for n(r). The boundary condition is n(Rs ) = no.

(b) Find the pressure of the corona as r ---+ co.

(c) The density and temperature at the base of the corona ote n6 = 3 x 107 cm I and

Io =1.5 x 106 K. Eventually, the solar corona meets the local interstellarmedium, composed of = 0.3 atoms/cm3 with a temperature of 7000 K. Is the

corona in static equilibrium?

You may need the following constants:

G:6.674x l0 l'*tkg-' s-2,ka:[.38x 102]m2kg K-l s-2,m=1.67 x l027kg

=_6 Mb■(r)m