qualifying examination, part 1 1:00 pm { 5:00 pm, …
TRANSCRIPT
QUALIFYING EXAMINATION, Part 1
1:00 pm – 5:00 pm, Thursday September 1, 2016
Attempt all parts of all four problems.
Please begin your answer to each problem on a separate sheet, write your 3 digit codeand the problem number on each sheet, and then number and staple together the sheetsfor each problem. Each problem is worth 100 points; partial credit will be given.
Calculators and cell phones may NOT be used.
1
Problem 1: Mathematical Methods
(a) (30 points) The operator implementing an infinitesimal Lorentz transformation in 1+1dimension is
U(ε) = 1− ετ (1)
where ε is the infinitesimal parameter and τ is the matrix
τ =
(0 −1−1 0
). (2)
What is U(θ), which implements a Lorentz transformation with a finite parameter θ?(You should give U as a 2× 2 matrix and write explicitly its four θ-dependent elements.You should also provide intermediate steps starting from what is given and not rely onprior knowledge of the answer.)
(b) (10 points) Argue that the integers (from −∞ to ∞) form a group under addition.Name the identity element. Do they form a group under multiplication? Explain youranswer.
(c) (30 points) Evaluate the following integral by contour integration∫ ∞0
1− cos 2x
1 + x2dx .
(d) (30 points) Given a basis of functions un = xn (n = 0, 1, ..) and a weight functionµ(x) = xe−x, find the two lowest-order normalized orthogonal polynomials in the interval0 ≤ x ≤ ∞. You can use the Gram-Schmidt procedure or any other method.
Hints: A weight function defines the scalar product of two real functions f(x) and g(x)by
〈f |g〉 =
∫ ∞0
f(x)g(x)µ(x)dx
A useful integral ∫ ∞0
xne−xdx = n!
2
Problem 2: Classical Mechanics
Consider a simple pendulum of mass M and length L suspended from a cart of mass4M . The cart can move freely along a horizontal track in the x-direction and the pendulumfeels a constant downward gravitational acceleration g (see figure).
4M
M
Lxθ
g
(a) (25 points) Write down the Lagrangian of this system using the position x of the cartand the angle θ between the pendulum and the vertical axis as generalized coordinates.
(b) (20 points) Write the Lagrangian you found in (a) in the limit of small oscillationsaround θ = 0, and derive the equations of motion for x and θ in this limit.
(c) (20 points) Find the equation of motion for x(t) in (b) for a solution with θ(t) = 0.Write down its general solution and describe the corresponding motion.
(d) (35 points) Find the normal frequencies and normal modes of this system. Write downthe most general solution to x(t) and θ(t) in the limit of small oscillations. How manyundetermined constants appear in this general solution?
3
Problem 3: Electromagnetism I
(a) (5 points) What is the electric field of a point charge q as a function of the position ~rwhere the charge is at the origin ~r = ~0 ?
(b) (10 points) What is the electric field of an infinite thin sheet of charge whose surfacecharge density [charge/area] is a constant σ.
(c) (25 points) What is the electric potential and electric field on the symmetry axis of athin disk of charge Q and radius R as a function of the displacement along the coordinatez?
(d) (15 points) For the disk of part (c), find the electric field in the limits when z � Rand z � R and compare with your results of parts (a) and (b).
(e) (30 points) Determine the electric potential V (r, θ) for the disk of charge Q and radiusR for any r > R and 0 ≤ θ ≤ π (see figure).
Q
x
y
z
R
r
θ
Hints: (i) The general solution of Laplace’s equation with azimuthal symmetry is
V (r, θ) =∞∑l=0
[Alr
l +Blr−(l+1)
]Pl(cos θ) ,
where Pl are Legendre polynomials satisfying Pl(1) = 1.(ii) Use your results from part (c).(iii) The binomial expansion for |x| < 1 is
(1 + x)α =∞∑k=0
(αk
)xk = 1 + αx+
α(α− 1)
2!x2 +
α(α− 1)(α− 2)
3!x3 + . . .
4
(f) (15 points) For an infinite plane of disks, each of charge Q and radius R, whose centersare separated by a > 2R (see figure), what is the magnitude and direction of the electricfield at a distance z >> a,R above the plane? Note that the answer to this part can befound without any detailed calculation.
Q
x
y
z
R
a
5
Problem 4: Electromagnetism II
yθ
x
SpecularReflecting Side
100% Absorbing Side
Shadow
Shadow
Reflected Light
IncidentPlaneWave
R r
Pivot
A massless rod can rotate without friction about the pivot point at its center (seefigure). Light, electromagnetic field propagating as a plane wave, propagates from left toright, along the x axis (crests are shown as lines, but are omitted for clarity in the regionof the rod). Its electric field is given by
~E(x, t) = E0y cos(kx− ωt) ,
where ~k = kx and E0 is a real number. The angle between the rod and y is denoted by θ.
Centered at the ends of the rod are disks, each with one side perfectly mirror (specu-larly) reflecting, and the other side 100% absorbing. The disks are oriented so that lightin the upper part of the rod (above the pivot) always strikes an absorptive surface, whilein the lower part, it strikes a reflective surface. Each of the disks have mass m and radiusr. Assume that the distance R from the pivot to the center of each disk satisfies R� r.The unit vector z is defined by z = x× y (a right-handed coordinate system).
The Poynting vector, which describes the energy flux density (i.e., the energy per unitarea per unit time) is given by
~S =
{1µ0
( ~E × ~B) (SI units)c4π
( ~E × ~B) (Gaussian units),
where c is the speed of light. The momentum density ~p carried by the wave is
~p =1
c2~S .
6
(a) (15 points) What are the frequency f , wavelength λ, and magnetic field ~B of the lightfield?
(b) (15 points) What is the time-average Poynting vector of the incident light?
(c) (30 points) What force does the light exert in the direction normal to the disk surfaceon the absorbing disk at a given angle θ? Similarly find the force exerted on the reflectingdisk.
Hint: consider the total linear momentum change of the electromagnetic field due to theabsorption and due to the reflection.
(d) (25 points) What is the total torque which is delivered by the light to the system ofrod plus disks around the pivot point? What is the average torque over a full rotation ofthe rod?
(e) (15 points) Determine the approximate angular acceleration of the rod, averaged overa rotation. Recall that R� r and the rod is massless.
7
N2\
€x'dlÈ;{ËUri
R
, v,¡=u'ff+u, iH-rH
å3 v'a.=",(} W-#)."(#-#)*" |($oau-ä),'v=I*o('#).##.#
Explicit Forms of Vector Operations
Let er, €2, €¡ b€ orthogonal unit vectors associated srith the coordinate directions
specified in the headings on the left, and A1, Az, A¡ be the corresponding
components of A. Then
vú=e, ffi*",ff*^ffiv.¡,:ðrA'+**&ôxt' ôxz- ðxg
:
v x a : e, (H-*)*' (#-#) -* (*-*)v'1,:ffi*ffi*ffi
vú=e' #*",1#** ^håËo .o=+$ tr a,l*ffiS rrt" oA,)*^hq#
vxÀ:er **[*n*oAt-H].., [^h #-] $ t,",r]*", + [#
('ot-#]
u',þ :+ *(t #).r..|6 * (""' #).*- #[No,",t ", ] $ (r S)=i $ t+il
.Ea9(Düìè
AND d FUNCTIONS
Note: A square-root sign is to be understood ovet elter|cocfficicnt, e.g., for -8/15 read -JW'
35. CLEBS CH-G ORDAN COEFFICIENTS, SPHERICAL HARMONIC S,
2x!/2
t- xt/+L +L/21
3/2
+r -r/2o +r/2
3/2 L/2-112 +Ll2I
4: \Æ"* o 2
"l =-l*sir.i,eiö
2x]-
L/3 2/32'/3 -r/3
2+
o -r/2-r +r/2
3
X1
3/2 r/2-L/2 -r/2
+1
F1 +11 1
32+2 +2I
2
+2
w:'/T^G"",'r-å)
",] : -ly- si,'l,eoso eió
"î :i'{*
"i,.2 s.2nó
0
+1
2/3 rl37/3 -2/3
+1 0
0+1
/3 2/3/3 -r/3
2
+1
-t -r/2
1
+1
L/2 r/2t/2 -t/2
+2 -1+1 0
0+1
3lZ-3/2
x1
321+1 +1 +1
35. Clebsch-Gordan coefficients 1
Yl^ = ?r) Ytr.
1/1sB/rs2/s
+t- -L00
-1 +1
2ll,2/,:
1
2L0000
r/3 3/s7/6 -3/ro
-L/2 rho
Ll6 1/z r/32/3 o-r/3r/6 -r/2 L/3
3/2xt
+z -Ll2+L +L/2
s/2 3/2-3/2+3/2
+al2 +Ll 1
+1 -100
-1 +1
rls 4/!4ls -Ll!
sl2
321000
-1 0
+3/2 O
+L/2 +1.
2
-1
+L-L/z
LlS r/2!/5 o
Lls -rl2
slz 3/2+112 +3/2
3/2xL/2
s/27/2
1
-t
./2 r/2
./2 -L/2
m1 ^rlm1 mz I Coefficients
t
312+L/2
2/s 3/s3/s -2/s
2/5 3ls3ls -zls
JJMM
3/10-2/53/ro
-1 -t
2
-2
+312-1+L/2 0
-7/2 +7
0 -rl2-r +L/2
+3/2 -!/2+Ll2 +L/2
0-1-1 0
-2 +L
s/2 3/2 t/2tr/2 +L/2 +L/2
321-1 -1 -1
I
s/2
a;,o: ffirvY"
L/2
LlLo zls3/s LlLs
311-0 -8115
2/s a/2 a/Lo8/rs -L/6-31r0LlLs -L/3 3/s
+
3/2Ll2
3lS 2/52lS -3/s
1
L/4 314rl4 -al4
+r/2 -L/2-r/z +L/2
-L -L/2-2 +L/2
L/2-r/3
Ll6
5/z 3/2-3/2 -3/2
2r00
+r/z -L-t/2 o
-3/2 +L
4/s rlsals -4ls
rl2 u2v2 -a/2
-24
s/2 312 u2-L/2 -L/2 -L/2
32-2 -2
-L/2 -L/2-3/2 +u2
-inø
2/3 L/3
3/LO s/rs3/s -r/Is
LlLo -zls
-2 -t/2
s/2
2!-t -1
3/4 L/41-/4-3/4
?
-t
1
r/6-a/3
L/2
-312-112
-!/2-L-3/2 O
-2
5/2 3/2-312 -3123ls 2/s2/s -3/5
-3/2 -!
s/2-s l2
1
QUALIFYING EXAMINATION, Part 2
1:00 pm – 5:00 pm, Friday September 2, 2016
Attempt all parts of all four problems.
Please begin your answer to each problem on a separate sheet, write your 3 digit codeand the problem number on each sheet, and then number and staple together the sheetsfor each problem. Each problem is worth 100 points; partial credit will be given.
Calculators and cell phones may NOT be used.
1
Problem 1: Quantum Mechanics I
Consider a particle of mass m confined in one dimension to the region −a ≤ x ≤ a.This describes an infinite square well potential of width 2a centered at the origin.
(a) (20 points) Write down the normalized expressions for the even and odd parity so-lutions to the stationary Schrodinger equation, and find the corresponding energy levelsEn (n = 1, 2, 3, . . .) in terms of m, a and ~.
(b) (40 points) Let this system be perturbed by a delta function potential at the origin
H ′ = βE1aδ(x),
where E1 is the unperturbed ground-state energy, and β is a small dimensionless param-eter. Find the corrections ∆En to the unperturbed energy levels through second order inβ.
Useful formula: for n odd,∑
1/(n2 − l2) = −1/4n2, where the sum is over all positiveodd integers l 6= n.
(c) (30 points) Find the exact energy eigenvalues in the presence of both the delta functionpotential and the infinite square well. To this end, use the piecewise solution for the wavefunction
ψ(x) =
{A+ sin(k(x− a)) for 0 < x < aA− sin(k(x+ a)) for − a < x < 0 .
Impose the continuity of ψ(x) at x = 0, and determine the discontinuity of the wave func-tion derivative ψ′(x) at x = 0 by integrating the Schrodinger equation in a small intervalaround the origin. Consider the even and odd parity solutions separately, and show thatthe energy eigenvalues of one class of solutions are not affected by the presence of thedelta function potential. For the other class, find the equation that determines implicitlythe exact energy eigenvalues.
(d) (10 points) Check that the exact energy eigenvalues you found in (c) agree with yourresult in (b) to first order in β.
2
Problem 2: Quantum Mechanics II
In the classical (optical) Kerr effect, the index of refraction is proportional to theintensity. In quantum mechanics, we may use the following Hamiltonian to characterizethe Kerr effect of the oscillator
HKerr =~χ2
(a†)2a2,
where a† and a are the creation and annihilation operators of the bosonic excitations ofthe oscillator, and χ is the Kerr frequency shift per excitation.For simplicity neglect theoscillator Hamiltonian Hosc = ~ω
(a†a+ 1/2
).
The system starts at time t = 0 in a coherent state
|ψ(t = 0)〉 = |α〉 ,
where |α〉 = e−|α|2/2∑∞
n=0αn√n!|n〉 is a superposition of number eigenstates a†a |n〉 = n |n〉,
and α is a complex number.
(a) (20 points) Verify that the number state |n〉 is also the eigenstate of HKerr and com-pute its eigenvalue.
(b) (30 points) At time t, the system evolves to |ψ (t)〉 = e−iHKerrt/~ |ψ (0)〉. Compute|ψ (t)〉 and its the overlap with the initial state 〈α|ψ (t)〉. (You may keep the summationin the final expression.)
(c) (25 points) Find the smallest positive time trevival such that |ψ (trevival)〉 = |α〉. Thisis called the first revival of the coherent state.
(d) (25 points) At time t = trevival/2, the system evolves to |ψ (trevival/2)〉. Show that|ψ (trevival/2)〉 can be written as a superposition of two coherent states, which is also calleda “Schrodinger cat” state.
Hint: The following formulas might be useful
eiπ2n = in
e−iπ2n2
=e−iπ/4 + (−1)n eiπ/4√
2,
where n is an integer.
3
Problem 3: Statistical Mechanics I
Consider a spherical drop of liquid water containing Nl molecules surrounded by N−Nl
molecules of water vapor. The total number of molecules N is fixed, and we assume thetemperature T is fixed at a given value throughout this problem. The drop and its sur-rounding vapor may be out of equilibrium.
(a) (15 points) Neglecting surface effects, write an expression for the Gibbs free energyof this system if the chemical potential of liquid water in the drop is µl and the chemicalpotential of water in the vapor is µv. Rewrite Nl in terms of the volume per molecule inthe liquid vl and the radius r of the drop. Note that vl is constant independent of the sizeof the drop.
Hint: The contribution to the Gibbs free energy from a single phase with Ni moleculesand chemical potential µi is Gi = µiNi.
(b) (15 points) The effect of the surface of the drop can be included by adding a contri-bution Gsurface = σA to the Gibbs free energy in part (a), where σ is the surface tension(σ > 0) and A is the surface area of the drop. Write Gtotal with the surface contributionexpressed in terms of r.
(c) (25 points) Make two qualitative sketches of Gtotal as a function of r: one sketch forµl − µv > 0 and a second sketch for µl − µv < 0. Starting from a drop with an arbitrarynon-equilibrium radius, describe qualitatively how the size of the drop evolves in its ap-proach to equilibrium in each of these two cases.
(d) (10 points) Under appropriate conditions, there is a critical radius rc that separatesdrops which grow in size from those that shrink. Determine this critical radius.
(e) (35 points) Assume that the vapor behaves as an ideal gas and recall that the chemicalpotential of an ideal gas with pressure p is given by
µv = µ0v + kT ln(p/p0) ,
where p0 and µ0v are, respectively, the pressure and chemical potential of a vapor in a
reference state taken to be its equilibrium with a large flat surface of water. Write thechemical potential difference µl − µv in terms of the vapor pressures p and p0 and thetemperature T . Derive the dependence of p/p0 (known as the relative humidity) on thecritical radius rc.
Do drops grow more easily in an environment with a higher relative humidity or with alower relative humidity? Explain your answer based on the expression you found for p/p0.
Hint to part (e): Consider first the equilibrium condition between vapor and water in thereference state.
4
Problem 4: Statistical Mechanics II
Consider N non-interacting and non-relativistic electrons of mass m that are restrictedto a 1D line of length L.
(a) (20 points) Determine the single-particle density of states D(ε) as a function of thesingle-particle energy ε.
(b) (15 points) Derive an expression for the Fermi energy εF of the system.
(c) (15 points) Determine the total energy per particle at temperature T = 0. Expressyour answer in terms of the Fermi energy.
(d) (30 points) Determine the temperature dependence of the chemical potential µ for thissystem to second order in T . The expansion coefficients should be expressed in terms of εF .
Hint: Use the following expansion∫ ∞0
f(ε)h(ε)dε ≈∫ µ
0
h(ε)dε+π2
6(kT )2h′(µ) + . . . ,
where h(ε) is a smooth function of ε, f(ε) is the Fermi-Dirac distribution at temperatureT and chemical potential µ, and h′ is the first derivative of h.
(e) (20 points) Does the chemical potential increase or decrease with temperature? Wouldyour answer have changed if we had considered electrons in 3D? Provide a physical ex-planation for your answer.
Hint: Consider the behavior of the single-particle density of states as a function of ε.
5
N2\
€x'dlÈ;{ËUri
R
, v,¡=u'ff+u, iH-rH
å3 v'a.=",(} W-#)."(#-#)*" |($oau-ä),'v=I*o('#).##.#
Explicit Forms of Vector Operations
Let er, €2, €¡ b€ orthogonal unit vectors associated srith the coordinate directions
specified in the headings on the left, and A1, Az, A¡ be the corresponding
components of A. Then
vú=e, ffi*",ff*^ffiv.¡,:ðrA'+**&ôxt' ôxz- ðxg
:
v x a : e, (H-*)*' (#-#) -* (*-*)v'1,:ffi*ffi*ffi
vú=e' #*",1#** ^håËo .o=+$ tr a,l*ffiS rrt" oA,)*^hq#
vxÀ:er **[*n*oAt-H].., [^h #-] $ t,",r]*", + [#
('ot-#]
u',þ :+ *(t #).r..|6 * (""' #).*- #[No,",t ", ] $ (r S)=i $ t+il
.Ea9(Düìè
AND d FUNCTIONS
Note: A square-root sign is to be understood ovet elter|cocfficicnt, e.g., for -8/15 read -JW'
35. CLEBS CH-G ORDAN COEFFICIENTS, SPHERICAL HARMONIC S,
2x!/2
t- xt/+L +L/21
3/2
+r -r/2o +r/2
3/2 L/2-112 +Ll2I
4: \Æ"* o 2
"l =-l*sir.i,eiö
2x]-
L/3 2/32'/3 -r/3
2+
o -r/2-r +r/2
3
X1
3/2 r/2-L/2 -r/2
+1
F1 +11 1
32+2 +2I
2
+2
w:'/T^G"",'r-å)
",] : -ly- si,'l,eoso eió
"î :i'{*
"i,.2 s.2nó
0
+1
2/3 rl37/3 -2/3
+1 0
0+1
/3 2/3/3 -r/3
2
+1
-t -r/2
1
+1
L/2 r/2t/2 -t/2
+2 -1+1 0
0+1
3lZ-3/2
x1
321+1 +1 +1
35. Clebsch-Gordan coefficients 1
Yl^ = ?r) Ytr.
1/1sB/rs2/s
+t- -L00
-1 +1
2ll,2/,:
1
2L0000
r/3 3/s7/6 -3/ro
-L/2 rho
Ll6 1/z r/32/3 o-r/3r/6 -r/2 L/3
3/2xt
+z -Ll2+L +L/2
s/2 3/2-3/2+3/2
+al2 +Ll 1
+1 -100
-1 +1
rls 4/!4ls -Ll!
sl2
321000
-1 0
+3/2 O
+L/2 +1.
2
-1
+L-L/z
LlS r/2!/5 o
Lls -rl2
slz 3/2+112 +3/2
3/2xL/2
s/27/2
1
-t
./2 r/2
./2 -L/2
m1 ^rlm1 mz I Coefficients
t
312+L/2
2/s 3/s3/s -2/s
2/5 3ls3ls -zls
JJMM
3/10-2/53/ro
-1 -t
2
-2
+312-1+L/2 0
-7/2 +7
0 -rl2-r +L/2
+3/2 -!/2+Ll2 +L/2
0-1-1 0
-2 +L
s/2 3/2 t/2tr/2 +L/2 +L/2
321-1 -1 -1
I
s/2
a;,o: ffirvY"
L/2
LlLo zls3/s LlLs
311-0 -8115
2/s a/2 a/Lo8/rs -L/6-31r0LlLs -L/3 3/s
+
3/2Ll2
3lS 2/52lS -3/s
1
L/4 314rl4 -al4
+r/2 -L/2-r/z +L/2
-L -L/2-2 +L/2
L/2-r/3
Ll6
5/z 3/2-3/2 -3/2
2r00
+r/z -L-t/2 o
-3/2 +L
4/s rlsals -4ls
rl2 u2v2 -a/2
-24
s/2 312 u2-L/2 -L/2 -L/2
32-2 -2
-L/2 -L/2-3/2 +u2
-inø
2/3 L/3
3/LO s/rs3/s -r/Is
LlLo -zls
-2 -t/2
s/2
2!-t -1
3/4 L/41-/4-3/4
?
-t
1
r/6-a/3
L/2
-312-112
-!/2-L-3/2 O
-2
5/2 3/2-312 -3123ls 2/s2/s -3/5
-3/2 -!
s/2-s l2
1