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PHYSICS AND ASTRONOMY
Ph.D. Preliminary Examination, Part I
Time: 9:00 a.m. -‐ 1:30 p.m. Tuesday, September 2, 2014 INSTRUCTIONS: 1. Answer a total of five questions. You must answer:
2 out of questions 1, 2, 3, 4 or 5 (Mathematical Methods and Astrophysics) 2 out of questions 6, 7 or 8 (Electrodynamics) 1 out of questions 9 or 10 (Mechanics/Relativity). 2. Use a separate answer book for each question. Write your name and the
question number on the front of each book. 3. Indicate below which five questions you wish to have graded,
sign your name, and turn in this cover sheet. ****** Circle the numbers of the five questions to be graded: 1 2 3 4 5 6 7 8 9 10
Signed ______________________
Problem #1 a) For 𝑐 < 1 show
11− 𝑐 − 1+ 𝑐 + 𝑐
! +⋯+ 𝑐! = 𝑐!!!
1− 𝑐 . b) Using Cauchy’s integral formula show that the Taylor series,
𝑓(𝑧) = 𝑓(𝑧!)+ 𝑓′(𝑧!)(𝑧 − 𝑧!) + 𝑓′′ (𝑧! )(𝑧 − 𝑧! )!/2!+⋯
=𝑓(!)(𝑧!)
𝑗!
!
!!!
(𝑧 − 𝑧!)!
converges to 𝑓(𝑧) in the disk 𝑧 − 𝑧! < 𝑅 .
Problem #2 Show that the equation !
!!!!"
𝑟! !"!"
= !"!"
admits solutions of the form f(r)g(t) which are bounded at r=0 and zero at r=a. Find the solution V(r,t) which is bounded at r = 0 and satisfies the conditions V( a, t ) = 0 (t > 0) V( r, 0 ) = 1 (0 ≤ r < a) . Show that 𝑉 0, 𝑡 = 2 (−1)!!! exp(−𝑛! 𝜋! 𝑡/𝑎!).!
!!!
Problem #3 Let 𝐴 be an 𝑛 × 𝑛 hermitean matrix. Consider the function
𝑓! 𝑧 = Tr 𝑧𝐼 − 𝐴 !!
where 𝐼 is the identity matrix.
a) What is the location and nature of the singularities (if any) of this function?
b) If it has poles, what is the meaning of the residue at each pole?
c) Find this function for the case 𝐴 = 1 22 4 .
Problem #4
a) Suppose the data 𝑥!,𝑥!,… , 𝑥! are drawn from a Gaussian distribution with unknown 𝜇 and 𝜎
𝑃 𝑥 = 12𝜋𝜎
exp − 𝑥 − 𝜇 !
2𝜎! .
Find the maximum likelihood estimators 𝜇 and 𝜎!. Show that the estimator for the mean is unbiased
𝐸 𝜇 = 𝜇,
and that the estimator for the variance is biased 𝐸 𝜎! ≠ 𝜎!, and is in fact underestimated.
b) Define the quantities chi-‐squared χ! and number of degrees of freedom 𝜈 that appear when a comparison is made between a set of data points and a model.
c) Explain how to evaluate the goodness-‐of-‐fit for both the maximum-‐likelihood and least-‐squares cases.
Problem #5
The figure represents a spherical shell of radius 𝑅 surrounding a star of radius 𝑟 << 𝑅 and effective temperature 𝜏! . The shell absorbs radiation from the star with absorption coefficient 𝛼(𝜔) ≤ 1. Outside of the shell is empty space into which the shell radiates.
a) If 𝛼 = 1 for all 𝜔, find the equilibrium temperature of the shell
𝜏! 𝜏! ,𝑅, 𝑟 . b) Now suppose that 𝛼 = 1 for 𝜔 < 𝜔! , but 𝛼 = 0 for 𝜔 ≥ 𝜔! .
Determine the total power (=energy/time) absorbed by the shell assuming that hω! 2𝜋 << 𝜏! .
c) Find the equilibrium temperature of the shell 𝜏! 𝜏! ,𝑅, 𝑟 assuming
that hω! 2𝜋 << 𝜏! .
Problem #6
A thin circular disk of radius 𝑅 lying in the 𝑥𝑦-‐plane and centered at the origin, has on it a fixed surface charge density
𝜎 𝑟,𝜑 = 𝐴𝑟sin2𝜑
where 𝑟 and 𝜑 are usual polar coordinates in the 𝑥𝑦-‐plane. Use the multipole expansion to compute the electrostatic potential at all points in space up through the electric quadrupole term. Express the answer in spherical coordinates.
Problem #7
Consider a semi-‐infinite dielectric with a real positive dielectric constant 𝜀 > 1 and magnetic permeability 𝜇 = 1. The surface of the dielectric is the 𝑥𝑦-‐plane at 𝑧 = 0 and the dielectric fills all of the space below this plane (𝑧 < 0). The region 𝑧 > 0 is a vacuum. A plane polarized simple harmonic electromagnetic wave of frequency 𝜔 is traveling inside the dielectric in the 𝑥-‐direction.
a) Give the boundary conditions that determine how the amplitudes of electromagnetic fields are related at the interface between the dielectric and the vacuum. What do the boundary conditions imply about the relation between the frequencies and wavevectors of the electromagnetic fields inside and outside the dielectric?
b) Show that the electromagnetic fields decay exponentially as one moves in the 𝑧-‐direction away from the surface of the dielectric into the vacuum.
vacuum
k dielectric
𝑧
𝑥
𝜀 > 1 𝜇 = 1
Problem #8
Consider a classical model of the Helium atom as shown below. Two electrons, each with charge – 𝑒, orbit together on opposite sides of a nucleus of charge +2𝑒. The radius of the orbit is 𝑎! and the electrons orbit with speed v, giving an angular velocity 𝜔! = v/𝑎!.
a) Show that there is no electric dipole radiation in this model.
b) Show that there is no magnetic dipole radiation in this model.
c) Compute the electric quadrupole tensor to show that there is electric quadrupole radiation. What is the frequency of this radiation?
+2e
v
-‐e -‐e
v
Problem # 9
Consider a particle of electric charge 𝑒 and mass 𝑚 interacting with a magnetic monopole of magnetic charge 𝑔 which is fixed at the origin. The magnetic field is
𝐁 𝐫 = 𝑔 𝐫
!! .
a) Obtain the equation of motion of the charged particle.
b) Show that the orbital angular momentum 𝐋 = 𝑚𝐫 × !𝐫
!" is not a
constant of the motion.
c) Find a modification of the angular momentum that is conserved. What is the physical meaning of this modificiation?
Problem #10
Consider a time-‐like curve in space-‐time. Define the proper time 𝜏 by the condition that !!
!
!" !!!!"
= 1. Define the acceleration by 𝑎! =!!!!!!!
. Find the trajectory in space-‐time of an observer moving with constant acceleration. What is the region of space-‐time not causally connected to this observer?
PHYSICS AND ASTRONOMY
Ph.D. Preliminary Examination, Part II
Time: 9:00 a.m. -‐ 1:30 p.m. Wednesday, September 3, 2014 INSTRUCTIONS: 1. Answer a total of five questions. You must answer: 3 out of questions 11, 12, 13, 14 or 15 (Quantum Mechanics) 2 out of questions 16, 17 or 18 (Statistical Mechanics) 2. Use a separate answer book for each question. Write your name and the
question number on the front of each book. 3. Indicate below which five questions you wish to have graded,
sign your name, and turn in this cover sheet. ****** Circle the numbers of the five questions to be graded: 11 12 13 14 15 16 17 18 Signed ______________________
Problem #11
Two noninteracting distinguishable particles of mass m are in the ground state of an infinite square well of width L. What is the mean square separation distance of the two particles? What is the average separation distance if the particles are indistinguishable bosons?
Problelm # 12
An isotropic two dimensional simple harmonic oscillator is described by the Hamiltonian
𝐻 = ℏ𝜔 1+ 𝑎!!𝑎!!!!,!
where 𝑎! ,𝑎!! = 𝛿!" and 𝑎! 0 >= 0.
a) Show that the kets
𝑎!!𝑎!!
!!!,!
!
𝑎! ! ± 𝑖𝑎!! ! 0 > 𝑛,𝑚 ≥ 0
are eigenkets of 𝐻. What are the eigenvalues?
b) What is the degeneracy (if any) of the eigenvalues?
c) What operator(s) could be included with 𝐻 to comprise a complete set of
commuting operators (CSCO)?
Problem #13
a) Prove the Hellmann-‐Feynman Theorem
𝜕𝜕𝜆 < 𝑛, 𝜆 𝐻 (𝜆) |𝑛, 𝜆 > = < 𝑛, 𝜆│
𝜕𝜕𝜆 𝐻 𝑛, 𝜆 >.
where the kets 𝑛, 𝜆 > are eigenvectors of H and where 𝜆 is a parameter of the system.
b) Use a) to evaluate the quantitites
𝑛ℓ𝓁𝑚 1 𝑟 𝑛ℓ𝓁𝑚
and
𝑛ℓ𝓁𝑚 1𝑟! 𝑛ℓ𝓁𝑚
for the Hydrogen atom.
Problem #14
Consider the Stark effect for the ground state of Hydrogen. The Hamiltonian is given by
𝐻 = 𝐻! − 𝑒ℰ𝑧
where 𝐻! is the usual Hamiltonian of the Hydrogen atom and ℇ denotes the magnitude of a uniform electric field.
a) Show that the lowest nonvanishing correction to the ground state energy must be of the form
Δ 𝐸 = −12 𝛼 ℰ
! where 𝛼 is a positive real number.
b) Derive an upper bound for 𝛼.
Problem #15
Consider scattering by a repulsive 𝛿 function potential of the form !!
ℏ!𝑉 𝑟 = 𝛾𝛿(𝑟 − 𝑅)
with 𝛾 > 0. Find the scattering cross section 𝜎 in the low energy limit 𝑘𝑅 << 1 .
Problem # 16
An ideal gas is confined to a cube of volume 𝑉 = 𝐿!. Each atom of the gas has an internal degree of freedom with one excited state at energy 𝜀 above the ground state. The gas is above a perfectly smooth surface of area 𝐿! . The atoms may attach to the surface with a binding energy 𝜑 and then are free to move on it. The whole system is in thermodynamic equilibrium at temperature T.
a) What is the fraction of the atoms on the surface? b) What fraction of the atoms is not on the surface and is in the excited
state?
c) Find the value of the fractions computed in a) and b) in the limits 𝑇 → 0 and 𝑇 → ∞ .
Problem # 17
Consider a container separated by a wall into two chambers. The wall is free to move but is a thermal insulator and impenetrable to particles. The gas in the left chamber (see picture below) is an ideal classical gas obeying Maxwell-‐Boltzmann statistics and consists of 𝑁! particles of mass 𝑚!. The gas in the right chamber is an ideal Bose gas consisting of 𝑁! particles of mass 𝑚!. The system reaches equilibrium keeping constant the temperatures 𝑇!and 𝑇!, of the two chambers respectively.
a) Assuming that 𝑇! is extremely high (such that the Bose gas behaves as an ideal classical gas) what will be the ratio 𝑉! 𝑉!?
b) 𝑇! is set to a somewhat lower temperature where quantum effects are present. What will then be the ratio 𝑉! 𝑉!? Should this ratio increase or decrease as 𝑇! is lowered? Explain.
c) Now assume that 𝑇! is lower than 𝑇! the critical temperature of the Bose gas. What is the ratio 𝑉! 𝑉! in this regime?
𝑉! 𝑉! 𝑁! 𝑁! 𝑇! 𝑇!
Problem # 18
A gas of volume 𝑉 consists of diatomic molecules A! and atoms A undergoing the chemical reaction
2A ↔ A!
The mass of an atom A is m and the binding energy of an A! molecule is 𝜀!. The gas is in thermodynamic equilibrium at temperature T. Find the number of dissociated atoms A relative to their total number (in atomic or molecular forms).