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Name:______________________________
Exam Number:_______
University of Michigan Physics Department
Graduate Qualifying Examination
Part II: Modern Physics Saturday 14 May 2016 9 :30 am – 2 :30 pm
This is a closed book exam, but a number of useful quantities and formulas are provided
in the front of the exam. ( Note that this list is more extensive than in past years. ) If you
need to make an assumption or estimate, indicate it clearly. Show your work in an
organized manner to receive partial credit for it. Answer the questions directly in this
exam booklet. If you need more space than there is under the problem, continue on the
back of the page or on additional blank pages that the proctor will provide. Please clearly
indicate if you continue your answer on another page. Label additional blank pages with
your exam number, found at the upper right of this page (but not with your name). Also
clearly state the problem number and “page x of y” (if there is more than one additional
page for a given question).
You must answer the first 8 required questions and 2 of the 4 optional questions.
Indicate which of the latter you wish us to grade (e.g. by circling the question number). We
will only grade the indicated optional questions. Good luck!!
Some integrals and series expansions
−∞^ exp
∞
∫ (−^ α x
(^2) ) d x =^ π α
−∞^ x^2
∞
∫ exp(−^ α x
(^2) ) d x = 1 2
π α^3
exp( x ) = 1 + x +
x^2 2
x^3 3!
x^4 4!
sin( x ) = x − x^3 3!
x^5 5!
cos( x ) = 1 − x^2 2
x^4 4!
ln( 1 + x ) = x −
x^2 2
x^3 3
x^4 4
( 1 + x )^ α^ = 1 + α x + α( α − 1 ) 2
x^2 + α( α − 1 )( α − 2 ) 3!
x^3 +
Some Fundamental Constants
speed of light c = 2. 998 × 108 m/s proton charge e = 1. 602 × 10 −^19 C Planck's constant = 6. 626 × 10 −^34 J·s = 4. 136 × 10 −^15 eV·s Rydberg constant R ∞ = 1. 097 × 107 m−^1 Coulomb constant k = ( 4 πε 0 )−^1 = 8. 988 × 109 N·m^2 / C^2 vacuum permeability μ 0 = 4 π × 10 −^7 T·m/A universal gas constant R = 8. 3 J / K·mol Avogadro! s number NA = 6. 02 × 1023 mol-^1 Boltzman n !s constant kB =R/NA = 1. 38 × 10 -^23 J/K= 8. 617 × 10 -^5 eV/K
Stefan-Boltzmann constant σ = 5. 67 × 10 −^8 W / m^2 K^4
radius of the sun R sun = 6. 96 × 108 m radius of the earth R earth = 6. 37 × 106 m radius of the moon R moon = 1. 74 × 106 m gravitational constant G = 6. 67 × 10 −^11 m^3 / (kg·s^2 )
- (Quantum Mechanics) A forced harmonic oscillator is described by the Hamiltonian H = (a†a + 12 )¯hω + f (t)a + f ∗(t)a†
where a and a†^ are the usual annihilation and creation operators satisfying [a, a†] = 1. Here f (t) is a time-dependent forcing function and f ∗(t) is its complex conjugate. The harmonic oscillator is initially in its ground state, and then the forcing function is turned on at time t 0. a) Find an expression for the transition probability Pn← 0 from the ground state to the n-th excited state in first-order time-dependent perturbation theory. b) What is the probability for the oscillator to be found in an excited state at time t = ∞ if the forcing function
f (t) = f 0 e−t
(^2) / 2 τ 2
(where f 0 and τ are constants) is turned on at time t 0 = −∞?
- (Quantum Mechanics) Consider a localized electron (spin operator S^ ~ = 1 2 ¯h~σ, where^ ~σ^ are the Pauli matrices) with magnetic moment^ ~μ^ and gyro- magnetic ratio g, whose Hamiltonian in the presence of an external magnetic field B~ is:
H = −~μ · B~ =
egh¯ 4 mc
~σ · B.~
The direction of the magnetic field is taken to define the z axis. Suppose that at time t = 0 the spin is an eigenstate of Sx with eigenvalue ¯h/2. What is the expectation value of Sx and Sy at a later time t? What is the physical interpretation of your result?
- (Statistical Mechanics) Consider a 1D quantum harmonic oscillator with eigen-energies En = (n + 1/2)¯hω, where n = 0, 1, 2,... a) At temperature T , what is the probability for the quantum oscillator to be at its ground state b) Prove that at high temperature, the probability is P ≈ ¯hω/(kB T ) c) Prove that at low temperature, the probability is P ≈ 1
- (Statistical Mechanics) A system consists of N weakly interacting par- ticles at a temperature T. Each particle has a mass m and performs one dimensional oscillations about its equilibrium position. Assuming the valid- ity of classical statistical mechanics calculate the heat capacity of this system for each of the following cases: a) The restoring force is proportional to the displacement x from equilibrium b) The restoring force is proportional to x^3.
- (Condensed Matter Physics) Consider a dimerized linear chain where all the atoms are identical. As shown in the figure, M is the mass of the atoms and k 1 , k 2 are the spring constants connecting two neighboring atoms. The spring constants are different so that k 1 6 = k 2. Calculate the sound velocity.
Optional (do 2 of of the final 4 problems)
- (Nuclear Physics) The semi-empirical mass formula (SEMF) provides a good working model for the binding energy EB of a nucleus and has the form
EB (A, Z) = CvA − CsA^2 /^3 − Cc
Z^2
A^1 /^3
− Ca
(A − 2 Z)^2
A
where A is the atomic weight and Z is the atomic number and the (Cv, Cs, Cc, Ca) are constants that are fit to nuclear data. (Note that we omit an asymmetry term). a) For a fixed (constant) atomic weight A, find the optimized number N of neutrons (where A = N + Z). Express your result in terms of the ratio N/Z. b) Using the result from the first part of the problem, derive an expression for the binding energy as a function of atomic weight A c) Now consider a simplified version of the problem where all nuclei have equal numbers of protons and neutrons, so that Z = N = A/2. Derive an expression for the atomic weight A∗ that has the highest binding energy per nucleon.
- (Particle Physics) A unit of radiation dose is the rad, which corre- sponds to an energy deposit of 0.01 J/kg. The annual safe dose for humans is 5 rad. Let us assume that 100 times this dose would lead to the extinction of life. Based on this, find a lower limit on the proton lifetime. (Assume that in proton decay all of the liberated energy is absorbed in tissue.) The mass of the proton is 1.67 × 10 −^27 kg.
- (Atomic Physics) The electron and positron have the same (absolute) magnetic moment, but opposite g-factors. Show that the “ground state” of the e+-e−^ atom (positronium), which is a 1 S 0 , 3 S 1 doublet, cannot have a linear Zeeman effect. Argue in terms of the total magnetic-moment operator.
