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The instructions and questions for the University of Michigan Physics Department Graduate Qualifying Examination Part II: Modern Physics. The exam covers topics such as integrals, series expansions, fundamental constants, and quantum mechanics. The exam is closed book, but a list of useful quantities and formulas is provided. Students must answer the required questions and choose two of the four optional questions to answer. integrals, series expansions, and fundamental constants.
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Name:______________________________
Exam Number:_______
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!!
−∞^ exp
∞
(^2) ) d x =^ π α
−∞^ x^2
∞
(^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 +
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 )
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 = −∞?
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?
Optional (do 2 of of the final 4 problems)
EB (A, Z) = CvA − CsA^2 /^3 − Cc
− Ca
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.