University of Michigan Physics Department Graduate Qualifying Examination

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1 Name: University of Michigan Physics Department Graduate Qualifying Examination Part II: Modern Physics Saturday 17 May :30 am 2:30 pm Exam Number: 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( αx 2 ) dx = π α x 2 exp( αx 2 ) dx = 1 2 π α 3 exp(x) =1+ x + x2 2 + x3 3! + x4 4! + sin(x) = x x3 3! + x5 5! + cos(x) =1 x2 2 + x4 4! + ln(1+ x) = x x2 2 + x3 3 x4 4 + (1+ x) α =1+αx + α(α 1) 2 x 2 α(α 1)(α 2) + x 3 + 3!

2 Some Fundamental Constants speed of light c = m/s 19 proton charge e = C Planck's constant h = J s = ev s Rydberg constant R = m 7 1 Coulomb constant k = (4 πú) = N m / C 7 vacuum permeability µ = 4π 10 T m/a universal gas constant R = 8.3 J / K mol Avogadroʹ s number N = mol 0 A B A Boltzmannʹ s constant k =R/N = J/K= ev/k Stefan-Boltzmann constant σ = W / m K radius of the sun R radius of the earth R radius of the moon R sun earth moon = m = m = m gravitational constant G = m / (kg s )

3 REQUIRED: DO ALL OF PROBLEMS (Quantum Mechanics) Consider the hydrogen atom. In this problem, neglect spin and relativistic corrections (fine-structure). This atom is placed in a non-central potential V 1 = ε f(r)xy, where f(r) is a radial function, and ε is a small parameter. Consider the n=2energy levels. In the presence of the perturbation, and to first order in ε, one of the eigenkets is shifted by+εa, where A is a constant. Given this fact, find the eigenenergies and eigenkets of the n=2 states to first order in ε (expressed in terms of A). (Hint: Don t evaluate any integrals you don t have to.) The following results may be useful: 1 3 Y0 0 = 4π, Y±1 1 = 8π e±iφ sinθ, Y 1,0 = 3 4π cosθ 1

4 2. (Quantum Mechanics) Tritium undergoes spontaneous beta decay emitting an electron of maximum energy of about 17 kev, 3 H 3 He + + e + ν e. Calculate the probability that the remaining helium ion is in the state with principle quantum number 2. Assume the tritium atom was initially in its ground state, and approximate the nucleus as infinitely heavy. For reference: R 1,0 = 2 ( ) Z 3/2 ( ) Z 3/2 ( e Zr/a, R 2,0 = 2 1 Zr ) e Zr/2a, a 2a 2a and R 2,1 = 1 ( ) Z 3/2 ( ) Zr e Zr/2a 3 2a 2a 2

5 3. (Quantum Mechanics) Let us assume that we have solved the Schrödinger problem exactly and that we have a list of all ortho-normalized eigenfunctions and of all the energy eigenvalues: H φ n = E n ψ n, (1) where we have assumed a discrete energy spectrum. We can obtain an approximation for the ground state energy E 0 using the variational principle, which requires the use of a trial wave function. If the trial function (by a lucky guess) turns out to be exactly the ground state wave function, the approximation would yield the exact value of E 0. In general, however, the trial wave function will not be exact and some approximation will result. (a) First consider the general case where the chosen trial function has the form ψ = ψ 0 +α ψ, (2) where ψ is a linear combination of states perpendicular to the ground state wave function ψ 0 and α is a number. Find the general expression for the approximate ground state energy in terms of the true ground state energy E 0, the parameter α, and the energy E. Use this result to find the relative error in the approximation. In terms of the quantity α = α α, what is the order of the approximation? (b) Now consider the case where the system is a (quantum) simple harmonic oscillator (this system has exact solutions, but we can still use it to illustrate the variational principle). Supppose that the trial state vector you choose corresponds to the state ψ trial = (3) 8 What is the estimate for the ground state energy? What is the corresponding relative error in the approximation? 3

6 4. (Statistical Mechanics) Consider a simple physical system with an infinite number of energy levels E n = ne 0 where n is any integer n 0 and where E 0 is a (constant) energy scale. The system is in contact with a reservoir of temperature T. Next assume that the degeneracy of each energy level is equal to n, i.e., there are n states for each energy level (each value of n). (a) Derive expressions for the partition function Z and the mean energy E as a function of temperature. (b) Evaluate the energy (find simpler expressions correct to leading order) in both the high temperature T and low temperature T 0 limits. (c) Find the dispersion of the energy ( E) 2, i.e., the second moment of E about its mean. 4

7 5. (Statistical Mechanics) Consider a zipper with N links which can open from one end. The energy of the link depends on whether or not it is closed. If open it has energy ε, if closed it has energy 0. (a) Compute the partition function of this system. Your answer should be in closed form. (b) In the limit εβ 1, find the average number of open links. This model is a very simplified model of the unwinding of two-stranded DNA molecules. (c) Find the average number of open links in the opposite limit εβ 1. 5

8 6. (Atomic Physics) The ammonia molecule can be modeled as a particle having mass m moving in an infinite one-dimensional square-well potential plus a delta function potential at the origin { V0 bδ(x) a/2 x a/2 V(x)= otherwise, where V 0, b, and a are positive constants. Assume that the delta function is strong, that is α= 2 π mv 0 ab 1. (a) Sketch the approximate wave functions for the two lowest energy states of the particle. Which one has a lower energy and why? (b) The actual splitting in the ammonia molecule of these two lowest states is about 24 GHz. Suppose we start the particle in a state that is mainly to the left of the barrier. Assuming that the initial state is some simple superposition of the two lowest eigenstates, solve the timedependent Schrodinger equation to determine the earliest time when the wave function will be mainly to the right of the barrier. 6

9 7. (Particle Physics) A pion traveling at a given speed β=v/c decays into a muon and an anti-neutrino through the reaction π µ + ν µ. If the anti-neutrino emerges at an angle of 90 o with respect to the original pion direction, find the angle at which the muon µ is emitted. Express your answer in terms of β and the masses of the muon and pion, m µ and m π, respectively. You can assume that the anti-neutrino is massless for purposes of this problem. 7

10 8. (Statistical Mechanics) Consider the following toy model of a gas of particles with hard-core repulsion (so that no two particles can occupy the same space): Divide the volume V containing the particles into M = V/a 3 cubic cells of side a (in 3 dimensions). Assume that there are N M particles in the volume and that each of the M cells can either be occupied by one particle or be empty. All allowed distributions of the N particles among the M cells have the same energy. (a) What is the entropy of the system as a function of N and the volume V = Ma 3? Assume that N and M are large and keep only extensive terms in your expression. (b) Derive the equation of state for this gas. Specifically, find an equation relating the pressure p to the volume V and the temperature T. (c) In what limit do you expect your equation of state to reduce to the ideal gas equation of state pv = Nk B T? Show that the equation behaves as expected. The figure below shows an example of a lattice gas in two dimensions. Here, space is divided into M = 36 cells of side a, some of which contain one of the N = 7 particles (filled circles) and some of which are empty. In the problem, N and M are arbitrary. a 8

11 OPTIONAL: DO 2 OF THE FOLLOWING 4 PROBLEMS 9. (Quantum Mechanics) Consider a one-dimensional quantum harmonic oscillator in the ground state 0 at t. Let the perturbation act between t and t +. H (t)= ee xe t2 /τ 2, (4) According to time-dependent perturbation theory, the wave function can be written in the form Ψ(t)= c n (t)ψ n e ient/, (5) n=0 where ψ n are the eigenfunctions of the unperturbed Hamiltonian, E n = ω(n+1/2) are the eigen energies of the unperturbed Hamiltonian. The coefficients c n (t), up to first order, and assuming the system was in the ground state are, c n (t)= i t H n0(t )e i(e n E 0 )t / dt, n 0. (6) (a) What is the probability that the oscillator is in the state n at t? (b) Determine the conditions required for the adiabatic approximation to hold. First state the content of the adiabatic approximation and then find the conditions on the parameters of the problem required for the approximation to be valid. If the adiabatic approximation is valid, what is the final state of the oscillator after the perturbation? 9

12 10. (Atomic Physics) Use simple arguments to explain why the n = 3,l=2 state in hydrogen splits into 10 levels in a weak static magnetic field aligned along the z direction. For strong magnetic fields, that is, when the magnetic field interaction is much stronger than the spin orbit interaction, are these 10 levels still non-degenerate? If not, how many non-degenerate levels are there if the spin-orbit coupling is neglected and if it is included? Explain. What are the good quantum numbers in both the weak and strong field regions? Explain. What are the exact constants of the motion in the presence of both spin-orbit and magnetic field interactions? 10

13 11. (Condensed Matter) In the 2D lattice below, each square (or triangle) represents an atom. We assume that the distance between two neighboring squares is a. (a) Find the two primitive translation vectors a 1 and a 2 (write down the x and y components for each vector. Here the horizontal and vertical directions are the x and y directions respectively). (b) Find the reciprocal lattice vectors (write down the x and y components for each vector. Here the horizontal and vertical directions are the x and y directions respectively). (c) Draw the first Brillouin zone, as well as the two reciprocal lattice vectors. 11

14 12. (Nuclear Physics) Suppose that the world consumes about BTU every year (where 1 BTU = 1055 Joule). (a) Estimate how much Uranium fuel you would need to meet the world s power needs (per year). Give both the mass in 235 U and the total mass of the Uranium. Express your answers in kilograms. (b) An asteroid that lands on the Earth s surface also delivers energy. Estimate the size (in meters) of an asteroid required to deliver the same energy (that the world uses per year). 12

University of Michigan Physics Department Graduate Qualifying Examination

Name: University of Michigan Physics Department Graduate Qualifying Examination Part I: Classical Physics Saturday 10 January 2015 9:30 am 2:30 pm Exam Number: This is a closed book exam, but a number