Department of Physics PRELIMINARY EXAMINATION 2015 Part II. Long Questions Friday May 15th, 2014, 14-17h Examiners: Prof. J. Cline, Prof. H. Guo, Prof. G. Gervais (Chair), and Prof. D. Hanna INSTRUCTIONS Answer 4 questions out of the choice of 8. You must attempt one problem in each sub-category, i.e. one in classical mechanics/special relativity, one in quantum mechanics, one in electromagnetic theory and one in statistical mechanics/thermo. This is a closed book exam. Approved calculators may be used (non-programmable ones), though approximate numerical results are valid. If you attempt more than four questions, you should clearly mark which ones should be graded. Write your name and student ID on the exam booklet. question number next to each answer. This exam has 9 pages, including this title page. Clearly indicate the
2015 Prelim Long Answers 2 Statistical Mechanics/Thermodynamics The N-atom crystal A cubic crystal contains N atoms. The atoms can exist at lattice sites or an atom may find itself displaced from its normal site into the centre of one of the eight adjacent unit cells. When an atom is displaced, its energy is increased by ɛ > 0 over its energy at the normal lattice site. Suppose that the crystal is in equilibrium at temperature τ. Assume τ ɛ so the chance that two atoms try to occupy the same cell is negligible. We would like you to calculate: (a) the partition function; (b) the entropy; (c) the energy (relative to the energy the crystal would have if all atoms were at their normal sites); (d) the number of atoms in displaced sites. Cereal box A breakfest cereal company puts small toys in its cereal boxes by mixing toys into bulk cereal, and then pouring the mix into boxes. In one production 10,000 boxes contain a total of 15,000 toys. What is the probability that a randomly chosen box will have three toys? You can give your answer as either a number or a formula. Hint! Use the maximum entropy method where S = n p nln(p n ), where p n is the probability that there are n toys in the box. Keep in mind that the system has two constraints and also that the probability of a box having 100 toys is likely to be negligible. 2
2015 Prelim Long Answers 3 Quantum Mechanics Beam in a magnetic field Consider a beam of spin polarized electrons moving along the x-direction with velocity v. The spin of the electrons is parallel to the x-axis (see figure below). The beam passes adiabatically into and out of a region 0 < x < L in which there is a small uniform magnetic field B o oriented along the z-axis. Clearly, in the region of the magnetic field, the Hamiltonian of a spin is Ĥ = ge 2mcŜzB o = ω o Ŝ z where Ŝz = ( h/2)σ z is the z-component of the spin operator and σ z its corresponding Pauli matrix. ω o gebo is a constant that depends on the g-factor g, 2mc electron charge e, mass m, speed of light c and the field B o. For this problem, we neglect the orbital part of the Hamiltonian, i.e. we neglect the Lorentz force, etc. (a) What is the normalized spin wave function ψ(x) at x = 0? Note: just before the B o region; we note that the spin is polarized along the x-axis. (b) After a time t, a spin arrives at the position x = L. Determine ψ(x, t) at x = L. Clearly, ψ(x = L, t) = Û(t, 0)ψ(x = 0) where Û = exp[ (i/ h)ĥt] is the time evolution operator. (c) For some strange reason, imagine the original beam splits into two equal beams just before x = 0: one beam passes through the B o region and the other does not. After traversing equal path and maintaining quantum coherence, the two beams are recombined just outside the B o region (i.e. at x = L). What is the wave function of the combined beam there? (d) Following (c), what is the intensity of the recombined beam as a function of B o? Roughly sketch the intensity versus B o and very briefly explain the 3
2015 Prelim Long Answers 4 physics indicated by this sketch (Intensity is proportional to ψ(l) 2 ). We provide the Pauli matrices: ( ) ( 0 1 0 i σ x =, σ 1 0 y = i 0 ) ( 1 0, σ z = 0 1 ). z Region with uniform magnetic field electron beam B o 0 L X Dipole-dipole interaction. The spin-spin interaction Hamiltonian between two neutrons is ( (1 ) ) V = µ B = µ2 nα( hc) 3 3 (σ (2m p c 2 ) 2 1 σ 2 3 σ 1 ˆr σ 2 ˆr) 8π r 3 σ 1 σ 2 δ 3 (r), where µ n is the magnetic moment in units of nuclear magnetons, (µ n = 1.4 for neutrons), α = 1/137 is the fine structure constant, m p is the proton mass, and the Pauli operators σ i are related to the respective spins of the two neutrons (i = 1, 2) in the usual way, S i = ( h/2)σ i. (a) To estimate the properties of possible bound states of two neutrons, consider the trial wave function ψ(r) = Ne r/a where r is the separation between 4
2015 Prelim Long Answers 5 the neutrons. Determine the normalization constant N. What is the angular momentum of this state? (b) Make an order of magnitude estimate of the expectation values of the kinetic and potential energy for the above state, ignoring the mass difference between the proton and the neutron. Assuming V is negative, estimate the magnitude of a that minimizes the total energy, as well as the magnitude of the binding energy. Use hc = 197 MeV fm, m p c 2 = 938 MeV. How do these compare to the size and mass energy of the neutron? Do we expect to see such bound states in nature? (c) Using rotational symmetry, show that the expectation value of V in the above trial state is given by only one of the three terms in V, and calculate it. (d) The result of (c) should depend only upon σ 1 σ 2. Evaluate the latter for the case where the two neutron spins are in the singlet or triplet state of the total spin. Which one leads to bound states? Draw a picture of the spin and dipole magnetic field from one neutron and the spin of the neighboring one to show that your answer agrees with classical expectations for the sign of V in this configuration. (e) Calculate the expectation value of the kinetic energy T. (Notice that reduced mass is relevant here.) Combine with the result of (c) to find the exact estimates for a and for the binding energy. 5
2015 Prelim Long Answers 6 Electromagnetic Theory Cylindrical capacitor A cylindrical capacitor of length L consists of an inner conductor wire of radius a, a thin outer conducting shell of radius b. the space in between is filled with nonconducting material of dielectric constant ɛ. (a) Determine the electric field as a function of radial position when the capacitor is charged with total charge Q (you can neglect end effects). (b) Determine the capacitance. (c) Suppose that the dielectric material is pulled part way out of the capacitor while the latter is connected to a battery of potential V. Find the force necessary to hold the dielectric in this position. In which direction must the force be applied? (neglect fringing fields). Polarized sphere A sphere of radius R has a spatially varying permanent polarization P = (R/r) 2 P 0ˆr. It rotates with angular frequency ω around its center. (a) Find the bound charge density on the surface (σ b ) and in the interior (ρ b ). Use Gauss s theorem to prove that ρ b 0. What is the functional form of ρ b? What is the total bound charge of the sphere? (b) Find the electric field inside and outside of the sphere. (c) Find the surface current due to the rotation. Take the axis of rotation to be ẑ. If you can t do it exactly, make an order of magnitude estimate. Check dimensions of your result. 6
2015 Prelim Long Answers 7 (d) Find the magnetic moment of the spinning sphere by integrating the contributions from concentric rings of current on the surface. From this, find the magnitude and direction of B at a position z R along the axis of rotation. (If you are not able to compute these exactly, make an order of magnitude estimate.) (e) Using a symmetry argument, describe how the situation [parts (a)-(d)] would differ if the polarization of the sphere was just a constant vector P, parallel to ω. Would there still be a dipole field? 7
2015 Prelim Long Answers 8 Classical Mechanics/Special Relativity Relativistic light emitter Consider two frames of reference, S and S. S moves with speed v along the x axis of frame S and its x axis is collinear with the x axis of S. A light beam is emitted at angle θ 0 with respect to the x axis in S. (a) Find the angle θ the beam makes with respect to the x axis in S. (b) A source which radiates light uniformly in all directions in its rest frame radiates strongly in the forward direction in a frame in which it is moving with speed v close to c. Using the result of part (a), find the speed of a source for which half the radiation is emitted in a cone subtending 10 3 rad. The Particle on a hemisphere A particle of mass m starts at rest on top of a smooth fixed hemisphere of radius a (see figure below). We are interested in finding the force of constraint and to determine at which angle the particle will leave the hemisphere as it falls. (a) Show that the Lagrangian is given by L = m 2 (ṙ2 + r 2 θ2 ) mgr cos θ. (b) Determine the equation of constraint f(r, θ) = 0 describing the mass in contact with the hemisphere. (c) Using r and θ as generalized coordinates, write down the Euler-Lagrange equations including the constraint and a single Lagrange multiplier λ. Show that these equations lead to θ = g sin θ. a (d) Show that θ 2 = g cos θ+ g, where the integration constant is g/a because 2 a a θ = 0 at t = 0 when θ = 0. 8
2015 Prelim Long Answers 9 Figure 1: The particle on a hemisphere. (e) Using the relation given in part d) show that the Lagrange multiplier is λ = mg(3 cos θ 2) and find at which angles the particle falls off the hemisphere. Show that it is given by θ 0 = cos 1 ( 2 3 ). 9