McGill University. Department of Physics. Ph.D. Preliminary Examination. Long Questions

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1 McGill University Department of Physics Ph.D. Preliminary Examination Long Questions Examiners: Date: Tuesday May 27, 2003 Time: 2:00 p.m. to 5:00 p.m. C. Burgess (Chairman) S. Das Gupta V. Kaspi J. Strom Olsen A. Warburton Instructions: 4 questions constitute a complete paper. You must answer any 3 questions from Part A and any 1 question from Part B. If you attempt more than 4 questions, indicate which 4 you wish to be considered for marking. All questions are of equal value. A data sheet is provided at the end of the paper. No textbooks or notes are allowed. Simple calculators are permitted. Use separate answer booklets for each problem. Write clearly your name and student number, as well as the question number, on each answer booklet.

2 Long Questions: Part A (Answer 3 of the following 5 questions.) A.1 (a) In cylindrical coordinates (ρ, φ, z) the position vector is given by r = ρ cos φˆx+ρ sin φŷ +zẑ. Express the velocity in cylindrical coordinates. (b) A particle of mass m is located on the frictionless inner surface of a paraboloid of revolution x 2 + y 2 = az. Under the influence of gravity acting in the ẑ direction, what are the equations of motion of the particle in cylindrical coordinates? (c) If the particle in (b) above is made to describe a horizontal circle in the plane z = h, show that the magnitude of its angular velocity needs to be ω = 2g/a. A.2 In the classical hydrogen atom, an electron of charge e orbits around a proton of opposite charge, +e. (a) Find the total energy E and the orbital frequency ω as a function of r, the distance between the electron and the proton. (b) Calculate the energy radiated per unit time as a function of r. (c) Using dr/dt = (dr/de)(de/dt), find the time it takes for a hydrogen atom to collapse from a radius of 10 4 cm to a radius of 0. What does your answer tell you about the classical model for the hydrogen atom? A.3 A pair of atoms, each with spin 1, interacts via the Heisenberg Hamiltonian 2 H = J S 1 S 2. A solid contains N such pairs; calculate their contribution to the magnetic susceptibility in a weak magnetic field (smaller than either J/µ B or k B T/µ B where T is the temperature, µ B the Bohr magneton and k B Boltzmann s constant), and to the specific heat in zero magnetic field. A.4 A particle of mass m moves in one dimension in the presence of a potential V (x) = with V 0 and a positive constants. V 0 (1 + x /a) 4, (a) Graph this potential as a function of x. (b) Compute, using the WKB approximation, the tunnelling rate for a particle to pass through the barrier described by V (x) in the limit that its energy E is vanishingly small. 2

3 A.5 Observer Alice sees constant crossed electric and magnetic fields, B = B ẑ and E = E ˆx in her rest frame. (a) What are the components of the electromagnetic field tensor, F µν, as seen by Alice? (b) Evaluate the quantities F µν F µν and ɛ µνλρ F µν F λρ in terms of E and B. (c) What are the fields, E and B, seen by observer Bob who moves with speed v along the positive y axis relative to Alice? (d) Evaluate E B and E 2 B 2 in terms of E and B. 3

4 Long Questions: Part B (Answer 1 of the following 5 questions.) B.1 The total binding energies of the nuclei 15 8 O, 16 8 O and 17 8O are MeV, MeV and MeV respectively. Deduce the energies of the last occupied state and the first unoccupied state of neutrons in 16 8 O. The total binding energy of 17 9F is MeV. Deduce the energy of the first unoccupied proton state in 16 8O. Why is it different from the energy of the corresponding neutron state? Account for the order of magnitude of the difference. The first excited state(j π = 1 + ) of O has an excited energy of 0.87 MeV. The corresponding state in 17 9F has an excitation energy of 0.5 MeV. Suggest a reason for the difference in excitation energy. B.2 Given Ḋ(15, 15, 80, Co) = 100 cgy/min calculate Ḋ(10, 20, 140, Co), where Ḋ(15, 15, 80, Co) = Ḋ(z, A, f, Co) stands for the dose rate in cgy/min at point Q in a water phantom at a depth z = 15 cm on the central axis of a cobalt beam with a field size A = cm 2 and source-surface distance SSD = f = 80 cm. (a) Present general answers for the dose rate at point Q using the SSD and the SAD technique. (b) Use data from the enclosed graphs and tables to obtain numerical answers for the dose rate at point Q. Please see attached table. B.3 Consider the Quantum Electrodynamics of electrons, photons and muons, described by the lagrangian density L = 1 4 F µνf µν ψ(/d + m e )ψ χ(/d + m µ )χ. Here ψ and χ are the usual Dirac spinors which respectively represent the electron and muon fields, and /D = γ µ ( µ + iea µ ) for both, with e being the electromagnetic coupling. (a) Suppose this lagrangian is supplemented by the effective interaction L = k (ψχ + χψ). The theory described by L+L contains two fermion mass eigenstates, which we again call the electron and muon. Is electron and muon 4

5 number conserved in this theory? (That is, does this theory allow these fermions to decay into one another through photon emission, as in µ e + γ?) (b) Imagine that instead of the effective interactions of part (a) we instead supplement the QED lagrangian with the effective interaction L = κ (ψ γ µ γ ν χ) F µν. Estimate the total decay rate for the process µ e + γ (in the muon rest frame) to linear order in the coupling κ in the limit m e m µ. (A basic estimate should give the dependence of the rate on the masses and couplings. A more thorough estimate should also track the powers of 2π.) (c) What is your estimate for this same total decay rate for a muon which is moving with speed v relative to the observer? B.4 Sometimes, observations of jets from astrophysical objects appear to imply super-luminal motion, i.e. motion of sources with speeds greater than that of light. In this problem, you will show how this is possible. (a) First, sketch the relevant geometry. Consider an astrophysical source that emits a radiating blob of gas at angle φ from the line-of-sight to (the very distant) Earth. Sketch that blob at two different times separated by t. Assume the blob is travelling with very high velocity v, where v < c. (b) Find an expression for t obs, the difference in arrival times of light from the blob at the two epochs as measured by an Earth-bound observer. (c) Now find an expression for the apparent velocity v obs of the blob as seen by the observer. Show that for e.g. φ = 11 and v = 0.999c, v obs = 10c. (d) Name a class of astrophysical object for which this effect has been observed. B.5 The diagram shows part of the Brillouin zone of the two-dimensional square lattice of side a, with its conventional labelling. (a) Draw the line corresponding to the free-electron Fermi energy, E F, appropriate to two valence electrons per atom and give the co-ordinates of the points A and B where this line intersects ΓM and MX. 5

6 (b) The free electron states are perturbed by a weak lattice potential V (r) = G V G e ig r, where G is a reciprocal lattice vector. Along and close to the line XM (but not too close to M) the component of lattice potential, V 10 corresponding to G = (±2π/a, 0), mixes the two degenerate (or nearly degenerate) plane wave states in the first and second zone. Assuming V 10 E F, derive an equation for the energy as a function of k around the line XM (but, again, not too close to M). (c) Hence determine the coordinates of the two points along XM in the first and second zone corresponding to energy E F. 6

7 Information which may be useful 1. Constants: e = Coulombs hc = 197 MeV-Fermi h = Joule-second = MeV-second c = meter/second k B = MeV/K = J/K N A = atoms/mole ɛ o = Coulombs/(Newton-meter 2 ) µ o = 4π 10 7 Newtons/Ampere 2 g = 9.8 m/s 2 Stefan-Boltzmann constant: σ = J/sec/m 2 /K 4 2. Maxwell s equations in linear isotropic media: ɛ E = ρ free E = µ H t 3. Cylindrical Coordinates: (r, φ, z) µ H = 0 ψ = ψ r ˆr + 1 ψ r φ ˆφ + ψ z ẑ, 2 ψ = 1 ( r ψ ) ψ r r r r 2 φ + 2 ψ 2 z 2 A = 1 r r (ra r) + 1 A φ r φ + A z z ( A 1 A z = r φ A ) ( φ Ar ˆr + z z A ) z r ˆφ + H = ɛ E t + J free ( 1 r r (ra φ) 1 r ) A r φ ẑ 4. Time-dependent Perturbation Theory: If H = H 0 + V, then S = U(, ) = 1 ī h where V (τ) = e ih0τ/ h V e ih0τ/ h. dτ V (τ) + O(V 2 ) 5. Hydrogen Wavefunctions: r, θ, φ n; l; m are given by: r 1; 0; 0 = 2 ( ) e r/a r 0 r 2; 1; ±1 = (2a 0 ) 3 2 a 0 8π a e r/(2a 0) sin θ e ±iφ 7

McGill University. Department of Physics. Ph.D. Preliminary Examination. Long Questions

McGill University. Department of Physics. Ph.D. Preliminary Examination. Long Questions McGill University Department of Physics Ph.D. Preliminary Examination Long Questions Examiners: Date: Wednesday June 8, 2005 Time: 2:00 p.m. to 5:00 p.m. D. Hanna (Chairman) R. Bennewitz G. Holder S. Lovejoy