Department of Physics PRELIMINARY EXAMINATION 2014 Part I. Short Questions

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1 Department of Physics PRELIMINARY EXAMINATION 2014 Part I. Short Questions Thursday May 15th, 2014, 14-17h Examiners: Prof. A. Clerk, Prof. M. Dobbs, Prof. G. Gervais (Chair), Prof. T. Webb, Prof. P. Wiseman INSTRUCTIONS Answer 10 questions out of the choice of 16. 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 ten 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 7 pages, including this title page. Clearly indicate the

2 2014 Prelim Short Answers 2! Brewster Window Laser Cavity Brewster Window Output Laser Light High Reflecting Mirror θ B Active Gas Medium n~n air θ B Output Mirror (~1% transmitting)! 1. Laser The laser cavity in a gas laser is defined by two end mirrors which reflect light back and forth repeatedly through the cavity and the active medium (the gas) allowing multiple rounds of stimulated emission and gain amplification. The output mirror is partially reflecting to allow a small component to leave the cavity as the laser beam. The active medium gas container has two transparent Brewster windows oriented at the Brewster angle to the propagating light in the cavity. Explain the physical reason for such a geometry of the Brewster windows in a gas laser highlighting its impact on gain in the laser cavity and properties of the transmitted laser light. 2. Two Spins Consider two quantum spin 1/2 particles. The Hamiltonian of the system only depends on spin, and is given by Ĥ = JŜ1 Ŝ2. 1. Without doing any calculations, explain why the energy eigenstates of this Hamiltonian can be chosen to be simultaneous eigenstates of Ŝ2 tot, where 2

3 2014 Prelim Short Answers 3 Ŝ tot is the total spin operator of the two particles. 2. Find the energy eigenstates and corresponding eigenvalues of this Hamiltonian. 3. Are there any degeneracies in the energy spectrum? In either case, give an intuitive explanation of why this was to be expected. 3. Relativistic Meson The average lifetime of a π-meson in its own frame of reference is 26.0 ns. (This is its proper lifetime.) If the π-meson moves with speed 0.95c with respect to the Earth, 1. what is its lifetime as measured by an observer at rest on Earth? 2. What is the average distance it travels before decaying as measured by an observer at rest on Earth? 4. Collisions In atomic and nuclear physics, when an unknown particle collides with a stationary target particle, if the two particles emerge traveling at 90 degrees then this is evidence that the two particles were equal in mass and the collision was elastic. Justify this from the point-of-view of classical physics. 3

4 2014 Prelim Short Answers 4 5. Quantum Mechanical Cat Assume h = 1J.s. Your 1kg cat suddenly starts running at a speed of 1m/s and gets through a door. How wide should the door be for your cat to interfere with itself? Assume the cat is a quantum particle. 6. From Lagrange to Hamilton Starting from the Lagrange function L = L(q, q, t), where q is a generalized coordinate and q the generalized velocity, show that you can construct from this Lagrangian a new function called the Hamiltonian where H = H(q, p, t) and q and p are the conjugate coordinate and associated canonical momentum. From this, derive Hamilton s canonical equations of motion. 7. Snell-Descartes from Maxwell s Using plane waves, show that for an electromagnetic wave hitting an interface separating two dialectric medium with dialectric constant n 1 and n 2, that Maxwell s equations implies that n 1 sinθ 1 = n 2 sinθ 2, i.e. the Snell-Descartes law. 8. Tides Using a few sentences, give a brief description of how the moon gives rise to tides on earth. Make a sketch showing the earth, moon, relevant forces, and locations of expected high tides for the configuration you choose. 4

5 2014 Prelim Short Answers 5 9. Spherical Charge Distribution The electrostatic potential associated with a certain spherically-symmetric charge distribution ρ(r) is φ(r) = q 4πɛ 0 r e r/d, where d > 0. Find ρ(r). 10. Heat Engine Derive the maximum possible efficiency, defined η = W Q H, for the heat engine shown schematically below. You will use two physical laws in your derivation and we want you to explicitly state them. 11. Fermi in a Car Thinking rather than Driving Enrico Fermi is in a car and rather than driving, he has a thought. He thinks: how far must a car travel to wear off one molecular layer of rubber from its tire tread? Please provide an answer with a justification. Hint! consider a tire 5

6 2014 Prelim Short Answers 6 that would be completely worn out after 100,000 km and make a simple ballpark estimate. 12. Bohr Model Show that in the Bohr model of hydrogen, the quantization of the angular momentum L in integer units of h is equivalent to considering a classical orbit quantized in units of the de Broglie wavelength. 13. Oscillations and Waves Two masses m 1 and m 2 are connected by a spring with constant k. Draw a representation of the one normal mode and find its angular frequency ω (show all your work!) 14. Van der Waals Gas Derive a formula for the thermodynamic internal energy of a monatomic van der Waals gas. We recall that the partition function of such gas is given by Z(N, V, T ) = 1 ( 2πmk BT ) 3N N! h 2 2 (V Nb) N e an 2 V k B T. 6

7 2014 Prelim Short Answers Uncertainty Principle Use the uncertainty principle to make an estimate of the ground state energy for the hydrogen atom. Express your answer in terms of fundamental constants. 16. Lagrange and Hamilton s Write down the Lagrange and Hamilton s function for a planar pendulum and derive the equations of motions using both Lagrange and Hamilton s formalism. 7

Department of Physics PRELIMINARY EXAMINATION 2014 Part II. Long Questions

Department of Physics PRELIMINARY EXAMINATION 2014 Part II. Long Questions Friday May 16th, 2014, 14-17h Examiners: Prof. A. Clerk, Prof. M. Dobbs, Prof. G. Gervais (Chair), Prof. T. Webb, Prof. P. Wiseman