Department of Physics PRELIMINARY EXAMINATION 2014 Part II. Long Questions

Transcription

1 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 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, on 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 8 pages, including this title page. Clearly indicate the

2 2014 Prelim Long Answers 2 Statistical Mechanics/Thermodynamics 1. Ideal Gas in a Gravitational Field An ideal monatomic classical gas of N particles, each of mass m, is in thermal equilibrium at absolute temperature T. The gas is contained in a cubical box of side L, whose top and bottom sides are parallel to the earth s surface. The effect of the earth s uniform gravitational field (acceleration g) should be considered. a) What is the average kinetic energy of a particle? b) What is the RMS speed v RMS of a particle? c) What is the average potential energy of a particle? 2. The Ensemble of Two-level Systems Consider a collection of N two-level systems in thermal equilibrium at a temperature T. Each system has only two states: a ground state of energy 0 and an excited state of energy ɛ. Using your knowledge of statistical mechanics, calculate the following quantities. a) The probability that a given system will be found in the excited state. b) The entropy of the entire collection. c) Make a sketch of the temperature dependence for both the probability and the entropy. 2

3 2014 Prelim Long Answers 3 Quantum Mechanics 3. Particle in a Potential Consider a particle of mass m moving in one dimension which experiences the potential V (x) = A(δ(x) + δ(x d)), where A > 0, d > 0. a) What is the general form of an energy eigenstate (energy E) that describes a particle incident on the potential from the left? b) Derive expressions for the transmission and reflection coefficients for this state. c) For what values of E is the particle perfectly transmitted? d) While energy eigenstates are time-independent, a real scattering experiment would be described by a time-dependent wavefunction ψ(x, t). At t = 0, we would start with a wavefunction describing a particle which is localized entirely to the left of the barriers. Explain in a few sentences how the properties of the energy eigenstates you found in (b) relate to this situation. Sketch how the probability distributions would evolve in time. 4. Quantum Dot Nanoparticles as Particles in a Potential Well Quantum dots are nanometer size semiconductor crystals that have optical properties that can be understood by quantum confinement effects. Absorption of photon by a quantum dot can promote an electron leaving a positively charged 3

4 2014 Prelim Long Answers 4 hole creating an exciton. If the physical size of the nanoparticle is below the Bohr excitonic radius, then the exciton behaves as a some kind of particle trapped in a potential well which has profound implications for the optical properties of the quantum dots. The optical properties of spherical nanoparticles can be understood from a model of particle confinement within a spherical potential. Consider the case of an infinite spherical well such that V (r) = 0 if r a and V (r) = otherwise. We want to find the wavefunctions as well as the energy levels of this system. a) In spherical coordinates for a time-independent potential, the wavefunction can be separated as ψ(r, θ, φ) = R(r)Y (θ, φ). Setup the time-independent Schrödinger equation for this potential and use this equation to obtain a simplified radial equation for the function u(r) defined as u(r) = rr(r). b) Solve the radial equation for the case l = 0 (l is the angular momentum for a given quantum state) when subject to the boundary condition u(a) = 0 so as to find the allow energies and the wavefunctions. Recall! The solutions for the radial equation are u(r) = ArJ l (kr) + BrN l (kr) which is a superposition of spherical Bessel J l (x) = ( x) l ( 1 d x dx )l ( sin(x) ) and x Neumann functions N l (x) = ( x) l ( 1 d x dx )l ( cos(x) ) of order l. x 4

5 2014 Prelim Long Answers 5 Electromagnetic Theory 5. The Conducting Sphere in an Electric Field Consider a conducting sphere (radius b, centred at the origin) which is subjected to a uniform electric field E 0 in the positive ẑ direction. a) To model the electric field, use two point charges ±Q placed at z = ±R. Argue that when R, this configuration will model the constant applied electric field. How is Q related to E 0? b) Calculate the electrostatic potential for positions r outside of the conducting sphere, keeping R finite. c) Take the limit R in such a way that the two charges we introduced in (a) model the uniform applied electric field. What is the electrostatic potential outside the sphere in this limit? Write you answer in terms of E 0. d) Use your results to calculate the polarizability of the sphere. 6. Plasma Propagation and Radiation The dispersion relation for an electromagnetic wave in a plasma is given by ω 2 = ωp 2 + c 2 k 2, with the plasma frequency ω p defined as ω p = Ne2 mɛ 0, where N is the electron density, and e and m are the electron charge and mass, respectively. a) Show that the index of refraction n for ω > ω p is n = (1 ω2 p ω 2 ) 1 2. b) Calculate the phase and group velocity for an electromagnetic wave prop- 5

6 2014 Prelim Long Answers 6 agating through this plasma. We recall that they are v φ = ω k and v g = dω dk, respectively. Compare your results with the speed of light and discuss. If special relativity forbids information to propagate faster than light, what do you conclude on the information mediated by an electromagnetic wave in a plasma, and its relation with the phase and group velocity? c) For the case where ω < ω p, discuss qualitatively the behavior of an electromagnetic wave propagating through the plasma. Provide arguments on your answer based on the dispersion relation and/or the index of refraction given above. d) In Rutherford s picture of the hydrogen atom, the electron is orbiting around the nucleus on classical orbits. Explain how this picture of the hydrogen atom cannot hold based on classical electrodymamics theory, and provide semiquantitative arguments and/or estimate to show that this picture is clearly not validated by observation. Hint! the power radiated by point charges undergoing circular motion is P = µ 0q 2 a 2 γ 4, where γ = 1 6πc (1 ( v and a is the acceleration. c )2 ) 6

7 2014 Prelim Long Answers 7 Classical Mechanics/Relativity 7. The Rolling Disk Problem Consider a rolling disk of mass m and radius R rolling down on an incline making an angle α with the horizontal. The problem is to find equations of motion and the forces of constraints on the rolling disk. We choose the coordinate y to be the distance from the top of the incline to the center of the disk, and the angle θ with respect to an axis perpendicular to y as the generalized coordinates. The disk is constrained to roll without slipping, and so the equation of constraint for this rolling disk is f(y, θ) = y Rθ = 0. The disk has a moment of inertia I = 1/2mR 2. a) Write down the Lagrangian for this system, showing explicitly the kinetic and potential energy terms. How many degrees-of-freedom are they in this problem and hence how many generalized coordinates? Using the Lagrange equations for y, derive the equation of motion. b) Write down the Hamiltonian of the system and rederive the equation of motion using Hamilton s equation. c) Using a single Lagrange multiplier λ and the equation of constraint f(y, θ) = y Rθ = 0, write down the Lagrange equation for both θ and y including the 7

8 2014 Prelim Long Answers 8 constraint. Solve for λ and θ and ÿ. Calculate the generalized forces of constraint Q y and Q θ and discuss their meaning. 8. Coupled pendula Consider two pendulum coupled by a spring as shown below, with a spring constant k. The angle θ 1 and θ 2 are not necessarily equal and they are kept sufficiently small that the vertical displacement of the masses can be neglected. The two pendulum are co-planar and we suppose that, at equilibrium, the potential energy of the spring is zero. The length of each pendulum is set to one. l = 1 l = 1 θ 1 θ 2 m m a) Choosing the angles θ 1 and θ 2 as generalized coordinates, calculate the kinetic and potential energy of this system. b) Expand the potential for small oscilllations to 2 nd order in displacements. c) Calculate the eigenmodes and eigenfrequencies associated with the small oscillations of this system. 8