QUALIFYING EXAMINATION, Part 1. 2:00 PM 5:00 PM, Thursday September 3, 2009

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1 QUALIFYING EXAMINATION, Part 1 2:00 PM 5:00 PM, Thursday September 3, 2009 Attempt all parts of all four problems. Please begin your answer to each problem on a separate sheet, write your 3 digit code and the problem number on each sheet, and then number and staple together the sheets for each problem. Each problem is worth 100 points; partial credit will be given. Calculators may NOT be used. 1

2 Problem 1: Mathematical Methods 1. The purpose of this first part is to evaluate the Fourier sum S = e 2πinτ a a 2 + (2πn) 2 n= for a > 0 and 1 < τ < 1 (a and τ are real numbers) using contour integration methods. (a) Show that S can be written as a contour integral dz S = C 2πi n(z) ezτ f(z), where n(z) = 1 e z 1 ; f(z) = a a 2 z 2 and the contour C encloses a narrow strip ( Re z < a) around the imaginary axis counterclockwise. (20 points) (b) Assuming 0 < τ < 1, show that we can replace the contour C by two contours: a contour Γ 1 in the l.h.s. (Re z < 0) of the complex plane composed of an infinitely large semi-circle plus a line parallel to the imaginary axis (that is the left part of contour C), and a similar contour Γ 2 in the r.h.s. (Re z > 0) of the complex plane. (20 points) (c) Evaluate the contour integrals over Γ 1 and Γ 2 to find a closed expression for S for 0 < τ < 1. (20 points) 2. Two radioactive nuclei decay successively in series such that their numbers N i (t) satisfy Initially N 1 (0) = n 1 and N 2 (0) = n 2. dn 1 /dt = λ 1 N 1 dn 2 /dt = λ 1 N 1 λ 2 N 2. (a) Use Laplace transforms to determine N 2 (s) (i.e., the Laplace transform of N 2 (t)). (20 points) (b) Use your result in (a) to find N 2 (t) for λ 1 λ 2 and for λ 1 = λ 2 = λ. (20 points) [Hint: The Laplace transforms of N i (t) are defined by N i (s) = 0 dt e st N i (t) with s > 0.] Note: Use Laplace transforms to solve this entire problem. No credit will be given for a direct solution of the differential equations. 2

3 Problem 2: Classical Mechanics A particle of mass m moves in a central field described by F(r) = V 0 a 1 + ( ) r a ( 2 r a) exp called a Yukawa force field, where V 0 and a are positive. ( r ), a (a) Write the equations of motion for a particle with energy E and angular momentum l. (40 points) (b) Reduce the equations to the equivalent one-dimensional problem in the radial coordinate and use the effective potential to discuss the qualitative nature of the orbits for different values of the energy and the angular momentum. [Hint: To study the nature of the orbits, it is convenient to introduce the dimensionless quantity α = l 2 /(ma 2 V 0 ) and consider some values of α, for example α = 0 (zero angular momentum), α = 1/2 (moderate angular momentum) and α = 1 (large angular momentum).] (30 points) (c) Consider the limit a, (V 0 a) k. What is the corresponding effective onedimensional potential? Derive analytically the condition for circular motion in terms of this effective potential. What is the radius of the orbit in units of β = l 2 /(mk)? For what value of the energy in units of k/β is the orbit circular? (30 points) 3

4 Problem 3: Electromagnetism 1 A charge q is located in the empty space above an infinite, flat, grounded conducting plate whose surface coincides with the plane z = 0. The charge s coordinates are (x, y, z) = (0, 0, a) with a > 0. (a) Calculate the force acting on the charge. (10 points) (b) Calculate the electric field in the half-space z 0. (15 points) (c) What is the surface charge density σ(x, y) induced on the surface of the plate? [Hint: it may be helpful to imagine a small Gaussian pillbox centered on the point (x, y, 0)]. (15 points) (d) By using a suitable Gaussian surface argument (easiest) or else by computing the result directly from your answer to Part (c) (more difficult) show that the total charge induced on the plate s surface is equal to q. (15 points) (e) How much work would be needed to (adiabatically) pull the charge q away to z = + along the z-axis? (10 points) (f) Now, suppose the charge is released from rest and moves vertically towards the plate. At some moment its distance from the plate is d and its (nonrelativistic) velocity is v. Ignoring any retardation or magnetic effects, find the distribution of surface current density, J(x), that the moving charge induces in the plate. (Hint: It may be useful to recall that in polar coordinates {r, θ} the divergence of a vector field W is given, in terms of its components W r and W θ, by W = (1/r)[ (rw r )/ r+ W θ / θ]). (25 points) (g) Now, still ignoring retardation, give an argument to show that magnetic fields (produced by currents in the plate) generate a vanishing magnetic force on the moving charge of Part (f). (10 points) 4

5 Problem 4: Electromagnetism 2 Toroidal currents are of great importance in plasma physics, electronics, even as models of parity-violating nuclear anapole moments. (a) Calculate the magnetic field inside a hollow toroid of square cross section wrapped with N closely-spaced turns of wire, through which a current I is flowing (see Figure). (25 points) (b) Calculate the self-inductance L of this ideal toroidal winding. The self-inductance is defined by E = 1 2 LI2, where E is the energy of the magnetic field created by the current carried by the wire. (25 points) I 2a R I Sketch of the top view of the toroid of square cross section and its wire winding. Cutaway side view of toroid of square cross section. (c) In the limit a R, expand L to first order in 1/R, and then express the selfinductance in terms of the cross-sectional area of the toroid and its radius R. (10 points) (d) In this limit, express the relative variation of the magnetic field inside the toroidal coil, 2(B max B min )/(B max + B min ), in terms of a/r. Here B max and B min are the largest and smallest field values inside the coil, respectively. (10 points) (e) What is the self-inductance of a toroid of circular cross section of radius b (see 5

6 Figure) in the limit b R? (10 points) b R Toroid of circular cross section Side view of toroid (f) An ideal inductor L with no resistance is found in a circuit consisting of a voltage source V 0, a resistor R, a capacitor C and a switch. The switch has been closed for a time long enough that the current I through the resistor at time t = 0 is V 0 /R. At t = 0, the switch is opened. Calculate the current I in the inductor as a function of time after the switch is opened (see Figure). (20 points) Switch R V0 C L 6

7 QUALIFYING EXAMINATION, Part 2 9:00 AM noon, Friday September 4, 2009 Attempt all parts of all four problems. Please begin your answer to each problem on a separate sheet, write your 3 digit code and the problem number on each sheet, and then number and staple together the sheets for each problem. Each problem is worth 100 points; partial credit will be given. Calculators may NOT be used. 1

8 Problem 1: Quantum Mechanics 1 1. An uncharged point particle of mass m is constrained to move on a circle of radius R. If you want, use units h = m = 1. (a) What are the normalized eigenfunctions as functions of θ along the circle, what are the energy eigenvalues, and what is the degree of degeneracy of each of the levels? (20 points) 2. Now the entire system is placed in a gravitational field g, with the plane of the circle parallel to g. (a) What is the potential energy as a function of θ? (5 points) (b) Assuming mgr << E n, where n labels the unperturbed energy levels, determine the first-order corrections to the levels E n, if any. (20 points) (c) Determine the correction to the ground state energy to order g 2. (20 points) 3. Now without the gravitational field, let the particle have spin 1/2 and add a second identical particle to the system. (a) What are the two-particle eigenfunctions in the absence of interaction between the particles, written as products of total spin (S = 0, 1) and spatial states, that satisfy the particle exchange symmetry requirements? (15 points) (b) Include an interaction of the form V = V 0 δ(θ 1 θ 2 ) between the two particles and assume that V 0 is small. What are the shifts in the total two-particle energies, for states with spin S = 1, due to V, to first order in V 0? (20 points) 2

9 Problem 2: Quantum Mechanics 2 Consider two s = 1/2 spins. Their interaction with each other is described by the Hamiltonian: H ex = A σ 1 σ 2, where A is a positive constant, and σ 1 and σ 2 are vectors with components given by the Pauli matrices. In addition, a magnetic field B is applied to spin #1 only, so that the Zeeman Hamiltonian of the system is H Z = gµ B B σ1. Here µ B is the Bohr magneton and g is the g-factor. (a) Assume that a static field is applied, B = Bẑ where ẑ is the unit vector along the z-axis. Find the eigenenergies of the system. Plot the spectrum as a function of B for fixed A, labeling all relevant features. Also find the eigenfunctions for B = 0 and in the limit of infinitely large B. (40 points) (b) Find the eigenenergies of the system, and plot them as functions of B as in part (a) but for the case when the vector B lies at a 30 angle from the z-axis. (20 points) (c) The static magnetic field B is set to zero, and the two-particle system is perturbed by an oscillating magnetic field of frequency ω applied along the x-axis: B(t) = B ac cos(ωt)ˆx, and acting on spin #1 only. Assume that the system is initially in its ground state. Assume also that B ac is small and that it is applied for a very long time. What is the total transition rate out of the ground state (i.e., the total transition rate from the ground state to all excited states) as a function of B ac, A, and ω? (40 points) 3

10 Problem 3: Statistical Mechanics 1 A one-dimensional chain of elongated molecules is being stretched with force F, as depicted below. Each molecule can be in one of two configurations: one of length a and another one of length b. The energies corresponding to these configurations are E a and E b, respectively, and there are N molecules in the chain. (a) Find the partition function for the molecular chain. (40 points) (b) Calculate the average length, L, of the chain as a function of temperature T and force F. (30 points) (c) Find the length of the chain in three limits described below assuming a < b and 0 < E a < E b. First, the zero temperature limit when the external force is zero. Second, the large temperature limit when the external force is zero. Third, the zero temperature limit when F 0 and assuming that (b a)f > E b E a, while the chain does not break. Succinctly explain the results you obtained in the three limits. (30 points) F b b a b a a b F b a 4

11 Problem 4: Statistical Mechanics 2 Consider a collection of fermions: electrons (charge e, effective mass m e, and spin 1/2) and protons (charge e, effective mass m p, and spin 1/2). The oppositely charged particles may pair up to form a gas of hydrogen atoms (presumed to be in their ground state), or the atoms may dissociate into an electron-proton gas. We assume that the electrons, protons, and atoms form ideal unpolarized classical gases in equilibrium with each other. (a) Calculate the free energy of a gas composed of N e electrons and N p protons, in volume V at temperature T, treating them as classical non-interacting particles with internal degrees of freedom (spins). (20 points) (b) Pairing an individual electron and proton at rest into an atom lowers the energy of the pair by ε. Calculate the free energy of a gas of N a atoms in volume V treating them as classical non-interacting particles of mass m e + m p ; the energy ε is independent of the internal states of these particles defined by the spins of the constituents. (20 points) (c) Calculate the chemical potentials µ e, µ p, and µ a of the electron, proton, and atom gases, respectively. (20 points) (d) Express the equilibrium condition between atoms and electrons/protons in terms of their chemical potentials. (15 points) (e) Assuming the charge neutrality of the system and high enough temperature, so that the density of atoms n a is small compared to the total density n of particles (i.e., electrons, protons, and atoms), find the density n a of atoms as a function of n, T, ε, m e, and m p. (25 points) 5

Qualifying Exam. Aug Part II. Please use blank paper for your work do not write on problems sheets!

Qualifying Exam. Aug Part II. Please use blank paper for your work do not write on problems sheets! Qualifying Exam Aug. 2015 Part II Please use blank paper for your work do not write on problems sheets! Solve only one problem from each of the four sections Mechanics, Quantum Mechanics, Statistical Physics