DEPARTMENT OF PHYSICS BROWN UNIVERSITY Written Qualifying Examination for the Ph.D. Degree January 26, 2007 READ THESE INSTRUCTIONS CAREFULLY

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1 DEPARTMENT OF PHYSICS BROWN UNIVERSITY Written Qualifying Examination for the Ph.D. Degree January 26, 2007 READ THESE INSTRUCTIONS CAREFULLY 1. The time allowed to complete the exam is 12:00 5:00 PM. 2. All work is to be done without the use of books or papers and without help from anyone. 3. Use a separate answer book for each question, or two books if necessary, labeling all books as described in (4) below. 4. Write your name in the upper right corner on the outside front cover of each exam book. Write the problem number in the center of the outside front cover. Write nothing else on the inside or outside of the front and back covers. Note that there are separate graders for each question. 5. Answer one (and only one) problem from each of the five pairs of questions. The pairs are labeled as follows: Classical Mechanics Electricity and Magnetism Statistical Mechanics Quantum Mechanics Quantum Mechanics CM1, CM2 EM1, EM2 SM1, SM2 QM1, QM2 QM3, QM4 Note that there are two pairs of Quantum Mechanics problems. You have to do one problem from each pair. 6. All problems have equal weight. 1

2 1. CM 1 Global Warming: As a result of global warming, an iceberg of the size of France with the average thickness of 1 km initially located at the North Pole, has broken away and drifts south, until it hits the coast of Brazil at the equator and gets stuck there. Ignoring melting of the iceberg en route, find the change in the duration of the day due to this catastrophic event. The area of France is 550, 000 km 2 ; the density of the ice is 0.9 g/cm 3, the radius of the Earth is 6,400 km, acceleration of free fall is 9.81 m/s 2, and the Newton s constant is 6.67 x m 3 /(kg s 2 ). What s the change in the angular momentum and mechanical energy of the iceberg-earth system as a result of this event? Explain which of the two is not conserved in this problem and why. Solve the same problem for the case of a bug of mass m located at the top of a solid sphere of mass M and radius R, in a constant gravitational field g. The sphere and bug initially rotate around a fixed frictionless vertical axis with a constant angular velocity Ω. The bug starts its route toward the equator and stops there. Find the change in the angular velocity, mechanical energy, and angular momentum of the bug-sphere system. 2. CM 2 (a) Find the Lagrangian for the coplanar double pendulum placed in a uniform gravitational field. (b) Determine the frequencies of small oscillations. l 1 ϕ 1 m 1 l 2 ϕ 2 m 2 2

3 3. EM 1 A model for an electron gun is sketched below. The electrons travel in the z direction from the planar anode at potential V 0 to the planar cathode at zero potential at a distance L away. There is cylindrical symmetry around the z-axis. Some of the electrons are allowed to pass through a very small hole of radius a L into the field free region on the right of the sketch. The velocity of the electrons is sufficiently high that inside the gun their radial position may be taken to be constant. V0 L Cathode (a) Use Gauss law to show that the radial component of the electric field near the z axis is E r (z 0, r) r ( ) dez 2 dz z 0 where z 0 is the location of the end of the gun and r L is the distance from the axis of the cylinder. (b) Show that p, the radial component of the electron momentum at r from the symmetry axis is p = qrv 0 2βcL where q is the charge of an electron, V 0 is the potential of the cathode, r is the radial deflection of the electron and β = v/c is the velocity of the electron. (c) We can define the deflection angle φ as φ = tan 1 ( p p z where p z is the momentum along the z axis. This can also be written in terms of the defocussing backward focal length f of the electron gun as ( ) r φ = tan 1 f Show that f = 2βγmc2 L qv 0 ) 3

4 4. EM 2 A conducting sphere of radius a with total charge Q is placed in an initially uniform field of magnitude E 0. (a) Find the potential V everywhere outside the sphere. (b) Find the charge density on the surface of the sphere. 5. SM 1 A system consists of a particle of mass m which can move in one dimension along the x axis. The potential energy is infinite if x < 0, and for x > 0 V (x) = αx where α is a constant. α is greater than zero. Assume that classical statistical mechanics is a good approximation. (a) What is the average energy of the particle if the particle is able to exchange energy with a heat reservoir of temperature T? (b) What is the average value of x if the particle is able to exchange energy with a heat reservoir of temperature T? (c) If the system has a definite energy E 0 what will be the average value of x? 4

5 6. SM 2 A DNA molecule consists of two linear strands held together by a sequence of N bonds evenly spaced at a distance d from one another. The energy of each bond is ε. Consider a situation where the two strands are pulled away from each other with a constant applied force F as indicated in the sketch. The strands can be pulled apart by breaking bonds in sequence. (a) Calculate the partition function, Z, in terms of N, ε, d, F, and the number of broken bonds, m. (b) Evaluate the average number of broken bonds. 5

6 7. QM 1 Consider a spin 1 2 quantum system with the spin angular momentum operators S given by S x = h ( ) S y = h ( ) i S z = h ) ( (a) Consider the operator R z e i S z φ h Show that the expectation value of S in a general state λ is rotated relative to the expectation value in the state R z λ. Demonstrate this by explicit use of the commutation relations of the above matrices. (b) Show how a state responds to a rotation through 2π. 8. QM 2 Follow the steps outlined below to find the eigenenergies of a particle of mass m and charge q moving in two spatial dimensions in a uniform magnetic field described by the symmetric vector gauge potential: A( r) = 1 2 B r. (Do not use any other choice for the gauge.) You may assume that the particle moves in the x-y plane, and that the magnetic field points in the z-direction: B = Bẑ. (a) (3 points) Show that the Hamiltonian may be written as the sum of two parts: Ĥ = Ĥsho + Ĥl where Ĥ sho = p2 x + p 2 y 2m + mω2 2 Ĥ l = qb 2m ˆl z, (x 2 + y 2 ), and ˆl z = xp y yp x. Solve for ω in terms of m, q, and B. (b) (2 points) The two parts commute: [Ĥsho, Ĥ l ] = 0. Explain why. Evidently the eigenenergies of Ĥ are the sum of the eigenenergies of the two dimensional simple harmonic oscillator (sho) and the eigenvalues of the z-component of the particle s angular momentum multiplied by a constant. The latter energy can be either positive or negative, so can the eigenenergies of the full Hamiltonian Ĥ also be negative? Why or why not? Explain. 6

7 (c) (3 points) Ĥsho can be diagonalized by employing the raising and lowering operators a x, a x, a y, and a y, where, for instance, a x = mω 1 2 h x i 2m hω p x. Show that by instead introducing the linear combinations a ± 1 2 (a x ± ia y) a ± 1 2 (a x ia y ) both Ĥsho and Ĥl may be simultaneously diagonalized. Hint: show that ˆl z = h(a + a + a a ). What are the allowed eigenenergies of the charged particle? Comment on the degeneracies (if any). (d) (2 points) If B = 3 T esla = gauss and the charged particle is an electron with m = kg and q = C = esu, what is the energy level spacing, in electron volts (ev)? Note that 1eV = J and h = J s. Produce an actual numerical answer complete with units, not an abstract algebraic expression. 7

8 9. QM 3 (a) Consider a particle of mass m moving in the 1-dimensional potential: V x V 0 x 0 Does a bound state exist for any values of x 0 and V 0? If you answer yes, then prove so. If you answer no, then find the conditions under which a bound state is possible. (b) The same as part (a), but now for the potential: V x V 0 x 0 8

9 10. QM 4 Consider a simple harmonic oscillator in one dimension (a) The ground state can be written H 0 = p2 2m mω2 x 2 ψ 0 (x) = Ne αx2 /2 What are N and α? What is the ground state energy? (b) At time t=0 a uniform electric field is switched on, adding a term to the Hamiltonian, H = e E x What is the new ground state energy? (c) If the system is initially in the ground state and the perturbation is switched on suddenly at t = 0, then what is the probability that the oscillator is in the new ground state at t > 0? 9

DEPARTMENT OF PHYSICS BROWN UNIVERSITY Written Qualifying Examination for the Ph.D. Degree January 23, 2009 READ THESE INSTRUCTIONS CAREFULLY

DEPARTMENT OF PHYSICS BROWN UNIVERSITY Written Qualifying Examination for the Ph.D. Degree January 23, 2009 READ THESE INSTRUCTIONS CAREFULLY 1. The time allowed to complete the exam is 12:00 5:00 PM.