Math Questions for the 2011 PhD Qualifier Exam 1. Evaluate the following definite integral 3" 4 where! ( x) is the Dirac! - function. # " 4 [ ( )] dx x 2! cos x 2. Consider the differential equation dx dt + x(t) = where f (t) is a periodic function with period T o and in the interval 0 < t < T o is given by: f (t) " f (t) =!1 0 < t < T 2 o #. $ +1 T o 2 < t < T o a. Determine the exponential Fourier Series for f (t). b. Assume x(t) can be expressed as an exponential Fourier Series and find its coefficients. c. What electronic circuit demonstrates the action of this differential equation? What type of electrical signals are represented by x(t) and f (t)? 3. Find the integrating factor and the general solution including the arbitrary constants for the following differential equation 1 dy y(x) dx + cot x = cos2 x y(x).

4. Convert the following integral 1 "!1 dx ( x! 2) 1! x 2 (where x is a pure real variable) to one in the complex z - plane and evaluate it using Cauchy techniques. 5. Three elements of a group are represented by the following 2! 2 matrices,! 1 0$ # & " 0 1%! w 0 $ # " 0 w 2 & %! 0 1$ # & " 1 0% where w = e i2! 3. a. Do these elements form a group? b. If they do explain why. If they don t determine what other elements are necessary to complete the group.

Closed Book, 1.5 hours AEP Physical Concepts PhD Qualifying Exam January 6, 2011 Show all work and explain your results when answering the seven questions. You may use only the simple numeric calculators provided and no other electronic devices. [1] (20 points) Give a short description of 5 out of the 6 terms below, using words and/or short equations. (a) Magnetic resonance (b) Lorentz force (c) Michelson interferometer (d) Stark effect (e) Doppler effect (f) Bose-Einstein condensation [2] (20 points) Give values in the proper units for the following physical quantities or constants. (a) k B T at room temperature (b) Mass of 1 cm 3 of mercury (c) The speed of light in water (d) Number of hydrogen atoms in 1 cm 3 of water (e) Estimated De Broglie wavelength of a rifle bullet as it leaves the muzzle [3] (15 points) Consider a cart rolling down an incline of slope θ as in the figure. The expected acceleration of the cart is g sin θ. We can measure the acceleration by timing the cart past two photocells. If the cart has length l and takes time t 1 to pass the first photocell, its speed there is v 1 = l/t 1, and likewise for the second cell. If s is the distance between the cells, the acceleration a can be obtained from the expression below.! $! a = # l2 1 & 2 " 2s% t ' 1 $ # 2 & " 2 t 1 %

The measured data from an experiment are l = 5.0 ± 0.02 cm s =100 ± 0.2 cm t 1 = 54 ±1 ms t 2 = 31 ±1 ms What is the value of a, and does it agree with the expected value? [4] (15 points) For the circuit shown, determine and sketch the gain of the circuit as a function of frequency. Assume that the operational amplifier is ideal. [5] (10 points) Calculate the radius of the orbit of a satellite in geosynchronous orbit around the earth. [6] (10 points) Estimate the force imparted to a 1 m 2 completely reflecting surface by sunlight at noon. [7] (10 points) The dispersion relation for an electromagnetic wave propagating in one dimension is.! 2 =! 0 2 + c 2 k 2 a) Calculate the phase and group velocities. b) Will a pulse (or wave packet) of electromagnetic field change its shape as it propagates?

Applied Physics Qualifying Examination Thursday, January 6, 2011 10:45 12:15 Quantum Mechanics This is a closed-book examination. Answer all questions and write your answers in the booklet provided. Each problem is worth 20 points. Handy Formula: x n e ax dx = n! 0 a n+1 1. Consider two non-interacting particles, each of mass m, in a 1-d harmonic oscillator potential. If one is in the ground state and the other is in the excited state, write down the combined wavefunction and calculate the separation <(x 1 x 2 ) 2 > for the cases in which the particles are: (a) distinguishable particles; (b) identical bosons; (c) identical fermions (assume they are in the same spin state). 2. Assume that a particle is initially in the first excited state (n = 2) of a one-dimensional infinite square-well potential V(0) = V(a) =, V(x) = 0, 0 < x < a. Assume that the wall at x = a is abruptly expanded to x = 2a. (a) What are the normalized eigenfunctions for the new, wider well? (b) What are the probabilities that the particle will be in ground state and first two excited states of the new well? 3. The Hamiltonian for the rotational energy of a diatomic molecule in a uniform magnetic field along the z-axis is given by H ˆ = 1 L ˆ 2 + ωl ˆ z 2I where I and ω are constants and ω << /I. (a) What are the energy eigenfunctions and eigenvalues? (b) Sketch the rotational energy-levels for the molecule for the four lowest-lying states. (c) Calculate the time evolution of the expectation values < L ˆ i > (i = x, y, z) and describe the motion of < L ˆ >.

4. Consider the ground state of hydrogen. (a) What are the possible spin orientations for the proton and electron? (b) The total spin angular momentum for the system is given by S ˆ = S ˆ e + S ˆ p, where S ˆ e and ˆ S p are the individual angular momenta for the electron and the proton, respectively. Write down the possible states s m s > for the system. Group together the values which have the same values of s and equate each state with the spin orientations given in (a). (c) As a result of spin-spin coupling, these states lose their degeneracy and are split. Calculate the shift for each states in (b) where the energy shift ΔE for each of the states is given by ΔE = K < ˆ S e ˆ S p > 5. Consider the case of a hydrogen atom in its ground state placed between two capacitor plates and subjected to a time-varying electric field such that it is abruptly turned on at t = 0 and then allowed to decay, that is, E(t ) = ˆ z E 0 e γt t > 0. Find the first-order probability for the atom to be in any of the n = 2 states in the long-time limit (i.e., t >> γ -1 ). The hydrogenic wavefunctions for the n = 1 and n = 2 states are: ψ 100 = 1 πa 0 3 e r /a 0, ψ 200 = 1 1 r e r / 2a 0, 3 8πa 2a 0 0 ψ 210 = 1 32πa 0 3 r a 0 e r / 2a 0 cosθ, ψ 21±1 = 1 64πa 0 3 r a 0 e r / 2a 0 sinθ e ±iφ.

Applied Physics PhD. Written Qualifying Examination Statistical and Thermal Physics Section Cornell University January 8, 2011 Possibly Useful information: Symbol Name Value! (ideal) Gas Constant 8.314!!!!!"#!!!! Avogadro s Number 6.022 10!"!! Boltzmann s Constant 1.38 10!!"!!!! h Planck s Constant 6.626 10!!"!!!! Stefan s Constant 5.67 10!!!!!!!!!! Electronic charge 1.602 10!!"!!! Proton Mass 1.673 10!!"!"!!" Thermal de Broglie Wavelength 1 atm = 101,325 Pa 1 ev = 1.602 x 10-19 J

Problem 1: State which statistics (Maxwell- Boltzmann; Fermi- Dirac, or Bose- Einstein) would be ap- propriate for the following problems and explain why (semi- quantitatively): a) Density of He 4 gas at room temperature and pressure. b) Density of conduction electrons in copper at room temperature. c) Density of carriers (electrons and holes) in semiconducting Ge at room temperature (Ge band- gap 1 volt). Problem 2: Consider a rigid lattice of distinguishable spin ½ atoms in a magnetic field. The spins have two states, with energies!!! and +!!! for spins up and down, respectively, relative to!. The system is at temperature!. a) Determine the canonical partition function! for this system. b) Determine the total magnetic moment! =!!!! of the system. c) Determine the entropy of the system. 2

Problem 3: Maximum mechanical work available from a photoconverter. A particular type of solar cell operates by absorbing short wavelength radiation from a hot body (the sun) at temperature T S and extracting some of the absorbed energy as work, W. (See figure for details) a) Assuming that both the sun and the solar cell are black bodies, calculate the net energy flux density re- ceived by the solar cell. b) Assuming that work is extracted from the solar cell via a heat engine, calculate the maximum amount of mechanical work which can be extracted. c) Calculate the power conversion efficiency, η the ratio of the extracted to the incident power. Schematic of an ideal solar energy converter. The incident radiation, at T S is fully concen- trated so that the cell absorbs and emits radia- tion in all directions. In this limit, the effective areas of the two bodies are equal. The cell, at T C,, is coupled to the environment at an ambi- ent temperature T A, through an ideal Carnot heat engine which extracts work W. 3

Problem 4: A molecular photo- converter can be modeled as a two level system with loss- less con- tacts from the upper (C) state to a negative terminal and from the lower (V) state to a positive terminal. Absorbed light (assumed monochromatic) of energy ħ! =!! =!!!! promotes electrons from the lower to the upper level, from which they are either collected or decay radiatively to the ground state. Occupation of the C and V levels is described by Fermi- Dirac statistics with occupation functions!! and!! and quasi Fermi levels!! and!! respectively. The output voltage of the photoconverter is given by!" =!!!!. The system is intrinsic so that!! =!! =!!!! +!! at equilibrium (i.e., in the dark). Two level system. Green arrows indicate processes of electron excitation, relaxation, and transfer to or from contacts. The photo- generated current density from a thin slab of thickness! can be written as! =!"!!!! +!!!!!!!!!!!!!!!!!! 1!! where!! is the flux density of photons of energy!! from the ambient,! is the flux density of photons of energy!! from the illuminating source,! is the mass density,!! is the absorption cross section/unit mass.! is a constant. a) Show, by considering the system in equilibrium and without illumination, that!! = b) Find an expression for the open circuit voltage!!" under illumination. c) Show, by considering the symmetry of the carrier populations, that! can be written as!!!!!!!!!.!! =!"!!!! +!!!! 1! 1! + 1! where! =!!!!!"!!!! with the cell voltage restricted to the range 0!!!". 4

d) Use the result of part c) to find an expression for the efficiency,!!, of this molecular photo- converter. 5