214_final_exam_solutions_fall_2007

214_final_exam_solutions_fall_2007 - Part 1: Multiple...

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Unformatted text preview: Part 1: Multiple Choice (4 points each for 10 questions) You do not need to show work. Just give your answer in the boxes. 1). Which of the following is an example of tunneling (list all that apply)? (A) An electron is liberated from a metal when an electric field is applied, even though its potential energy should have it classically trapped according to the value of the material’s work function. (B) A beam of light strikes a piece of metal and sends an electron speeding away from its surface. (C) By means of electrons, electricity is conducted through a single piece of metal. (D) A flash of light appears on an old glow—in-the—dark clock dial indicating that an alpha particle has left a nucleus. 2) Which of the following is not true of a photon? (A)p=h/A (B)f=c//l (C) Its dispersion relation is not linear. (D) E = P C “I; C: W i Page 2 of M 3) Light in air is reflected from a lens that has an antireflective coating of thickness t with index of refraction n = 1.38. The glass under the coating has n’ = 1.50. What is the condition for destructive interference in the reflected light? 10 is the wavelength inlair and m is any positive integer. (Assume normal incidence.) (A) t = mnlo (B) t = m/to/n (C) t= (m ~— ‘/2)n/lo (D) t = (m — ‘/2)flo/n (E) 2t = mn/lo (F) 2t = mic/n (G) 21 = (m — Van/h) (H) 2! = (m —- ‘/z)/io/n 4) A transmission grating with slit spacing d = 2.1 pm and slit width 0 = 0.7 pm is illuminated with light of wavelength 0.6 pm. What is the total number of principal maxima that can be seen? (A) 2 (B) 3 ($0 4 (E) 6 (F) 7 (G) 8 5) In a photoelectric effect experiment, the stopping potential depends on What metal is being used as target the frequency of the incident light (C) the intensity of the incident light (D) both (A) and (C) but not (B) (E) both (A) and (B) but not (C) (F) both (E) and (C) but not (A) Page 3 of 14 6) In a photoelectric effect experiment, which of the following quantities is directly proportional to the work function of the target? List all that apply. (A) threshold frequency ,3: (B) threshold wavelength (C) stopping potential (D) intensity (E) current . E Mg 57%”ij it? 7) In a Compton scattering experiment with incident x-rays of wavelength 1.00 pm, what is the maximum possible wavelength of a scattered photon? (A) 1.00 pm (B) 2.00 pm (C) 2.43 pm (D) 3.43 pm (E) 4.43 pm (F) 4.86 pm (G) 5.86 pm (H) 6.86 pm )‘l 1 ’* r 8) A particle is confined to a 1D quantum well: U(x) = 0 inside the well (0 < x < L) and U(x) = U0 > 0 outside. Which statements are true for bound states both for finite U0 and for U0 = 00? List all that apply. ?(A)}the probability density inside the well is non-uniform (I?) the probability density outside the well is zero the energy levels are discrete (i. anot continuous) the number of bound states is infinite the minimum possible energy is nonzero / (F)Z the wavefunction inside the well is sinusoidal the uncertainty in px is zero Page 4 of 14 9) The three lines (violet, blue-green, red) in the hydrogen spectrum that you saw in Lab 6 result from transitions from m = 3, 4, or 5 to nf = m 2. What is the wavelength of the {Ed line? (E 1 represents the ground state energy for hydrogen). the 1 gig f (B) ‘(hC/El)*6 it, (C) «(he/E1)*(100/21) (D) ~(hc/E1)*(10/3) , (E) ~(E1/hc)*(36/5) a I (F) —(Ei/hc)*(6) (G) —(E1/hc)*(100/21) (H) —(E1/hc)*(10/3) 10) Imagine that a radar system with a dish type antenna of 1 m diameter aperture size uses 3 cm wavelength microwaves to scan for targets, Approximately how wide is the beam at a distance of 10 km (E) 4,00 ‘ H i" r «W N , a; m Page 5 of 14 Part 11: Using an Interferometer to Weigh Neutrons (24 points) You do need to Show work. . Consider the following design for an interferometer to measure the quantum phase shift of a neutron due to gravity. As shown below, slow moving (i.e. nonrelativistic) neutrons of mass m from a reactor having initial kinetic energy T0 * enter the interferometer at beam splitter A and either take the high road with vertical height y = 0 along the path AB or the low road with vertical height y = —H along path CD. The two beams recombine at D and proceed to a neutron detector which gives a signal proportional to beam intensity. Treat the beams as plane waves of definite wavelength and frequency in the following discussion (you shouldn’t have to work with wavefunctions to give proper answers). [*We are using the symbol T for kinetic energy to avoid confusion with the wavenumber.] Page 6 of 14 (a) Explain why despite the variation in potential energy and hence kinetic energy with height, the quantum phase shift along AC is the same as along BD. along the high road than along the low road. (Circle your choice (greater or less) in the sentence) A/ (c) Give a qualitative sketch of how the intensity in the detector should behave as a function of H assuming that the neutron beam from the reactor is very intense. (Here one somehow keeps the interferometer perfectly aligned while varyingH . The actual experiment done involved a fixed reactor beam and rotation of the interferometer/detector assembly about a horizontal direction along the incoming beam.) WWMLWMWWWWy.a. HM (d)%Regarding the difference-in wavenumber of part (b), Ak as slight, differentiate the relation between kinetic energy and wavenurnber to show that AT / TO Ak / k and find the constant of proportionality. Page 8 of 14 (e) Assuming that all that matters is the change in wavenumber use the result of part ((1) to estimate the slight phase difference (13¢) in the two paths as a function of L, H, m, g (the acceleration of gravity at the earth’s surface), T0 and any constants needed. The best answer is the simplest possible. a ,,,_ it {3/ .lérggg, ’ (f) Suppose youriioureeissewweaglethagneutronsafrive one at a time in the detector (heralded by individual clicks in a loudspeaker hooked up to the detector). Suppose that you get an experimental answer to (c) by scanning H back and forth in little steps covering the same range in H over and over again while spending equal time at each step. Qualitativer explain briefly how your answer to (c) will develop as time passes and you accumulate data. .. Awfw'ww’” ow //<> g < a 5 m a HEW“; Page 9 of 14 Part III: A Quantum Particle Encounters a Barrier (24 points) You do need to Show work. Consider the behavior of a beam of particles each of mass m traveling along a single spatial dimension (x) given potential energy equal to 0 for x < 0 and U 0 for x > 0. The particles are streaming in from a source at x = —-00. E is the energy of the incident particles and we will begin by assuming that E < U 0 . (a) Sketch the absolute value of the wavefunction as a function of x at a fixed time t for X > 0. I : WM” 1 may as W I N w t, »» Page 10 of 14 (b) Assuming that the incident beam is a plane wave of unit amplitude, find the complete wavefunction THC, t) for all space and time, (Note that it will be necessary to give different expressions for x < O and x > 0), There is much work to be done in this part of the problem. Make judicious choices for the form of the wavefimction, introducing and defining as needed new parameters. Your final answer needs to be expressed in terms of the starting quantities m , E , Page 11 ofl4 [Even if you did not get part ( b), you may be able to answer parts (0) and (aDJ (0) Explain the qualitative change that occurs in the wavefunction for x > 0 if E starts to exceed ((1) Without going into a long mathematical discussion, completely describe the wavefunction over all space in the limitE / U 0 -4> 00. Page 12 of 14 Part IV: The Case of the Lucky Radio Engineer (12 points) You do need to show work. You’vebeen tasked with the design and assembly of a linear antenna array to be used to beam radio broadcasts from a transmitting station to two distant cities. You go traditional and drive in phase a line of N vertical wire antennas spaced 6] = fl / 2 apart laid out over a total end to end distanceL . Your array is positioned exactly between the two cities along a perpendicular as shown below (hence the name “broadside” array). /c= flare/W”: (a) Estimate the angular width of the beam A 6 directed at either city in the limit that N is large. Use a well drawn and labeled phasor diagram to derive and explain your result. Page 13 of 14 (b) Expecting a bonus for your work you are outraged when it doesn’t come through. In anger that night, you go to the array, dig up all the antennas and replant them. While leaving them along the original line and spanning the original distance L you fiendishly use a random spacing betweenthe antennas that had been formerly exactly half a wavelength apart. The next morning you receive field engineering reports that the signal strength in both your target cites is still strong. Explain what’s going on. Are people at work goofing with you? Your answer should be short and concise. m ) l V 1% err—to M EC": ail wows WUCJ cc éaw’fifi diameter, Mn arrive Mimi lens m9» wflfl, ...
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214_final_exam_solutions_fall_2007 - Part 1: Multiple...

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