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Physics 214* Sample Final Exam Spring 2010 Last Name: First Name ID Discussion Section: Discussion TA Name: Exam Room & Building ______________________ Seat Number___________ Instructions— Turn off your cell phone and put it away. This is a closed book exam. You have two (2) hours to complete it. I. Fill in ALL the information requested on the lines above and sign the Formula Sheet. II. At the end of this exam, you must return this Exam Booklet complete with all pages, including the formula sheet, along with your answer sheet. Note that this is a different policy than the one applying to midterm exams. III. If you do not turn in a complete Exam Booklet, including the formula sheet, your Answer Sheet will not be graded and you will receive the grade AB (Absent) for this exam. Kindly paper clip the Answer Sheet to the Exam Booklet 1. Use a #2 pencil; do not use a mechanical pencil or a pen. Fill in completely (until there is no white space visible) the circle for each intended input – both on the identification side of your answer sheet and on the side on which you mark your answers. If you decide to change an answer, erase vigorously; the scanner sometimes registers incompletely erased marks as intended answers; this can adversely affect your grade. Light marks or marks extending outside the circle may be read improperly by the scanner. 2. Print your last name in the YOUR LAST NAME boxes on your answer sheet and print the first letter of your first name in the FIRST NAME INI box. Mark (as described above) the corresponding circle below each of these letters. 3. Print your NetID in the NETWORK ID boxes, and then mark the corresponding circle below each of the letters or numerals. Note that there are different circles for the letter “I” and the numeral “1” and for the letter “O” and the numeral “0”. Do not mark the hyphen circle at the bottom of any of these columns. 4. This Exam Booklet is Version A. Mark the A circle in the TEST FORM box at the bottom of the front side of your answer sheet. 5. Stop now and double-check that you have bubbled-in all the information requested in 2 through 4 above and that your marks meet the criteria in 1 above. Check that you do not have more than one circle marked in any of the columns. 6. Do not write in or mark any of the circles in the STUDENT NUMBER or SECTION boxes. 7. On the SECTION line, print your DISCUSSION SECTION. (You need not fill in the COURSE or INSTRUCTOR lines.) 8. Sign (DO NOT PRINT) your name on the STUDENT SIGNATURE line. CHECK NOW THAT YOU HAVE COMPLETED ALL THE ABOVE STEPS. YOUR GRADE DEPENDS ON IT! Before starting work, check to make sure that your test booklet is complete. You should have 15 pages (36 problems), excluding the Formula Sheets at the end. Grading policy is explained on page 2. Academic Integrity—Giving assistance to or receiving assistance from another student or using unauthorized materials during a University Examination can be grounds for disciplinary action, up to and including dismissal from the university. 1 of 15 pages 36 problems Physics 214* Sample Final Exam Spring 2010 Exam Grading Policy— The exam is worth a total of ___ points; it is composed of two types of questions. MC5: multiple-choice-five-answer questions, each worth 6 points. Partial credit will be granted as follows. (a) If you mark only one answer and it is the correct answer, you earn 6 points. (b) If you mark two answers, one of which is the correct answer, you earn 3 points. (c) If you mark three answers, one of which is the correct answer, you earn 2 points. (d) If you mark no answers, or more than three, you earn 0 points. MC3: multiple-choice-three-answer questions, each worth 3 points. No partial credit. (a) If you mark only one answer and it is the correct answer, you earn 3 points. (b) If you mark a wrong answer or no answers, you earn 0 points. 2 of 15 pages 36 problems Physics 214* Sample Final Exam Spring 2010 1. A beam of photons with wavelength 150 nm and beam of electrons having the same energy as the photons go through the same slit of width 355 nm. You observe the diffraction pattern on a distant screen. All angles are measured from the centerline. The photons produce their first dark band at an angle α. Is the magnitude of α bigger than, equal to, or smaller than the magnitude the angle, β, where the electrons produce their first dark band? a. | α | > | β | b. | α | = | β | c. | α | < | β | 2. It takes 3.0 eV of energy to excite an electron in a 1-dimensional infinite well from the ground state to the first excited state. What is the width, L, of the box? a. b. c. d. e. L = 0.25 nm L = 0.61 nm L = 2.10 nm L = 10.6 nm L = 109 nm The next two questions are related to the following situation: A particle of unknown mass is in a 1-dimensional box of width L = 3.0 x 10-10 m with infinitely high potential walls at x = 0 and at x = L, and zero potential for 0 < x < L. The particle is in the second excited state of the box. 3. What is the de Broglie wavelength, λ, of the particle? a. b. c. d. e. λ = 1.0 x 10-10 m λ = 2.0 x 10-10 m λ = 3.0 x 10-10 m λ = 6.0 x 10-10 m Not enough information is given. 4. For which values of x is the probability of finding the particle largest? a. b. c. d. e. Only at x = L/6 and x =5 L/6 (two values of x) Only at x = L/4 (one value of x) Only at x = L/6 and at x = L/2 and at x = 5L/6 (three values of x) The probability is everywhere the same. Not enough information is given. 3 of 15 pages 36 problems Physics 214* Sample Final Exam Spring 2010 The next two questions are related: 5. How many distinct (n,l,m) states of the hydrogen atom with n = 3 are there? Neglect electron spin. a. 1 distinct state b. 5 distinct states c. 9 distinct states 6. Which statement about the energies of the (n,l,m) states of the hydrogen atom with n = 3 is correct? a. They are all equal to -1.51 eV. b. The highest (most positive) energy is equal to -1.51 eV, the others are all smaller. c. At least one energy is equal to -2.44 eV. d. At least one energy is equal to +3.55 eV. e. States with larger l have higher energies. 4 of 15 pages 36 problems Physics 214* Sample Final Exam Spring 2010 The next two questions pertain to the following situation: A particle is in a bound energy state of the finite depth potential well shown. V Left half Left barrier Right half Right barrier x 0 L 7. If we measure the particle’s position in the well, which is the more likely result? a. The particle has a larger wavelength in the left half of the well. b. The particle has a larger wavelength in the right half of the well. c. The particle wavelength is the same on both sides of the well. 8. At which values of x is the total energy of the particle the largest? a. At x = 0 b. At x= L c. The energy is the same everywhere. 9. The longest wavelength of light that can be absorbed by a particular harmonic oscillator is λ = 1000 nm. What is the second longest wavelength that can be absorbed? a. λ = 188 nm b. λ = 300 nm c. λ = 500 nm d. Every λ ≤ 1000 nm can be absorbed. e. No shorter wavelengths can be absorbed. 10. Consider a well that has an adjustable shape, so that we can vary its two lowest energy levels, E1 (the ground state) and E2. We will put a particle into a superposition of the two energy states (i.e., Ψ = a ψ 1 + b ψ 2). Which of these manipulations will increase the frequency of oscillation of the particle’s spatial probability density? a. b. c. d. e. Increase E1, keeping E2 constant. Increase |E2 – E1|. Increase a, keeping b constant. Increase |b - a|. None of the above. 5 of 15 pages 36 problems Physics 214* Sample Final Exam Spring 2010 The next three questions pertain to the following situation: An electron is confined to an infinite 1-dimensional well of width L = 1.5 nm. At t = 0, it is in a superposition of the ground state and second excited state: Ψ(x, t=0) = aψ1 + bψ3, where ψ1 and ψ3 are individually normalized. 11. What is the frequency of oscillation, f, of the spatial probability density? a. b. c. d. e. f = 1.27 x 1014 Hz f = 2.54 x 1014 Hz f = 3.24 x 1014 Hz f = 3.55 x 1015 Hz f = 2.29 x 1015 Hz 12. Which of these pairs of values of a and b correctly normalizes the wave function and can result, at some instant of time, in a zero probability density at the middle of the well (i.e., at x = L/2)? a. b. c. d. e. a = b = 1/√2 a = b = 1/2 a = √(2/3), b = √(1/3) a = √(2/3), b = -√(1/3) No values of a and b will ever result in a zero probability density at the middle of the well. 13. Suppose, now, that a = 0.399 and b = 0.917. If we measure the energy, what is the probability, P3, of obtaining E3? a. P3 = 1.00 b. P3 = 0.399 c. P3 = 0.518 d. P3 = 0.841 e. P3 = 0.917 6 of 15 pages 36 problems Physics 214* Sample Final Exam Spring 2010 The next two questions pertain to the following situation: The work function (energy needed to remove an electron) of gold is 5.1 eV. Two pieces of gold (at the same potential) are separated by a distance, L. L φ =5.1eV 14. For what value of L will the transmission probability for an electron to cross from one to the other be T ≈ 10-3? Assume that G = 1 in the formula for the tunneling probability. a. b. c. d. e. L = 0.001 nm L = 0.02 nm L = 0.1 nm L = 0.3 nm L = 4 nm 15. Suppose we increase L by a factor of two from the value required for the transmission probability, T, to be ≈ 10-3. What is the new value of T? a. b. c. d. e. T ≈ 10-6 T ≈ 0.5 x 10-3 T ≈ 1 x 10-3 T ≈ 2 x 10-3 T ≈ 0.03 7 of 15 pages 36 problems Physics 214* Sample Final Exam Spring 2010 The next two questions are related: 16. An electron is confined in a 3-dimensional rectangular box (V = 0 inside, and V = ∞ outside) with sides L = 4 nm, 4 nm, and 5 nm. What minimum energy, E, must a photon have in order to excite the electron out of its ground state. (The electron absorbs the photon.) a. b. c. d. e. E = 0.015 eV E = 0.045 eV E = 0.071 eV E = 0.125 eV E = 0.241 eV 17. Now we put 10 electrons into the box. Assuming the total energy of the system is as low as possible, and including the effects of spin, how many electrons have the energy of the second excited state? a. b. c. d. e. None 1 2 4 All 10 18. Consider a wave function that is a spherical blob with diameter, D, as shown. The average value of its kinetic energy (if one makes a large number of measurements of identical systems) is <KE0>. Now, shrink the wave function in all three dimensions, so that it is a similar spherical blob with diameter half D. What will be the new average kinetic energy, <KE>? D a. <KE> = <KE0> b. <KE> = 2<KE0> c. <KE> = 4<KE0> 19. A hydrogen atom is initially at rest (approximately), in an excited state of unknown quantum number, n. The electron, drops to the next lower energy level, emitting a photon. Estimate the largest possible recoil velocity of the atom. a. 0 (the atom does not recoil when it emits a photon) b. 3.26 m/s c. 121.6 m/s 8 of 15 pages 36 problems Physics 214* Sample Final Exam Spring 2010 The next two questions pertain to the following situation: You are sitting 10 meters from a musician (left-most dot) playing an instrument with a steady note at some unknown frequency f. You are facing in a direction perpendicular to the direction of the musician, as shown in the figure below (not to scale): 15 cm 10 m right ear left ear 20. The speed of sound in air is 346 m/sec, while the speed in your head depends on whether the sound propagates directly across (i.e., through your brain, with diameter 15 cm) at vbrain = 1540 m/s, or propagates around the periphery through the skull bone (assumed to be spherical for this problem) at vskull = 4080 m/s. Assuming the individual intensities from these two paths are the same, for what frequency will there be destructive interference at your left eardrum? a. f = 12.6 kHz b. f = 8.3 kHz c. f = 2.3 kHz 21. A second musician (“X”) now joins in and they both play a steady middle-C note (f = 278 Hz) in phase. The second musician sits 2 meters away from the first musician in the perpendicular direction from the line between you and the first musician, as shown in the figure below. X 2m 10 m right ear The intensity that each musician produces at your right eardrum when they play alone is 2 W/m2. What is the net intensity, I, at your right eardrum? a. b. c. d. e. I = 0 W/m2 I = 0.8 W/m2 I = 1.5 W/m2 I = 6.2 W/m2 I = 7.0 W/m2 9 of 15 pages 36 problems Physics 214* Sample Final Exam Spring 2010 22. The Apollo 11 mission landed on the Moon in 1969. Moon hoax enthusiasts often cite a lack of teloscopic evidence for hardware they left behind. The Hubble Space Telescope (HST) is the largest orbiting telescope, with a 2.4meter aperture mirror. At closest approach, the HST-moon distance is 376,000 km. By approximately what factor would we have to increase the HST mirror aperture to resolve the lunar rover left by the Apollo 11 astronauts under the most optimistic circumstances? Use a wavelength of 700 nm for the light being detected by the HST and require that we need to resolve 15 cm features in order to identify it. a. b. c. d. e. Factor = 10 Factor = 100 Factor = 1000 Factor = 10000 Factor = 100000 23. Light of unknown wavelength falls on a tiny pinhole. The pattern created by the pinhole has its first diffraction minimum at an angle 2° from the center line. In a second experiment light falls on a pinhole with a diameter that is half of that in the first experiment. What is the angle, θmin, of the first diffraction minimum for the second (smaller) pinhole? a. b. c. d. e. θmin = 1° θmin = 2° θmin = 4° θmin = 8° θmin = 16° 10 of 15 pages 36 problems Physics 214* Sample Final Exam Spring 2010 The next two questions pertain to the following situation: In a two-slit interference experiment, a viewing screen is placed 5 meters directly behind two slits separated by 4 µm. Coherent, monochromatic light of wavelength λ = 700 nm emerges (in phase) from the slits. (Assume the slit width is very small compared to the wavelength λ.) x 4 µm 5m 24. At what value of x on the screen does the largest-order intensity maximum occur (i.e., the one that is the farthest from the center line)? a. b. c. d. e. x = 1.2 m x = 4.4 m x = 5.0 m x = 9.0 m x = 15.7 m 25. What happens to the separation between the existing intensity maxima on the screen as we increase the slit spacing? a. The separation decreases. b. The separation increases. c. There is no change. 11 of 15 pages 36 problems Physics 214* Sample Final Exam Spring 2010 The next two questions pertain to the following situation: Consider doubly ionized Lithium (Li++), which has 1 electron orbiting a charge +3 nucleus. 26. Compare ao,Li, the most likely distance for the electron in a Li++ atom to be from the nucleus, with ao,H, the most likely distance for the electron in a hydrogen atom to be from the nucleus, assuming the electron is in the ground state in both cases. Which of the following is true? a. ao,Li < ao,H b. ao,Li = ao,H c. ao,Li > ao,H 27. What is the maximum wavelength, λmax, of light that would completely ionize the Li++ (i.e., free the electron from the nucleus)? a. b. c. d. e. photon λmax = 0.665 nm λmax = 10.1 nm λmax = 21 nm λmax = 137 nm λmax = 487 nm n=3 n=2 n=1 28. Which of the following statements is true? a. An ideal metal conducts because each electron ‘belongs’ to every nucleus in the crystal lattice, while in an ideal insulator, each electron is localized to a single nucleus. b. Metals conduct much better than insulators because, per gram of material, metals have orders of magnitude more electrons. c. At sufficiently low temperatures, (intrinsic) semi-conductors are effectively insulators. 12 of 15 pages 36 problems Physics 214* Sample Final Exam Spring 2010 29. A Ca atom (atomic number = 20) is in the ground state. Light is shined on the atom, exciting the most energetic electron. Which of the following are possible quantum numbers (n,l,m) of this excited electron? We omit the electron spin quantum number. a. b. c. d. e. (4, 0, 0) (4, 1, -1), (4, 1, 0), and (4, 1, 1) (5, 2, +1) and (5, 2, -1) All of the above None of the above 30. You are given a collection of quantum dots (treat as 1-D infinite square wells) and a collection of diatomic molecules (treat as simple harmonic oscillators). You find that the same wavelength of light will excite both the quantum dots and the molecules from their ground states to their first excited states. What is the ratio of the photon energies required to further excite each system from the first excited state to the second excited state? a. b. c. d. e. Equantum dot/Emolecule = 1 Equantum dot/Emolecule = 6/5 Equantum dot/Emolecule = 5/3 Equantum dot/Emolecule = 15/2 The information given is not sufficient to answer this question. 31. Electron spin resonance (ESR) is a technique much like nuclear magnetic resonance (NMR), except that it relies on using radio-frequency photons to flip the spin of an electron in a magnetic field. If the magnetic field applied to the electrons is increased from 0.010 Tesla to 0.011 Tesla, how much, Δf, will the transition frequency change? a. Δf = 7 MHz b. Δf = 14 MHz c. Δf = 28 MHz 13 of 15 pages 36 problems Physics 214* Sample Final Exam Spring 2010 The next three questions pertain to the following situation: An electron is moving in the potential, U(x), shown to the right. This potential is an even function of x. The energies, E1 = -1.5 eV and E2 = +0.5 eV, of the two lowest energy states are indicated by the dashed lines. Five x values are labeled A-F in the figure. ψ1(x) is the wave function of the ground state, and ψ2(x) is the wave function of the first excited state. U(x) A B C D F E2 = +0.5 eV x E1 = -1.5 eV 32. Which one of the following statements is not true? a. ψ1(A) = ψ1(F) b. ψ2(A) = -ψ2(F) c. Both ψ1 and ψ2 are proportional to sin(kx) in the regions A < x < B and D < x < F (though ‘k’ might be different for ψ1 and ψ2). d. dψ1 =0 d x x=C d ψ2 e. dx 2 >0 x<A 33. Which wave function penetrates farther into the region x > F? That is, for which wave function is ψ(x) / ψ(F) larger when x > F? a. ψ1 b. ψ2 c. They are the same. 34. If the system is prepared in a superposition of ψ1 and ψ2, at what frequency does the probability density oscillate? a. f = 4.8 x 1014 Hz b. f = 3.6 x 1014 Hz c. f = 1.2 4 x 1014 Hz 14 of 15 pages 36 problems Physics 214* Sample Final Exam Spring 2010 The following two questions refer to this situation: The graph at the right shows the results of a photoelectric effect experiment in which the stopping voltage was measured for several frequencies of light. Vstop(Volts) The slope of the line is -15 14 4.0×10 Volts/Hz, and the x-intercept is f0 = 6.0×10 Hz. 35. What is the work function, Φ, of the material? a. b. c. d. e. Φ = -6.0 eV Φ = -4.0 eV Φ = +2.4 eV Φ = +4.0 eV Φ = +6.0 eV f(Hz) f0 36. What value of Planck’s constant, h, does this data yield? a. b. c. d. e. h = 6.0×10-34 J.s h = 6.2×10-34 J.s h = 6.4×10-34 J.s h = 6.6×10-34 J.s h = 6.8×10-34 J.s Did you bubble in your name, exam version and network-ID? Check to make sure you have bubbled in all your answers. 15 of 15 pages 36 problems