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Chapter 28 Problems 1, 2, 3 = straightforward, intermediate, challenging = full solution available in Student Solutions Manual/Study Guide = coached solution with hints available at www.pop4e.com = computer useful in solving problem 3. Figure P28.3 shows the spectrum of light emitted by a firefly. Determine the temperature of a black body that would emit radiation peaked at the same wavelength. Based on your result, would you say that firefly radiation is blackbody radiation? = paired numerical and symbolic problems = biomedical application Section 28.1 Blackbody Radiation and Planck’s Hypothesis 1. With young children and the elderly, use of a traditional fever thermometer has risks of bacterial contamination and tissue perforation. The radiation thermometer shown in Figure 28.5 works fast and avoids most risks. The instrument measures the power of infrared radiation from the ear canal. This cavity is accurately described as a black body and is close to the hypothalamus, the body’s temperature control center. Take normal body temperature as 37.0°C. If the body temperature of a feverish patient is 38.3°C, what is the percentage increase in radiated power from his ear canal? 2. The radius of our Sun is 6.96 × 108 m, and its total power output is 3.85 × 1026 W. (a) Assuming that the Sun’s surface emits as a black body, calculate its surface temperature. (b) Using the result of part (a), find λmax for the Sun. Figure P28.3 4. Calculate the energy, in electron volts, of a photon whose frequency is (a) 620 THz, (b) 3.10 GHz, and (c) 46.0 MHz. (d) Determine the corresponding wavelengths for these photons and state the classification of each on the electromagnetic spectrum. 5. An FM radio transmitter has a power output of 150 kW and operates at a frequency of 99.7 MHz. How many photons per second does the transmitter emit? 6. The average threshold of darkadapted (scotopic) vision is 4.00 × 10−11 W/m2 at a central wavelength of 500 nm. If light having this intensity and wavelength enters the eye and the pupil is open to its maximum diameter of 8.50 mm, how many photons per second enter the eye? 7. A simple pendulum has a length of 1.00 m and a mass of 1.00 kg. The amplitude of oscillations of the pendulum is 3.00 cm. Estimate the quantum number for the pendulum. 8. Review problem. This problem is about how strongly matter is coupled to radiation, the subject with which quantum mechanics began. For a very simple model, consider a solid iron sphere 2.00 cm in radius. Assume that its temperature is always uniform throughout its volume. (a) Find the mass of the sphere. (b) Assume that it is at 20°C and has emissivity 0.860. Find the power with which it is radiating electromagnetic waves. (c) If it were alone in the Universe, at what rate would its temperature be changing? (d) Assume that Wien’s law describes the sphere. Find the wavelength λmax of electromagnetic radiation it emits most strongly. Although it emits a spectrum of waves having all different wavelengths, model its whole power output as carried by photons of wavelength λmax. Find (e) the energy of one photon and (f) the number of photons it emits each second. The answer to part (f) gives an indication of how fast the object is emitting and also absorbing photons when it is in thermal equilibrium with its surroundings at 20°C. Section 28.2 The Photoelectric Effect 9. Molybdenum has a work function of 4.20 eV. (a) Find the cutoff wavelength and cutoff frequency for the photoelectric effect. (b) What is the stopping potential if the incident light has a wavelength of 180 nm? 10. Electrons are ejected from a metallic surface with speeds ranging up to 4.60 × 105 m/s when light with a wavelength of 625 nm is used. (a) What is the work function of the surface? (b) What is the cutoff frequency for this surface? 11. Two light sources are used in a photoelectric experiment to determine the work function for a particular metal surface. When green light from a mercury lamp (λ = 546.1 nm) is used, a stopping potential of 0.376 V reduces the photocurrent to zero. (a) Based on this measurement, what is the work function for this metal? (b) What stopping potential would be observed when using the yellow light from a helium discharge tube (λ = 587.5 nm)? 12. From the scattering of sunlight, J. J. Thomson calculated the classical radius of the electron as having a value of 2.82 × 10−15 m. Sunlight with an intensity of 500 W/m2 falls on a disk with this radius. Calculate the time interval required to accumulate 1.00 eV of energy. Assume that light is a classical wave and that the light striking the disk is completely absorbed. How does your result compare with the observation that photoelectrons are emitted promptly (within 10−9 s)? 13. Review problem. An isolated copper sphere of radius 5.00 cm, initially uncharged, is illuminated by ultraviolet light of wavelength 200 nm. What charge will the photoelectric effect induce on the sphere? The work function for copper is 4.70 eV. Section 28.3 The Compton Effect 14. Calculate the energy and momentum of a photon of wavelength 700 nm. 15. X-rays having an energy of 300 keV undergo Compton scattering from a target. The scattered rays are detected at 37.0° relative to the incident rays. Find (a) the Compton shift at this angle, (b) the energy of the scattered x-ray, and (c) the energy of the recoiling electron. 16. A 0.110-nm photon collides with a stationary electron. After the collision, the electron moves forward and the photon recoils backward. Find the momentum and the kinetic energy of the electron. 17. A 0.001 60-nm photon scatters from a free electron. For what (photon) scattering angle does the recoiling electron have kinetic energy equal to the energy of the scattered photon? 18. After a 0.800-nm x-ray photon scatters from a free electron, the electron recoils at 1.40 × 106 m/s. (a) What was the Compton shift in the photon’s wavelength? (b) Through what angle was the photon scattered? Section 28.4 Photons and Electromagnetic Waves 19. An electromagnetic wave is called ionizing radiation if its photon energy is larger than about 10.0 eV so that a single photon has enough energy to break apart an atom. With reference to Figure 24.12, identify what regions of the electromagnetic spectrum fit this definition of ionizing radiation and what do not. Section 28.5 The Wave Properties of Particles 20. Calculate the de Broglie wavelength for a proton moving with a speed of 1.00 × 106 m/s. 21. (a) An electron has kinetic energy 3.00 eV. Find its wavelength. (b) A photon has energy 3.00 eV. Find its wavelength. 22. In the Davisson–Germer experiment, 54.0-eV electrons were diffracted from a nickel lattice. If the first maximum in the diffraction pattern was observed at = 50.0° (Fig. P28.22), what was the lattice spacing a between the vertical rows of atoms in the figure? (It is not the same as the spacing between the horizontal rows of atoms.) currently accepted age of the Universe, which is 4 × 1017 s. (c) Should this student worry about being diffracted? Figure P28.22 23. The nucleus of an atom is on the order of 10−14 m in diameter. For an electron to be confined to a nucleus, its de Broglie wavelength would have to be on this order of magnitude or smaller. (a) What would be the kinetic energy of an electron confined to this region? (b) Make also an order-of-magnitude estimate of the electric potential energy of a system of an electron inside an atomic nucleus. Would you expect to find an electron in a nucleus? Explain. 24. After learning about de Broglie’s hypothesis that particles of momentum p have wave characteristics with wavelength λ = h/p, an 80.0-kg student has grown concerned about being diffracted when passing through a 75.0-cm-wide doorway. Assume that significant diffraction occurs when the width of the diffraction aperture is less than 10.0 times the wavelength of the wave being diffracted. (a) Determine the maximum speed at which the student can pass through the doorway so as to be significantly diffracted. (b) With that speed, how long will it take the student to pass through the doorway if it is in a wall 15.0 cm thick? Compare your result to the 25. The resolving power of a microscope depends on the wavelength used. If one wished to “see” an atom, a resolution of approximately 1.00 × 10−11 m would be required. (a) If electrons are used (in an electron microscope), what minimum kinetic energy is required for the electrons? (b) If photons are used, what minimum photon energy is needed to obtain the required resolution? Section 28.6 The Quantum Particle 26. Consider a freely moving quantum particle with mass m and speed u. Its energy is E = K = ½mu2. Determine the phase speed of the quantum wave representing the particle and show that it is different from the speed at which the particle transports mass and energy. 27. For a free relativistic quantum particle moving with speed v, the total energy is E = hf = ħω = p 2 c 2 m 2 c 4 and the momentum is p = h/λ = ħk = γmv. For the quantum wave representing the particle, the group speed is vg = dω/dk. Prove that the group speed of the wave is the same as the speed of the particle. Section 28.7 The Double-Slit Experiment Revisited 28. A modified oscilloscope is used to perform an electron interference experiment. Electrons are incident on a pair of narrow slits 0.060 0 μm apart. The bright bands in the interference pattern are separated by 0.400 mm on a screen 20.0 cm from the slits. Determine the potential difference through which the electrons were accelerated to give this pattern. 29. Neutrons traveling at 0.400 m/s are directed through a pair of slits having a 1.00-mm separation. An array of detectors is placed 10.0 m from the slits. (a) What is the de Broglie wavelength of the neutrons? (b) How far off axis is the first zerointensity point on the detector array? (c) When a neutron reaches a detector, can we say which slit the neutron passed through? Explain. 30. In a certain vacuum tube, electrons evaporate from a hot cathode at a slow, steady rate and accelerate from rest through a potential difference of 45.0 V. Then they travel 28.0 cm as they pass through an array of slits and fall on a screen to produce an interference pattern. If the beam current is below a certain value, only one electron at a time will be in flight in the tube. What is this value? In this situation, the interference pattern still appears, showing that each individual electron can interfere with itself. Section 28.8 The Uncertainty Principle 31. An electron (me = 9.11 × 10−31 kg) and a bullet (m = 0.020 0 kg) each have a velocity with a magnitude of 500 m/s, accurate to within 0.010 0%. Within what limits could we determine the position of the objects along the direction of the velocity? 32. Suppose Fuzzy, a quantummechanical duck, lives in a world in which h = 2π J · s. Fuzzy has a mass of 2.00 kg and is initially known to be within a pond 1.00 m wide. (a) What is the minimum uncertainty in the component of the duck’s velocity parallel to the width of the pond? (b) Assuming that this uncertainty in speed prevails for 5.00 s, determine the uncertainty in the duck’s position after this time interval. 33. An air rifle is used to shoot 1.00-g particles at 100 m/s through a hole of diameter 2.00 mm. How far from the rifle must an observer be to see the beam spread by 1.00 cm because of the uncertainty principle? Compare this answer with the diameter of the visible Universe (2 × 1026 m). 34. A π0 meson is an unstable particle produced in high-energy particle collisions. Its rest energy is about 135 MeV, and it exists for an average lifetime of only 8.70 × 10−17 s before decaying into two gamma rays. Using the uncertainty principle, estimate the fractional uncertainty Δm/m in its mass determination. 35. A woman on a ladder drops small pellets toward a point target on the floor. (a) Show that, according to the uncertainty principle, the average miss distance must be at least 2 x f m 1/ 2 2H g 1/ 4 where H is the initial height of each pellet above the floor and m is the mass of each pellet. Assume that the spread in impact points is given by Δxf = Δxi + (Δvx)t. (b) If H = 2.00 m and m = 0.500 g, what is Δxf? Section 28.9 An Interpretation of Quantum Mechanics 36. The wave function for a particle is ( x) a (x a 2 ) 2 for a > 0 and −∞ < x < +∞. Determine the probability that the particle is located somewhere between x = −a and x = +a. 37. A free electron has a wave function ( x) Ae i ( 5.0010 10 x) where x is in meters. Find (a) its de Broglie wavelength, (b) its momentum, and (c) its kinetic energy in electron volts. Section 28.10 A Particle in a Box 38. An electron that has an energy of approximately 6 eV moves between rigid walls 1.00 nm apart. Find (a) the quantum number n for the energy state that the electron occupies and (b) the precise energy of the electron. 39. An electron is contained in a one-dimensional box of length 0.100 nm. (a) Draw an energy level diagram for the electron for levels up to n = 4. (b) Find the wavelengths of all photons that can be emitted by the electron in making downward transitions that could eventually carry it from the n = 4 state to the n = 1 state. 40. The nuclear potential energy that binds protons and neutrons in a nucleus is often approximated by a square well. Imagine a proton confined in an infinitely high square well of length 10.0 fm, a typical nuclear diameter. Calculate the wavelength and energy associated with the photon emitted when the proton moves from the n = 2 state to the ground state. In what region of the electromagnetic spectrum does this wavelength belong? 41. A photon with wavelength λ is absorbed by an electron confined to a box. As a result, the electron moves from state n = 1 to n = 4. (a) Find the length of the box. (b) What is the wavelength of the photon emitted in the transition of that electron from the state n = 4 to the state n = 2? Section 28.11 The Quantum Particle Under Boundary Conditions Section 28.12 The Schrödinger Equation 42. The wave function of a particle is given by ψ(x) = A cos(kx) + B sin(kx) where A, B, and k are constants. Show that ψ is a solution of the Schrödinger equation (Eq. 28.31), assuming the particle is free (U = 0), and find the corresponding energy E of the particle. 43. Show that the wave function ψ = i(kx − ωt) Ae is a solution to the Schrödinger equation (Eq. 28.31) where k = 2π/λ and U = 0. 44. Prove that the first term in the Schrödinger equation, −(ħ2/2m)(d2ψ/dx2), reduces to the kinetic energy of the particle multiplied by the wave function (a) for a freely moving particle, with the wave function given by Equation 28.21, and (b) for a particle in a box, with the wave function given by Equation 28.36. 45. A particle in an infinitely deep square well has a wave function given by 2 ( x) 2 2x sin L L for 0 ≤ x ≤ L and zero otherwise. (a) Determine the expectation value of x. (b) Determine the probability of finding the particle near L/2 by calculating the probability that the particle lies in the range 0.490L ≤ x ≤ 0.510L. (c) Determine the probability of finding the particle near L/4 by calculating the probability that the particle lies in the range 0.240L ≤ x ≤ 0.260L. (d) Argue that the result of part (a) does not contradict the results of parts (b) and (c). 46. The wave function for a particle confined to moving in a one-dimensional box is nx L ( x) A sin Use the normalization condition on ψ to show that A 2 L (Suggestion: Because the box length is L, the wave function is zero for x < 0 and for x > L, so the normalization condition, Equation 2 28.23, reduces to 0L dx 1 .) 47. The wave function of an electron is ( x) 2 2x sin L L Calculate the probability of finding the electron between x = 0 and x = L/4. 48. A particle of mass m moves in a potential well of length 2L. The potential energy is infinite for x < −L and for x > +L. Inside the region −L < x < L, the potential energy is given by 2 x2 U ( x) mL2 ( L2 x 2 ) In addition, the particle is in a stationary state that is described by the wave function ψ(x) = A(1 − x2/L2) for −L < x < +L and by ψ(x) = 0 elsewhere. (a) Determine the energy of the particle in terms of ħ, m, and L. (Suggestion: Use the Schrödinger equation, Eq. 28.31.) (b) Show that A = (15/16L)1/2. (c) Determine the probability that the particle is located between x = −L/3 and x = +L/3. Section 28.13 Tunneling Through a Potential Energy Barrier 49. An electron with kinetic energy E = 5.00 eV is incident on a barrier with thickness L = 0.200 nm and height U = 10.0 eV (Fig. P28.49). What is the probability that the electron (a) will tunnel through the barrier and (b) will be reflected? 51. An electron has a kinetic energy of 12.0 eV. The electron is incident upon a rectangular barrier of height 20.0 eV and thickness 1.00 nm. By what factor would the electron’s probability of tunneling through the barrier increase if the electron absorbs all the energy of a photon of green light (with wavelength 546 nm) just as it reaches the barrier? Section 28.14 Context Connection—The Cosmic Temperature Problems 24.14 and 24.59 in Chapter 24 can be assigned with this section. 52. Review problem. A star moving away from the Earth at 0.280c emits radiation that we measure to be most intense at the wavelength 500 nm. Determine the surface temperature of this star. Figure P28.49 Problems 28.49 and 28.50. 50. An electron having total energy E = 4.50 eV approaches a rectangular energy barrier with U = 5.00 eV and L = 950 pm as shown in Figure P28.49. Classically, the electron cannot pass through the barrier because E < U. Quantum-mechanically, however, the probability of tunneling is not zero. Calculate this probability, which is the transmission coefficient. 53. The cosmic background radiation is blackbody radiation from a source at a temperature of 2.73 K. (a) Determine the wavelength at which this radiation has its maximum intensity. (b) In what part of the electromagnetic spectrum is the peak of the distribution? 54. Find the intensity of the cosmic background radiation, emitted by the fireball of the Big Bang at a temperature of 2.73 K. Additional Problems 55. Review problem. Design an incandescent lamp filament. Specify the length and radius a tungsten wire can have to radiate electromagnetic waves with power 75.0 W when its ends are connected across a 120-V power supply. Assume that its constant operating temperature is 2 900 K and that its emissivity is 0.450. Assume that it takes in energy only by electrical transmission and loses energy only by electromagnetic radiation. From Table 21.1, you may take the resistivity of tungsten at 2 900 K as 5.6 × 10−8 Ω · m × [1 + (4.5 × 10−3/°C)(2 607 °C)] = 7.13 × 10−7 Ω · m. 56. Figure P28.56 shows the stopping potential versus the incident photon frequency for the photoelectric effect for sodium. Use the graph to find (a) the work function, (b) the ratio h/e, and (c) the cutoff wavelength. The data are taken from R. A. Millikan, Physical Review 7:362 (1916). experiment. (a) Using these data, make a graph similar to Active Figure 28.9 that plots as a straight line. From the graph, determine (b) an experimental value for Planck’s constant (in joule-seconds) and (c) the work function (in electron volts) for the surface. (Two significant figures for each answer are sufficient.) Wavelength Maximum Kinetic Energy (nm) of Photoelectrons (eV) 588 0.67 505 0.98 445 1.35 399 1.63 58. Review problem. Photons of wavelength λ are incident on a metal. The most energetic electrons ejected from the metal are bent into a circular arc of radius R by a magnetic field having a magnitude B. What is the work function of the metal? 59. Johnny Jumper’s favorite trick is to step out of his 16th-story window and fall 50.0 m into a pool. A news reporter takes a picture of 75.0-kg Johnny just before he makes a splash, using an exposure time of 5.00 ms. Find (a) Johnny’s de Broglie wavelength at this moment, (b) the uncertainty of his kinetic energy measurement during such a period of time, and (c) the percent error caused by such an uncertainty. Figure P28.56 57. The following table shows data obtained in a photoelectric 60. A particle of mass 2.00 × 10−28 kg is confined to a one-dimensional box of length 1.00 × 10−10 m. For n = 1, what are (a) the particle’s wavelength, (b) its momentum, and (c) its ground-state energy? 61. An electron is represented by the time-independent wave function Ae x Ae x ( x) for x 0 for x 0 (a) Sketch the wave function as a function of x. (b) Sketch the probability density representing the likelihood that the electron is found between x and x + dx. (c) Argue that ψ(x) can be a physically reasonable wave function. (d) Normalize the wave function. (e) Determine the probability of finding the electron somewhere in the range x1 particles. Proceed as follows. Show that the ik x ik x wave function 1 Ae 1 Be 1 satisfies the Schrödinger equation in region 1, for x < ik x 0. Here Ae 1 represents the incident beam ik x and Be 1 represents the reflected ik x particles. Show that 2 Ce 2 satisfies the Schrödinger equation in region 2, for x > 0. Impose the boundary conditions ψ1 = ψ2 and dψ1/dx = dψ2/dx at x = 0 to find the relationship between B and A. Then evaluate R = B2/A2. (b) A particle that has kinetic energy E = 7.00 eV is incident from a region where the potential energy is zero onto one in which U = 5.00 eV. Find its probability of being reflected and its probability of being transmitted. 1 1 to x 2 2 2 62. Particles incident from the left are confronted with a step in potential energy shown in Figure P28.62. Located at x = 0, the step has a height U. The particles have energy E > U. Classically, we would expect all the particles to continue on, although with reduced speed. According to quantum mechanics, a fraction of the particles are reflected at the barrier. (a) Prove that the reflection coefficient R for this case is (k k2 )2 R 1 ( k1 k 2 ) 2 where k1 = 2π/λ1 and k2 = 2π/λ2 are the wave numbers for the incident and transmitted Figure P28.62 63. For a particle described by a wave function ψ(x), the expectation value of a physical quantity f(x) associated with the particle is defined by f ( x) * f ( x) dx For a particle in a one-dimensional box extending from x = 0 to x = L, show that 65. x2 2 A particle has a wave function 2 L L 2 2 3 2n 64. A particle of mass m is placed in a one-dimensional box of length L. Assume that the box is so small that the particle’s motion is relativistic, so K = p2/2m is not valid. (a) Derive an expression for the kinetic energy levels of the particle. (b) Assume that the particle is an electron in a box of length L = 1.00 × 10−12 m. Find its lowest possible kinetic energy. By what percent is the nonrelativistic equation in error? (Suggestion: See Eq. 9.18.) © Copyright 2004 Thomson. All rights reserved. 2 x / a e a ( x) 0 for x 0 for x 0 (a) Find and sketch the probability density. (b) Find the probability that the particle will be at any point where x < 0. (c) Show that ψ is normalized, and then find the probability that the particle will be found between x = 0 and x = a.