1. X rays scattered from a crystal have a first-order diffraction
peak at θ = 12.5°. At what angle will the second- and third-order peaks
appear? Get solution
1q. In 1900, did it seem clear that x rays were electromagnetic radiation? Give reasons for your answer. Was it important to perform further experiments to verify the characteristics of x rays? Get solution
2. X rays of wavelength 0.207 nm are scattered from NaCl. What is the angular separation between first and second-order diffraction peaks? Assume scattering planes that are parallel to the surface. Get solution
2q. In the early 1900s it was found that x rays were more difficult to refract or diffract than visible light. Why did this lead researchers to suppose that the wavelengths of x rays were shorter rather than longer than those of light? Get solution
3. Potassium chloride is a crystal with lattice spacing of 0.314 nm. The first peak for Bragg diffraction is observed to occur at 12.8°. What energy x rays were diffracted? What other order peaks can be observed (θ ≤ 90°)? Get solution
3q. What determines whether a given photon is an x ray? Could an x ray have a wavelength longer than some ultraviolet light? Get solution
4. A cubic crystal with interatomic spacing of 0.24 nm is used to select γ rays of energy 100 keV from a radioactive source containing a continuum of energies. If the incident beam is normal to the crystal, at what angle do the 100-keV γ rays appear? Get solution
4q. For a single crystal, transmission x-ray scattering will produce dots. However, if there are randomly oriented crystals, as in powder, concentric rings appear. Explain the difference qualitatively. Get solution
5. Calculate the de Broglie wavelength of a 1.2-kg rock thrown with a speed of 6.0 m/s into a pond. Is this wavelength similar to that of the water waves produced? Explain. Get solution
5q. How many particles do you think might be shown experimentally to exhibit wavelike properties? List at least four and discuss possible experiments to show this behavior. Get solution
6. Calculate the de Broglie wavelength of a typical nitrogen molecule in the atmosphere on a hot summer day (37°C). Compare this with the diameter (less than 1 nm) of the molecule. Get solution
6q. Why are neutrons more widely used than protons for studying crystal structure? What about using a hydrogen atom? Get solution
7. Transmission electron microscopes that use high energy electrons accelerated over a range from 40 to 100 kV are employed in many applications including the study of biological samples (like a virus) and nanoscience research and development (alloy particles and carbon nanotubes, for example). What would be the spatial limitation for this range of electrons? It is often true that resolution is limited by the optics of the lens system, not by the intrinsic limitation due to the de Broglie wavelength. Get solution
7q. Why are “cold” neutrons useful for studying crystal structure? How could one obtain “cold” neutrons? Get solution
8. A 3.0-MV transmission electron microscope has been in operation at Osaka University in Japan for several years. The higher-energy electrons allow for deeper sample penetration and extremely high resolution. What is the resolution limit for these electrons? Get solution
8q. It has been said that many experimental discoveries are made as a result of accidents (for example, that of Davisson and Germer). This statement may have some truth, but what traits and abilities must good scientists possess to take advantage of their accidents? Get solution
9. Work out Example 5.2b strictly using SI units of m, J, kg, and so on, and compare with the method of the example using eV units. ... ... Get solution
9q. Images taken with transmission electron microscopes are produced by passing very high energy electrons (40–100 keV) through matter. Why are the images always in black and white (or any two colors)? Get solution
10. Assume that the total energy E of an electron greatly exceeds its rest energy E0. If a photon has a wavelength equal to the de Broglie wavelength of the electron, what is the photon’s energy? Repeat the problem assuming E = 2E0 for the electron. Get solution
10q. Are the following phenomena wave or particle behaviors? Give your reasoning. (a) Television picture, (b) rainbows, (c) football sailing between goal posts, (d) telescope observing the moon, (e) police radar. Get solution
11. Determine the de Broglie wavelength of a particle of mass m and kinetic energy K. Do this for both (a) a relativistic and (b) a nonrelativistic particle. Get solution
11q. The experiment by Jönsson that showed the wavelike properties of electrons passing through a double slit is considered a pedagogically interesting experiment but not a landmark experiment. Why do you suppose this is true? Get solution
12. The Stanford Linear Accelerator accelerated electrons to an energy of 50 GeV. What is the de Broglie wavelength of these electrons? What fraction of a proton’s diameter ...can such a particle probe? Get solution
12q. Can you think of an experiment other than those mentioned in this chapter that might show the wavelike properties of particles? Discuss it. Get solution
13. Find the kinetic energy of (a) photons, (b) electrons, (c) neutrons, and (d) α particles that have a de Broglie wavelength of 0.13 nm. Get solution
13q. Why doesn’t the uncertainty principle restriction apply between the variables pz and x? Get solution
14q. How does the uncertainty principle apply to a known stable atomic system that apparently has an infinite lifetime? How well can we know the energy of such a system? Get solution
15. An electron initially at rest is accelerated across a potential difference of 3.00 kV. What are its wavelength, momentum, kinetic energy, and total energy? An electron initially at rest is accelerated across a potential difference of 3.00 kV. What are its wavelength, momentum, kinetic energy, and total energy? Get solution
15q. According to the uncertainty principle, can a particle having a kinetic energy of exactly zero be confi ned somewhere in a box of length L? Explain. Get solution
16. What is the wavelength of an electron with kinetic energy (a) 40 eV, (b) 400 eV, (c) 4.0 keV, (d) 40 keV, (e) 0.40 MeV, and (f) 4.0 MeV? Which of these energies are most suited for study of the NaCl crystal structure? Get solution
16q. What is similar about the conjugate variable pairs (px, x), (E, t), (L, θ), and (I, ω)? Get solution
17. Calculate the de Broglie wavelength of (a) an oxygen (O2) molecule darting around the room at 480 m/s and (b) an Escherichia coli bacterium of mass 6.5 × 10-13 kg, which has been measured to move at a speed of 1.0 × 10-5 m/s. Get solution
17q. What are the dimensions of the wave function ψ (x, t) that describes matter waves? Give your reasoning. Get solution
18. (a) What is the de Broglie wavelength of the 1.0-TeV protons accelerated in the Fermilab Tevatron accelerator? These high-energy protons are needed to probe elementary particles (see Chapter 14). (b) Repeat for the 7.0-TeV protons produced at CERN. Get solution
18q. Soon after their discovery, Davisson and Germer were using their experimental technique to describe new crystal structures of nickel. Do you think they were justified? Explain how you think their results allowed them to make such statements. Get solution
19. In an electron-scattering experiment, an intense reflected beam is found at Ø = 32° for a crystal with an interatomic distance of 0.23 nm. What is the lattice spacing of the planes responsible for the scattering? Assuming first-order diffraction, what are the wavelength, momentum, kinetic energy, and total energy of the incident electrons? Get solution
19q. Albert Einstein was a dissenter to the Copenhagen Interpretation and what it represented until the day he died. In a letter to Max Born in December 1926, Einstein wrote, “The theory yields a lot, but it hardly brings us any closer to the secret of the Old One. In any case I am convinced that God does not throw dice.” What do you think Einstein meant by this statement? Who or what is the Old One? Get solution
20. Davisson and Germer performed their experiment with a nickel target for several electron bombarding energies. At what angles would they find diffraction maxima for 48-eV and 64-eV electrons? Get solution
20q. It has been said that the energy-time version of the uncertainty principle allows a violation of the conservation of energy. The argument is that the uncertainty ...E allows the possibility that we may not know that energy conservation is violated during a small time ...t. Discuss arguments pro and con concerning this possibility. Get solution
21. A beam of 2.0-keV electrons incident on a crystal is refracted and observed (by transmission) on a screen 35 cm away. The radii of three concentric rings on the screen, all corresponding to first-order diffraction, are 2.1 cm, 2.3 cm, and 3.2 cm. What is the lattice-plane spacing corresponding to each of the three rings? Get solution
21q. The Fifth Solvay Congress, held in Brussels in October 1927, was dedicated to the quantum theory. A photo taken of the famous participants is often reproduced. Identify at least 10 participants and discuss what their contributions were to quantum physics, either experimentally or theoretically. Get solution
22. A beam of thermal neutrons (kinetic energy = 0.025 eV) scatters from a crystal with interatomic spacing 0.45 nm. What is the angle of the first-order Bragg peak? Get solution
22q. Summarize the discussions that Einstein and Bohr had at the 1927 Solvay Congress. List at least three objections that Einstein had to the Copenhagen interpretation of quantum mechanics and give Bohr’s explanation. Get solution
23. Generating plants in some power systems drop 10% of their load when the AC frequency changes by 0.30 Hz from the standard of 60 Hz. How often must the reading be monitored in order for the automatic operating system to be able to take corrective action? Let the time between measurements be at least half that determined by the bandwidth relation. Get solution
24. Consider electrons of kinetic energy 6.0 eV and 600 keV. For each electron, find the de Broglie wavelength, particle speed, phase velocity (speed), and group velocity (speed). Get solution
25. A wave, propagating along the x direction according to Equation (5.18), has a maximum displacement of 4.0 cm at x = 0 and t = 0. The wave speed is 5.0 cm/s, and the wavelength is 7.0 cm. (a) What is the frequency? (b) What is the wave’s amplitude at x = 10 cm and t = 13 s? Get solution
26. A wave of wavelength 4.0 cm has a wave speed of 4.2 cm/s. What is its (a) frequency, (b) period, (c) wave number, and (d) angular frequency? Get solution
27. Two waves are traveling simultaneously down a long Slinky. They can be represented by ψ1 (x, t) = 0.0030 sin(6.0x - 300t) and ψ2 (x, t) = 0.0030 sin(7.0x - 250t). Distances are measured in meters and time in seconds. (a) Write the expression for the resulting wave. (b) What are the phase and group velocities? (c) What is Δx between two adjacent zeros of ψ? (d) What is Δk Δx? Get solution
28. A wave packet describes a particle having momentum p. Starting with the relativistic relationship ...,show that the group velocity is c and the phase velocity is c/ β (where β = v/c). How can the phase velocity physically be greater than c ? Get solution
29. For waves in shallow water the phase velocity is about equal to the group velocity. What is the dependence of the phase velocity on the wavelength? Get solution
30. Find the group and phase velocities of 10-MeV protons and 10-MeV electrons (see Problem 28). Problem 28 A wave packet describes a particle having momentum p. Starting with the relativistic relationship ...,show that the group velocity is c and the phase velocity is c/ β (where β = v/c). How can the phase velocity physically be greater than c ? Get solution
31. Use Equation (5.25) with Ã(k) = A0 for the range of k = k0 - Δk/2 to k0 + Δk/2, and Ã(k) = 0 elsewhere, to determine ψ(x, 0), that is, at t = 0. Sketch the envelope term, the oscillating term, and |ψ(x, 0)|2. Approximately what is the width Δx over the full-widthhalf- maximum part of |ψ(x, 0)|2? What is the value of Δk Δx? ... Get solution
32. Use Equation (5.31) to show that ugr correctly represents the velocity of the particle both relativistically and classically. ... Get solution
33. Light of intensity I0 passes through two sets of apparatus. One contains one slit and the other two slits. The slits have the same width. What is the ratio of the out going intensity amplitude for the central peak for the two-slit case compared to the single slit? Get solution
34. Design a double-slit electron-scattering experiment using 1.0-keV electrons that will provide the first maximum at an angle of 1°. What will be the slit separation d? Get solution
35. You want to design an experiment similar to the one done by Jönsson that does not require magnification of the interference pattern in order to be seen. Let the two slits be separated by 2000 nm. Assume that you can discriminate visually between maxima that are as little as 0.3 mm apart. You have at your disposal a lab that allows the screen to be placed 80 cm away from the slits. What energy electrons will you require? Do you think such low-energy electrons will represent a problem? Explain. Get solution
36. A proton is confined in a uranium nucleus of radius 7.2 × 10-15 m. Determine the proton’s minimum kinetic energy according to the uncertainty principle if the proton is confined to a one-dimensional box that has length equal to the nuclear diameter. Get solution
37. A neutron is confined in a deuterium nucleus (deuteron) of diameter 3.1 × 10-15 m. Use the energy level calculation of a one-dimensional box to calculate the neutron’s minimum kinetic energy. What is the neutron’s minimum kinetic energy according to the uncertainty principle? Get solution
38. Show that the uncertainty principle can be expressed in the form ..., where θ is the angle and L the angular momentum. For what uncertainty in L will the angular position of a particle be completely undetermined? Get solution
39. Show that the uncertainty principle can be expressed in the form ..., where θ is the angle and L the angular momentum. For what uncertainty in L will the angular position of a particle be completely undetermined? Get solution
40. Some physics theories indicate that the lifetime of the proton is about 1036 years. What would such a prediction say about the energy of the proton? Get solution
41. What is the bandwidth Δ ω of an amplifier for radar if it amplifies a pulse of width 2.4 μs? Get solution
42. Find the minimum uncertainty in the speed of a bacterium having mass 3.0 × 10-15 kg if we know the position of the bacterium to within 1.0 μm, that is, to about its own size. Get solution
43. An atom in an excited state of 4.7 eV emits a photon and ends up in the ground state. The lifetime of the excited state is 1.0 × 10-13 s. (a) What is the energy uncertainty of the emitted photon? (b) What is the spectral line width (in wavelength) of the photon? Get solution
44. An electron microscope is designed to resolve objects as small as 0.14 nm. What energy electrons must be used in this instrument? Get solution
45. Rayleigh’s criterion is used to determine when two objects are barely resolved by a lens of diameter d. The angular separation must be greater than θR where ... In order to resolve two objects 4000 nm apart at a distance of 20 cm with a lens of diameter 5 cm, what energy (a) photons or (b) electrons should be used? Is this consistent with the uncertainty principle? Get solution
46. Calculate the de Broglie wavelength of a 5.5-MeV α particle emitted from an 241Am nucleus. Could this particle exist inside the 241Am nucleus (diameter ... 1.6 × 10-14 m)? Explain. Get solution
48. The wave function of a particle in a one-dimensional box of width L is ψ(x) = A sin(πx/L). If we know the particle must be somewhere in the box, what must be the value of A? Get solution
49. Write a cogent description of the Schrödinger cat paradox. Discuss variations of the paradox and the current status of its experimental verification. Get solution
50. Write a cogent description of the Einstein-Podolsky- Rosen paradox. Discuss variations of the paradox and the current status of its experimental verification. Get solution
51. Write a cogent description of the Bell inequality. Discuss variations and the current status of its experimental verification. Get solution
52. Write down the normalized wave functions for the first three energy levels of a particle of mass m in a one-dimensional box of width L. Assume there are equal probabilities of being in each state. Get solution
53. A particle in a one-dimensional box of length L has a kinetic energy much greater than its rest energy. What is the ratio of the following energy levels En: E2/E1, E3/ E1, E4/E1? How do your answers compare with the nonrelativistic case? Get solution
54g. Consider a wave packet having the product Δp Δx = h at a time t = 0. What will be the width of such a wave packet after the time m(Δx)2/ h? Get solution
55g. Analyze the Gaussian wave packet carefully and show that Δk Δx = 1/2. You must justify the assumptions you make concerning uncertainties in k and x. Take the Gaussian form given in Equation (5.26). (Hint: the linear “spread” of the wave packet Δx is given by one standard deviation, at which point the probability amplitude (|ψ|2) has fallen to one half its peak value.) ... Get solution
56g. An electron emitted in the beta decay of bismuth-210 has a mean kinetic energy of 390 keV. (a) Find the de Broglie wavelength of the electron. (b) Would such an electron be useful in a Davisson-Germer type scattering experiment? Address this question by deter- mining the angle at which a first-order diffraction maximum would be found using the same nickel target as Davisson and Germer. Get solution
57g. Electrons produced at the Thomas Jefferson National Accelerator Facility have a maximum energy of 6.0 GeV. (a) What is the de Broglie wavelength of each electron? (b) In what part of the electromagnetic spectrum do you fi nd a photon of comparable wavelength? Get solution
58g. The artificially created nuclear isotope tritium (3H) is important in many applications. This isotope undergoes beta decay, emitting an electron with a mean kinetic energy of 5.7 keV. (a) What is the de Broglie wavelength of such an emitted electron? (b) Is it likely that the electron existed inside the 3.4-fm-diameter nucleus just before it was emitted? Explain. Get solution
59g. As you saw in Chapter 4, the size of the hydrogen atom grows in proportion to n2, where n is the quantum state. For an atom in the n =10 state, model the electron as confined to a one-dimensional box of length equal to the atom’s diameter. Find the minimum energy of the electron is this box. Get solution
60g. An oboe player tunes the orchestra with the “Concert A” note, which has a frequency of 440 Hz. If she plays the note for 2.5 s, what minimum range of frequencies is heard during this time? Get solution
61g. As we learned in Section 3.9, an electron and positron can annihilate each other completely and form two gamma rays. (a) If the electron and positron were initially at rest, what are the wavelengths of the two emitted gamma rays? (b) Repeat if the electron and positron were each traveling at a speed of 0.30c measured in the lab and collided head-on. Get solution
62g. Most of the particles known to physicists are unstable. For example, the lifetime of the neutral pion, π0, is about 8.4 × 10-17 s. Its mass is 135.0 MeV/c2. a) What is the energy width of the π0 in its ground state? b) What is the relative uncertainty Δm/m of the pion’s mass? Get solution
63g. The range of the nuclear strong force is believed to be about 1.2 ×10-15 m. An early theory of nuclear physics proposed that the particle that “mediates” the strong force (similar to the photon mediating the electromagnetic force) is the pion. Assume that the pion moves at the speed of light in the nucleus, and calculate the time Δt it takes to travel between nucleons. Assume that the distance between nucleons is also about 1.2 ×10-15 m. Use this time Δt to calculate the energy ΔE for which energy conservation is violated during the time Δt. This ΔE has been used to estimate the mass of the pion. What value do you determine for the mass? Compare this value with the measured value of 135 MeV/c2 for the neutral pion. Get solution
64g. The planes of atoms in a particular cubic crystal lie parallel to the surface, 0.80 nm apart. X rays having wavelength 0.50 nm are directed at an angle θ to the surface. (a) For what values of θ will there be a strong reflection? (b) What energy electrons could give the same result? Get solution
65g. Aliens visiting Earth are fascinated by baseball. They are so advanced that they have learned how to vary ... to make sure that a pitcher cannot throw a strike with any confidence. Assume the width of the strike zone is 0.38 m, the speed of the baseball is 35 m/s, the mass of the baseball is 145 g, and the ball travels a distance of 18 m. What value of ... is required? (Hint: there are two uncertainties here: the width of the strike zone and the transverse momentum of the pitched ball.) Get solution
66g. Neutrons from nuclear reactors are used in neutron diffraction experiments to measure interplanar spacings of a crystal lattice. The interplanar spacing can be measured as an indication of strain in the sample. Neutrons are particularly useful because they are less destructive than x rays and are able to penetrate deep into the sample. Their magnetic moment allows their use to study magnetic properties of matter. To study a particular polycrystalline sample with a planar spacing of 0.156 nm, a detector is mounted at an angle of 26° from the incident neutron beam. What energy neutrons from the reactor must be used in this experiment? An accelerator-based spallation neutron source is in operation at Oak Ridge National Laboratory. Get solution
67g. Use a computer program to produce a wave packet using the function ψn = An cos (2 πnx) where the integer n ranges from 9 to 15. Let the amplitude A12 = 1 with the amplitudes An decreasing symmetrically by 1/2, 1/3, 1/4 on either side of A12 (for example, A10 = 1/3 and A15 = 1/4). (a) Plot the wave packet ...versus x and each wave ψn over a wide enough range in x to see repeatable behavior for the wave packet. (b) Where is the wave packet centered? Over what value of x is the wave packet repeated? Get solution
68g. Most elementary particles (see Chapter 14) are not stable, and physicists have measured their mean lifetime .... Consider the uncertainty that this places on their mass-energy. The energy spread is the full width of the particle’s energy distribution at half its maximum value. (a) If we relate ... = Δt and ... = 2 ΔE, what is the relation ... in terms of the uncertainty principle? (b) What is the energy spread in the massenergy of the following particles with their mean lifetimes in parentheses: neutron (886 s), charged pion π- or π+ (2.6 × 10-8 s), and upsilon (1.2 × 10-20 s)? Get solution
69g. “Ultrafast” lasers produce bursts of light that last only on the order of 10 fs. Because of the uncertainty principle, such short bursts have a relatively large uncertainty in frequency and wavelength. A particular ultrafast laser produces a 10-fs burst of light from a 532-nm laser. (a) Find the uncertainty Δf in the light’s frequency and the ratio Δf/f. (b) What is the range Δ λ of wavelengths produced? (c) Compare your answer to part (b) with the original wavelength and with the length of the light pulse that is generated in 10 fs. Get solution
70g. An ultrafast laser (see the preceding problem) has a central wavelength of 550 nm. What pulse duration would result in a spread of wavelengths that just covered the visible spectrum, 400 nm to 700 nm? Get solution
1q. In 1900, did it seem clear that x rays were electromagnetic radiation? Give reasons for your answer. Was it important to perform further experiments to verify the characteristics of x rays? Get solution
2. X rays of wavelength 0.207 nm are scattered from NaCl. What is the angular separation between first and second-order diffraction peaks? Assume scattering planes that are parallel to the surface. Get solution
2q. In the early 1900s it was found that x rays were more difficult to refract or diffract than visible light. Why did this lead researchers to suppose that the wavelengths of x rays were shorter rather than longer than those of light? Get solution
3. Potassium chloride is a crystal with lattice spacing of 0.314 nm. The first peak for Bragg diffraction is observed to occur at 12.8°. What energy x rays were diffracted? What other order peaks can be observed (θ ≤ 90°)? Get solution
3q. What determines whether a given photon is an x ray? Could an x ray have a wavelength longer than some ultraviolet light? Get solution
4. A cubic crystal with interatomic spacing of 0.24 nm is used to select γ rays of energy 100 keV from a radioactive source containing a continuum of energies. If the incident beam is normal to the crystal, at what angle do the 100-keV γ rays appear? Get solution
4q. For a single crystal, transmission x-ray scattering will produce dots. However, if there are randomly oriented crystals, as in powder, concentric rings appear. Explain the difference qualitatively. Get solution
5. Calculate the de Broglie wavelength of a 1.2-kg rock thrown with a speed of 6.0 m/s into a pond. Is this wavelength similar to that of the water waves produced? Explain. Get solution
5q. How many particles do you think might be shown experimentally to exhibit wavelike properties? List at least four and discuss possible experiments to show this behavior. Get solution
6. Calculate the de Broglie wavelength of a typical nitrogen molecule in the atmosphere on a hot summer day (37°C). Compare this with the diameter (less than 1 nm) of the molecule. Get solution
6q. Why are neutrons more widely used than protons for studying crystal structure? What about using a hydrogen atom? Get solution
7. Transmission electron microscopes that use high energy electrons accelerated over a range from 40 to 100 kV are employed in many applications including the study of biological samples (like a virus) and nanoscience research and development (alloy particles and carbon nanotubes, for example). What would be the spatial limitation for this range of electrons? It is often true that resolution is limited by the optics of the lens system, not by the intrinsic limitation due to the de Broglie wavelength. Get solution
7q. Why are “cold” neutrons useful for studying crystal structure? How could one obtain “cold” neutrons? Get solution
8. A 3.0-MV transmission electron microscope has been in operation at Osaka University in Japan for several years. The higher-energy electrons allow for deeper sample penetration and extremely high resolution. What is the resolution limit for these electrons? Get solution
8q. It has been said that many experimental discoveries are made as a result of accidents (for example, that of Davisson and Germer). This statement may have some truth, but what traits and abilities must good scientists possess to take advantage of their accidents? Get solution
9. Work out Example 5.2b strictly using SI units of m, J, kg, and so on, and compare with the method of the example using eV units. ... ... Get solution
9q. Images taken with transmission electron microscopes are produced by passing very high energy electrons (40–100 keV) through matter. Why are the images always in black and white (or any two colors)? Get solution
10. Assume that the total energy E of an electron greatly exceeds its rest energy E0. If a photon has a wavelength equal to the de Broglie wavelength of the electron, what is the photon’s energy? Repeat the problem assuming E = 2E0 for the electron. Get solution
10q. Are the following phenomena wave or particle behaviors? Give your reasoning. (a) Television picture, (b) rainbows, (c) football sailing between goal posts, (d) telescope observing the moon, (e) police radar. Get solution
11. Determine the de Broglie wavelength of a particle of mass m and kinetic energy K. Do this for both (a) a relativistic and (b) a nonrelativistic particle. Get solution
11q. The experiment by Jönsson that showed the wavelike properties of electrons passing through a double slit is considered a pedagogically interesting experiment but not a landmark experiment. Why do you suppose this is true? Get solution
12. The Stanford Linear Accelerator accelerated electrons to an energy of 50 GeV. What is the de Broglie wavelength of these electrons? What fraction of a proton’s diameter ...can such a particle probe? Get solution
12q. Can you think of an experiment other than those mentioned in this chapter that might show the wavelike properties of particles? Discuss it. Get solution
13. Find the kinetic energy of (a) photons, (b) electrons, (c) neutrons, and (d) α particles that have a de Broglie wavelength of 0.13 nm. Get solution
13q. Why doesn’t the uncertainty principle restriction apply between the variables pz and x? Get solution
14q. How does the uncertainty principle apply to a known stable atomic system that apparently has an infinite lifetime? How well can we know the energy of such a system? Get solution
15. An electron initially at rest is accelerated across a potential difference of 3.00 kV. What are its wavelength, momentum, kinetic energy, and total energy? An electron initially at rest is accelerated across a potential difference of 3.00 kV. What are its wavelength, momentum, kinetic energy, and total energy? Get solution
15q. According to the uncertainty principle, can a particle having a kinetic energy of exactly zero be confi ned somewhere in a box of length L? Explain. Get solution
16. What is the wavelength of an electron with kinetic energy (a) 40 eV, (b) 400 eV, (c) 4.0 keV, (d) 40 keV, (e) 0.40 MeV, and (f) 4.0 MeV? Which of these energies are most suited for study of the NaCl crystal structure? Get solution
16q. What is similar about the conjugate variable pairs (px, x), (E, t), (L, θ), and (I, ω)? Get solution
17. Calculate the de Broglie wavelength of (a) an oxygen (O2) molecule darting around the room at 480 m/s and (b) an Escherichia coli bacterium of mass 6.5 × 10-13 kg, which has been measured to move at a speed of 1.0 × 10-5 m/s. Get solution
17q. What are the dimensions of the wave function ψ (x, t) that describes matter waves? Give your reasoning. Get solution
18. (a) What is the de Broglie wavelength of the 1.0-TeV protons accelerated in the Fermilab Tevatron accelerator? These high-energy protons are needed to probe elementary particles (see Chapter 14). (b) Repeat for the 7.0-TeV protons produced at CERN. Get solution
18q. Soon after their discovery, Davisson and Germer were using their experimental technique to describe new crystal structures of nickel. Do you think they were justified? Explain how you think their results allowed them to make such statements. Get solution
19. In an electron-scattering experiment, an intense reflected beam is found at Ø = 32° for a crystal with an interatomic distance of 0.23 nm. What is the lattice spacing of the planes responsible for the scattering? Assuming first-order diffraction, what are the wavelength, momentum, kinetic energy, and total energy of the incident electrons? Get solution
19q. Albert Einstein was a dissenter to the Copenhagen Interpretation and what it represented until the day he died. In a letter to Max Born in December 1926, Einstein wrote, “The theory yields a lot, but it hardly brings us any closer to the secret of the Old One. In any case I am convinced that God does not throw dice.” What do you think Einstein meant by this statement? Who or what is the Old One? Get solution
20. Davisson and Germer performed their experiment with a nickel target for several electron bombarding energies. At what angles would they find diffraction maxima for 48-eV and 64-eV electrons? Get solution
20q. It has been said that the energy-time version of the uncertainty principle allows a violation of the conservation of energy. The argument is that the uncertainty ...E allows the possibility that we may not know that energy conservation is violated during a small time ...t. Discuss arguments pro and con concerning this possibility. Get solution
21. A beam of 2.0-keV electrons incident on a crystal is refracted and observed (by transmission) on a screen 35 cm away. The radii of three concentric rings on the screen, all corresponding to first-order diffraction, are 2.1 cm, 2.3 cm, and 3.2 cm. What is the lattice-plane spacing corresponding to each of the three rings? Get solution
21q. The Fifth Solvay Congress, held in Brussels in October 1927, was dedicated to the quantum theory. A photo taken of the famous participants is often reproduced. Identify at least 10 participants and discuss what their contributions were to quantum physics, either experimentally or theoretically. Get solution
22. A beam of thermal neutrons (kinetic energy = 0.025 eV) scatters from a crystal with interatomic spacing 0.45 nm. What is the angle of the first-order Bragg peak? Get solution
22q. Summarize the discussions that Einstein and Bohr had at the 1927 Solvay Congress. List at least three objections that Einstein had to the Copenhagen interpretation of quantum mechanics and give Bohr’s explanation. Get solution
23. Generating plants in some power systems drop 10% of their load when the AC frequency changes by 0.30 Hz from the standard of 60 Hz. How often must the reading be monitored in order for the automatic operating system to be able to take corrective action? Let the time between measurements be at least half that determined by the bandwidth relation. Get solution
24. Consider electrons of kinetic energy 6.0 eV and 600 keV. For each electron, find the de Broglie wavelength, particle speed, phase velocity (speed), and group velocity (speed). Get solution
25. A wave, propagating along the x direction according to Equation (5.18), has a maximum displacement of 4.0 cm at x = 0 and t = 0. The wave speed is 5.0 cm/s, and the wavelength is 7.0 cm. (a) What is the frequency? (b) What is the wave’s amplitude at x = 10 cm and t = 13 s? Get solution
26. A wave of wavelength 4.0 cm has a wave speed of 4.2 cm/s. What is its (a) frequency, (b) period, (c) wave number, and (d) angular frequency? Get solution
27. Two waves are traveling simultaneously down a long Slinky. They can be represented by ψ1 (x, t) = 0.0030 sin(6.0x - 300t) and ψ2 (x, t) = 0.0030 sin(7.0x - 250t). Distances are measured in meters and time in seconds. (a) Write the expression for the resulting wave. (b) What are the phase and group velocities? (c) What is Δx between two adjacent zeros of ψ? (d) What is Δk Δx? Get solution
28. A wave packet describes a particle having momentum p. Starting with the relativistic relationship ...,show that the group velocity is c and the phase velocity is c/ β (where β = v/c). How can the phase velocity physically be greater than c ? Get solution
29. For waves in shallow water the phase velocity is about equal to the group velocity. What is the dependence of the phase velocity on the wavelength? Get solution
30. Find the group and phase velocities of 10-MeV protons and 10-MeV electrons (see Problem 28). Problem 28 A wave packet describes a particle having momentum p. Starting with the relativistic relationship ...,show that the group velocity is c and the phase velocity is c/ β (where β = v/c). How can the phase velocity physically be greater than c ? Get solution
31. Use Equation (5.25) with Ã(k) = A0 for the range of k = k0 - Δk/2 to k0 + Δk/2, and Ã(k) = 0 elsewhere, to determine ψ(x, 0), that is, at t = 0. Sketch the envelope term, the oscillating term, and |ψ(x, 0)|2. Approximately what is the width Δx over the full-widthhalf- maximum part of |ψ(x, 0)|2? What is the value of Δk Δx? ... Get solution
32. Use Equation (5.31) to show that ugr correctly represents the velocity of the particle both relativistically and classically. ... Get solution
33. Light of intensity I0 passes through two sets of apparatus. One contains one slit and the other two slits. The slits have the same width. What is the ratio of the out going intensity amplitude for the central peak for the two-slit case compared to the single slit? Get solution
34. Design a double-slit electron-scattering experiment using 1.0-keV electrons that will provide the first maximum at an angle of 1°. What will be the slit separation d? Get solution
35. You want to design an experiment similar to the one done by Jönsson that does not require magnification of the interference pattern in order to be seen. Let the two slits be separated by 2000 nm. Assume that you can discriminate visually between maxima that are as little as 0.3 mm apart. You have at your disposal a lab that allows the screen to be placed 80 cm away from the slits. What energy electrons will you require? Do you think such low-energy electrons will represent a problem? Explain. Get solution
36. A proton is confined in a uranium nucleus of radius 7.2 × 10-15 m. Determine the proton’s minimum kinetic energy according to the uncertainty principle if the proton is confined to a one-dimensional box that has length equal to the nuclear diameter. Get solution
37. A neutron is confined in a deuterium nucleus (deuteron) of diameter 3.1 × 10-15 m. Use the energy level calculation of a one-dimensional box to calculate the neutron’s minimum kinetic energy. What is the neutron’s minimum kinetic energy according to the uncertainty principle? Get solution
38. Show that the uncertainty principle can be expressed in the form ..., where θ is the angle and L the angular momentum. For what uncertainty in L will the angular position of a particle be completely undetermined? Get solution
39. Show that the uncertainty principle can be expressed in the form ..., where θ is the angle and L the angular momentum. For what uncertainty in L will the angular position of a particle be completely undetermined? Get solution
40. Some physics theories indicate that the lifetime of the proton is about 1036 years. What would such a prediction say about the energy of the proton? Get solution
41. What is the bandwidth Δ ω of an amplifier for radar if it amplifies a pulse of width 2.4 μs? Get solution
42. Find the minimum uncertainty in the speed of a bacterium having mass 3.0 × 10-15 kg if we know the position of the bacterium to within 1.0 μm, that is, to about its own size. Get solution
43. An atom in an excited state of 4.7 eV emits a photon and ends up in the ground state. The lifetime of the excited state is 1.0 × 10-13 s. (a) What is the energy uncertainty of the emitted photon? (b) What is the spectral line width (in wavelength) of the photon? Get solution
44. An electron microscope is designed to resolve objects as small as 0.14 nm. What energy electrons must be used in this instrument? Get solution
45. Rayleigh’s criterion is used to determine when two objects are barely resolved by a lens of diameter d. The angular separation must be greater than θR where ... In order to resolve two objects 4000 nm apart at a distance of 20 cm with a lens of diameter 5 cm, what energy (a) photons or (b) electrons should be used? Is this consistent with the uncertainty principle? Get solution
46. Calculate the de Broglie wavelength of a 5.5-MeV α particle emitted from an 241Am nucleus. Could this particle exist inside the 241Am nucleus (diameter ... 1.6 × 10-14 m)? Explain. Get solution
48. The wave function of a particle in a one-dimensional box of width L is ψ(x) = A sin(πx/L). If we know the particle must be somewhere in the box, what must be the value of A? Get solution
49. Write a cogent description of the Schrödinger cat paradox. Discuss variations of the paradox and the current status of its experimental verification. Get solution
50. Write a cogent description of the Einstein-Podolsky- Rosen paradox. Discuss variations of the paradox and the current status of its experimental verification. Get solution
51. Write a cogent description of the Bell inequality. Discuss variations and the current status of its experimental verification. Get solution
52. Write down the normalized wave functions for the first three energy levels of a particle of mass m in a one-dimensional box of width L. Assume there are equal probabilities of being in each state. Get solution
53. A particle in a one-dimensional box of length L has a kinetic energy much greater than its rest energy. What is the ratio of the following energy levels En: E2/E1, E3/ E1, E4/E1? How do your answers compare with the nonrelativistic case? Get solution
54g. Consider a wave packet having the product Δp Δx = h at a time t = 0. What will be the width of such a wave packet after the time m(Δx)2/ h? Get solution
55g. Analyze the Gaussian wave packet carefully and show that Δk Δx = 1/2. You must justify the assumptions you make concerning uncertainties in k and x. Take the Gaussian form given in Equation (5.26). (Hint: the linear “spread” of the wave packet Δx is given by one standard deviation, at which point the probability amplitude (|ψ|2) has fallen to one half its peak value.) ... Get solution
56g. An electron emitted in the beta decay of bismuth-210 has a mean kinetic energy of 390 keV. (a) Find the de Broglie wavelength of the electron. (b) Would such an electron be useful in a Davisson-Germer type scattering experiment? Address this question by deter- mining the angle at which a first-order diffraction maximum would be found using the same nickel target as Davisson and Germer. Get solution
57g. Electrons produced at the Thomas Jefferson National Accelerator Facility have a maximum energy of 6.0 GeV. (a) What is the de Broglie wavelength of each electron? (b) In what part of the electromagnetic spectrum do you fi nd a photon of comparable wavelength? Get solution
58g. The artificially created nuclear isotope tritium (3H) is important in many applications. This isotope undergoes beta decay, emitting an electron with a mean kinetic energy of 5.7 keV. (a) What is the de Broglie wavelength of such an emitted electron? (b) Is it likely that the electron existed inside the 3.4-fm-diameter nucleus just before it was emitted? Explain. Get solution
59g. As you saw in Chapter 4, the size of the hydrogen atom grows in proportion to n2, where n is the quantum state. For an atom in the n =10 state, model the electron as confined to a one-dimensional box of length equal to the atom’s diameter. Find the minimum energy of the electron is this box. Get solution
60g. An oboe player tunes the orchestra with the “Concert A” note, which has a frequency of 440 Hz. If she plays the note for 2.5 s, what minimum range of frequencies is heard during this time? Get solution
61g. As we learned in Section 3.9, an electron and positron can annihilate each other completely and form two gamma rays. (a) If the electron and positron were initially at rest, what are the wavelengths of the two emitted gamma rays? (b) Repeat if the electron and positron were each traveling at a speed of 0.30c measured in the lab and collided head-on. Get solution
62g. Most of the particles known to physicists are unstable. For example, the lifetime of the neutral pion, π0, is about 8.4 × 10-17 s. Its mass is 135.0 MeV/c2. a) What is the energy width of the π0 in its ground state? b) What is the relative uncertainty Δm/m of the pion’s mass? Get solution
63g. The range of the nuclear strong force is believed to be about 1.2 ×10-15 m. An early theory of nuclear physics proposed that the particle that “mediates” the strong force (similar to the photon mediating the electromagnetic force) is the pion. Assume that the pion moves at the speed of light in the nucleus, and calculate the time Δt it takes to travel between nucleons. Assume that the distance between nucleons is also about 1.2 ×10-15 m. Use this time Δt to calculate the energy ΔE for which energy conservation is violated during the time Δt. This ΔE has been used to estimate the mass of the pion. What value do you determine for the mass? Compare this value with the measured value of 135 MeV/c2 for the neutral pion. Get solution
64g. The planes of atoms in a particular cubic crystal lie parallel to the surface, 0.80 nm apart. X rays having wavelength 0.50 nm are directed at an angle θ to the surface. (a) For what values of θ will there be a strong reflection? (b) What energy electrons could give the same result? Get solution
65g. Aliens visiting Earth are fascinated by baseball. They are so advanced that they have learned how to vary ... to make sure that a pitcher cannot throw a strike with any confidence. Assume the width of the strike zone is 0.38 m, the speed of the baseball is 35 m/s, the mass of the baseball is 145 g, and the ball travels a distance of 18 m. What value of ... is required? (Hint: there are two uncertainties here: the width of the strike zone and the transverse momentum of the pitched ball.) Get solution
66g. Neutrons from nuclear reactors are used in neutron diffraction experiments to measure interplanar spacings of a crystal lattice. The interplanar spacing can be measured as an indication of strain in the sample. Neutrons are particularly useful because they are less destructive than x rays and are able to penetrate deep into the sample. Their magnetic moment allows their use to study magnetic properties of matter. To study a particular polycrystalline sample with a planar spacing of 0.156 nm, a detector is mounted at an angle of 26° from the incident neutron beam. What energy neutrons from the reactor must be used in this experiment? An accelerator-based spallation neutron source is in operation at Oak Ridge National Laboratory. Get solution
67g. Use a computer program to produce a wave packet using the function ψn = An cos (2 πnx) where the integer n ranges from 9 to 15. Let the amplitude A12 = 1 with the amplitudes An decreasing symmetrically by 1/2, 1/3, 1/4 on either side of A12 (for example, A10 = 1/3 and A15 = 1/4). (a) Plot the wave packet ...versus x and each wave ψn over a wide enough range in x to see repeatable behavior for the wave packet. (b) Where is the wave packet centered? Over what value of x is the wave packet repeated? Get solution
68g. Most elementary particles (see Chapter 14) are not stable, and physicists have measured their mean lifetime .... Consider the uncertainty that this places on their mass-energy. The energy spread is the full width of the particle’s energy distribution at half its maximum value. (a) If we relate ... = Δt and ... = 2 ΔE, what is the relation ... in terms of the uncertainty principle? (b) What is the energy spread in the massenergy of the following particles with their mean lifetimes in parentheses: neutron (886 s), charged pion π- or π+ (2.6 × 10-8 s), and upsilon (1.2 × 10-20 s)? Get solution
69g. “Ultrafast” lasers produce bursts of light that last only on the order of 10 fs. Because of the uncertainty principle, such short bursts have a relatively large uncertainty in frequency and wavelength. A particular ultrafast laser produces a 10-fs burst of light from a 532-nm laser. (a) Find the uncertainty Δf in the light’s frequency and the ratio Δf/f. (b) What is the range Δ λ of wavelengths produced? (c) Compare your answer to part (b) with the original wavelength and with the length of the light pulse that is generated in 10 fs. Get solution
70g. An ultrafast laser (see the preceding problem) has a central wavelength of 550 nm. What pulse duration would result in a spread of wavelengths that just covered the visible spectrum, 400 nm to 700 nm? Get solution