9. Insulators (nonmetals) have a higher BE than metals, and it is more difficult for photons to eject electrons from insulators. Discuss how this relates
to the free charges in metals that make them good conductors.
10. If you pick up and shake a piece of metal that has electrons in it free to move as a current, no electrons fall out. Yet if you heat the metal,
electrons can be boiled off. Explain both of these facts as they relate to the amount and distribution of energy involved with shaking the object as
compared with heating it.
29.3 Photon Energies and the Electromagnetic Spectrum
11. Why are UV, x rays, and γ rays called ionizing radiation?
CHAPTER 29 | INTRODUCTION TO QUANTUM PHYSICS 1059
12. How can treating food with ionizing radiation help keep it from spoiling? UV is not very penetrating. What else could be used?
13. Some television tubes are CRTs. They use an approximately 30-kV accelerating potential to send electrons to the screen, where the electrons
stimulate phosphors to emit the light that forms the pictures we watch. Would you expect x rays also to be created?
14. Tanning salons use “safe” UV with a longer wavelength than some of the UV in sunlight. This “safe” UV has enough photon energy to trigger the
tanning mechanism. Is it likely to be able to cause cell damage and induce cancer with prolonged exposure?
15. Your pupils dilate when visible light intensity is reduced. Does wearing sunglasses that lack UV blockers increase or decrease the UV hazard to
your eyes? Explain.
16. One could feel heat transfer in the form of infrared radiation from a large nuclear bomb detonated in the atmosphere 75 km from you. However,
none of the profusely emitted x rays or γ rays reaches you. Explain.
17. Can a single microwave photon cause cell damage? Explain.
18. In an x-ray tube, the maximum photon energy is given by hf = qV. Would it be technically more correct to say hf = qV + BE, where BE is
the binding energy of electrons in the target anode? Why isn’t the energy stated the latter way?
19. Which formula may be used for the momentum of all particles, with or without mass?
20. Is there any measurable difference between the momentum of a photon and the momentum of matter?
21. Why don’t we feel the momentum of sunlight when we are on the beach?
29.6 The Wave Nature of Matter
22. How does the interference of water waves differ from the interference of electrons? How are they analogous?
23. Describe one type of evidence for the wave nature of matter.
24. Describe one type of evidence for the particle nature of EM radiation.
29.7 Probability: The Heisenberg Uncertainty Principle
25. What is the Heisenberg uncertainty principle? Does it place limits on what can be known?
29.8 The Particle-Wave Duality Reviewed
26. In what ways are matter and energy related that were not known before the development of relativity and quantum mechanics?
In what ways are matter and energy related that were not known before the development of relativity and quantum mechanics?
1060 CHAPTER 29 | INTRODUCTION TO QUANTUM PHYSICS
Problems & Exercises
Red light having a wavelength of 700 nm is projected onto magnesium
metal to which electrons are bound by 3.68 eV. (a) Use
KE
e = hf – BE to calculate the kinetic energy of the ejected electrons.
(b) What is unreasonable about this result? (c) Which assumptions are
1. A LiBr molecule oscillates with a frequency of 1.7×1013 Hz. (a)
unreasonable or inconsistent?
What is the difference in energy in eV between allowed oscillator states?
19. Unreasonable Results
(b) What is the approximate value of n for a state having an energy of
(a) What is the binding energy of electrons to a material from which
1.0 eV?
4.00-eV electrons are ejected by 400-nm EM radiation? (b) What is
2. The difference in energy between allowed oscillator states in HBr
unreasonable about this result? (c) Which assumptions are unreasonable
molecules is 0.330 eV. What is the oscillation frequency of this
or inconsistent?
molecule?
3. A physicist is watching a 15-kg orangutan at a zoo swing lazily in a tire
29.3 Photon Energies and the Electromagnetic
at the end of a rope. He (the physicist) notices that each oscillation takes
3.00 s and hypothesizes that the energy is quantized. (a) What is the
difference in energy in joules between allowed oscillator states? (b) What
20. What is the energy in joules and eV of a photon in a radio wave from
is the value of n for a state where the energy is 5.00 J? (c) Can the
an AM station that has a 1530-kHz broadcast frequency?
quantization be observed?
21. (a) Find the energy in joules and eV of photons in radio waves from
an FM station that has a 90.0-MHz broadcast frequency. (b) What does
this imply about the number of photons per second that the radio station
must broadcast?
4. What is the longest-wavelength EM radiation that can eject a
photoelectron from silver, given that the binding energy is 4.73 eV? Is this 22. Calculate the frequency in hertz of a 1.00-MeV γ -ray photon.
in the visible range?
23. (a) What is the wavelength of a 1.00-eV photon? (b) Find its
5. Find the longest-wavelength photon that can eject an electron from
frequency in hertz. (c) Identify the type of EM radiation.
potassium, given that the binding energy is 2.24 eV. Is this visible EM
radiation?
24. Do the unit conversions necessary to show that
hc = 1240 eV ⋅ nm, as stated in the text.
6. What is the binding energy in eV of electrons in magnesium, if the
longest-wavelength photon that can eject electrons is 337 nm?
25. Confirm the statement in the text that the range of photon energies
7. Calculate the binding energy in eV of electrons in aluminum, if the
for visible light is 1.63 to 3.26 eV, given that the range of visible
longest-wavelength photon that can eject them is 304 nm.
wavelengths is 380 to 760 nm.
8. What is the maximum kinetic energy in eV of electrons ejected from
26. (a) Calculate the energy in eV of an IR photon of frequency
sodium metal by 450-nm EM radiation, given that the binding energy is
2.00×1013 Hz. (b) How many of these photons would need to be
2.28 eV?
absorbed simultaneously by a tightly bound molecule to break it apart?
9. UV radiation having a wavelength of 120 nm falls on gold metal, to
(c) What is the energy in eV of a γ ray of frequency 3.00×1020 Hz?
which electrons are bound by 4.82 eV. What is the maximum kinetic
energy of the ejected photoelectrons?
(d) How many tightly bound molecules could a single such γ ray break
10. Violet light of wavelength 400 nm ejects electrons with a maximum
apart?
kinetic energy of 0.860 eV from sodium metal. What is the binding energy 27. Prove that, to three-digit accuracy, h = 4.14×10−15 eV ⋅ s, as
of electrons to sodium metal?
stated in the text.
11. UV radiation having a 300-nm wavelength falls on uranium metal,
ejecting 0.500-eV electrons. What is the binding energy of electrons to
28. (a) What is the maximum energy in eV of photons produced in a CRT
uranium metal?
using a 25.0-kV accelerating potential, such as a color TV? (b) What is
their frequency?
12. What is the wavelength of EM radiation that ejects 2.00-eV electrons
from calcium metal, given that the binding energy is 2.71 eV? What type
29. What is the accelerating voltage of an x-ray tube that produces x rays
of EM radiation is this?
with a shortest wavelength of 0.0103 nm?
13. Find the wavelength of photons that eject 0.100-eV electrons from
30. (a) What is the ratio of power outputs by two microwave ovens having
potassium, given that the binding energy is 2.24 eV. Are these photons
frequencies of 950 and 2560 MHz, if they emit the same number of
visible?
photons per second? (b) What is the ratio of photons per second if they
have the same power output?
14. What is the maximum velocity of electrons ejected from a material by
80-nm photons, if they are bound to the material by 4.73 eV?
31. How many photons per second are emitted by the antenna of a
microwave oven, if its power output is 1.00 kW at a frequency of 2560
15. Photoelectrons from a material with a binding energy of 2.71 eV are
MHz?
ejected by 420-nm photons. Once ejected, how long does it take these
electrons to travel 2.50 cm to a detection device?
32. Some satellites use nuclear power. (a) If such a satellite emits a
1.00-W flux of γ rays having an average energy of 0.500 MeV, how
16. A laser with a power output of 2.00 mW at a wavelength of 400 nm is
projected onto calcium metal. (a) How many electrons per second are
many are emitted per second? (b) These γ rays affect other satellites.
ejected? (b) What power is carried away by the electrons, given that the
How far away must another satellite be to only receive one γ ray per
binding energy is 2.31 eV?
second per square meter?
17. (a) Calculate the number of photoelectrons per second ejected from a 33. (a) If the power output of a 650-kHz radio station is 50.0 kW, how
1.00-mm 2 area of sodium metal by 500-nm EM radiation having an
many photons per second are produced? (b) If the radio waves are
intensity of 1.30 kW/m2 (the intensity of sunlight above the Earth’s
broadcast uniformly in all directions, find the number of photons per
atmosphere). (b) Given that the binding energy is 2.28 eV, what power is
second per square meter at a distance of 100 km. Assume no reflection
carried away by the electrons? (c) The electrons carry away less power
from the ground or absorption by the air.
than brought in by the photons. Where does the other power go? How
34. How many x-ray photons per second are created by an x-ray tube
can it be recovered?
that produces a flux of x rays having a power of 1.00 W? Assume the
18. Unreasonable Results
average energy per photon is 75.0 keV.
CHAPTER 29 | INTRODUCTION TO QUANTUM PHYSICS 1061
35. (a) How far away must you be from a 650-kHz radio station with
power 50.0 kW for there to be only one photon per second per square
29.6 The Wave Nature of Matter
meter? Assume no reflections or absorption, as if you were in deep outer
49. At what velocity will an electron have a wavelength of 1.00 m?
space. (b) Discuss the implications for detecting intelligent life in other
solar systems by detecting their radio broadcasts.
50. What is the wavelength of an electron moving at 3.00% of the speed
of light?
36. Assuming that 10.0% of a 100-W light bulb’s energy output is in the
visible range (typical for incandescent bulbs) with an average wavelength
51. At what velocity does a proton have a 6.00-fm wavelength (about the
of 580 nm, and that the photons spread out uniformly and are not
size of a nucleus)? Assume the proton is nonrelativistic. (1 femtometer =
absorbed by the atmosphere, how far away would you be if 500 photons
10−15 m. )
per second enter the 3.00-mm diameter pupil of your eye? (This number
easily stimulates the retina.)
52. What is the velocity of a 0.400-kg billiard ball if its wavelength is 7.50
cm (large enough for it to interfere with other billiard balls)?
37. Construct Your Own Problem
53. Find the wavelength of a proton moving at 1.00% of the speed of
Consider a laser pen. Construct a problem in which you calculate the
light.
number of photons per second emitted by the pen. Among the things to
be considered are the laser pen’s wavelength and power output. Your
54. Experiments are performed with ultracold neutrons having velocities
instructor may also wish for you to determine the minimum diffraction
as small as 1.00 m/s. (a) What is the wavelength of such a neutron? (b)
spreading in the beam and the number of photons per square centimeter
What is its kinetic energy in eV?
the pen can project at some large distance. In this latter case, you will
55. (a) Find the velocity of a neutron that has a 6.00-fm wavelength
also need to consider the output size of the laser beam, the distance to
(about the size of a nucleus). Assume the neutron is nonrelativistic. (b)
the object being illuminated, and any absorption or scattering along the
What is the neutron’s kinetic energy in MeV?
way.
56. What is the wavelength of an electron accelerated through a 30.0-kV
potential, as in a TV tube?
57. What is the kinetic energy an electron in a TEM having a 0.0100-nm
38. (a) Find the momentum of a 4.00-cm-wavelength microwave photon.
wavelength?
(b) Discuss why you expect the answer to (a) to be very small.
58. (a) Calculate the velocity of an electron that has a wavelength of
39. (a) What is the momentum of a 0.0100-nm-wavelength photon that
1.00 μm. (b) Through what voltage must the electron be accelerated to
could detect details of an atom? (b) What is its energy in MeV?
have this velocity?
40. (a) What is the wavelength of a photon that has a momentum of
5.00×10−29 kg ⋅ m/s
59. The velocity of a proton emerging from a Van de Graaff accelerator is
? (b) Find its energy in eV.
25.0% of the speed of light. (a) What is the proton’s wavelength? (b)
What is its kinetic energy, assuming it is nonrelativistic? (c) What was the
41. (a) A γ -ray photon has a momentum of 8.00×10−21 kg ⋅ m/s .
equivalent voltage through which it was accelerated?
What is its wavelength? (b) Calculate its energy in MeV.
60. The kinetic energy of an electron accelerated in an x-ray tube is 100
42. (a) Calculate the momentum of a photon having a wavelength of
keV. Assuming it is nonrelativistic, what is its wavelength?
2.50 μm . (b) Find the velocity of an electron having the same
61. Unreasonable Results
momentum. (c) What is the kinetic energy of the electron, and how does
(a) Assuming it is nonrelativistic, calculate the velocity of an electron with
it compare with that of the photon?
a 0.100-fm wavelength (small enough to detect details of a nucleus). (b)
43. Repeat the previous problem for a 10.0-nm-wavelength photon.
What is unreasonable about this result? (c) Which assumptions are
unreasonable or inconsistent?
44. (a) Calculate the wavelength of a photon that has the same
momentum as a proton moving at 1.00% of the speed of light. (b) What is
the energy of the photon in MeV? (c) What is the kinetic energy of the
29.7 Probability: The Heisenberg Uncertainty Principle
proton in MeV?
62. (a) If the position of an electron in a membrane is measured to an
45. (a) Find the momentum of a 100-keV x-ray photon. (b) Find the
accuracy of 1.00 μm , what is the electron’s minimum uncertainty in
equivalent velocity of a neutron with the same momentum. (c) What is the velocity? (b) If the electron has this velocity, what is its kinetic energy in
neutron’s kinetic energy in keV?
eV? (c) What are the implications of this energy, comparing it to typical
molecular binding energies?
46. Take the ratio of relativistic rest energy, E = γmc 2 , to relativistic
63. (a) If the position of a chlorine ion in a membrane is measured to an
momentum, p = γmu , and show that in the limit that mass approaches
accuracy of 1.00 μm , what is its minimum uncertainty in velocity, given
zero, you find E / p = c .
its mass is 5.86×10−26 kg ? (b) If the ion has this velocity, what is its
47. Construct Your Own Problem
kinetic energy in eV, and how does this compare with typical molecular
Consider a space sail such as mentioned in Example 29.5. Construct a
binding energies?
problem in which you calculate the light pressure on the sail in N/m2
64. Suppose the velocity of an electron in an atom is known to an
produced by reflecting sunlight. Also calculate the force that could be
accuracy of 2.0×103 m/s (reasonably accurate compared with orbital
produced and how much effect that would have on a spacecraft. Among
velocities). What is the electron’s minimum uncertainty in position, and
the things to be considered are the intensity of sunlight, its average
how does this compare with the approximate 0.1-nm size of the atom?
wavelength, the number of photons per square meter this implies, the
area of the space sail, and the mass of the system being accelerated.
65. The velocity of a proton in an accelerator is known to an accuracy of
0.250% of the speed of light. (This could be small compared with its
48. Unreasonable Results
velocity.) What is the smallest possible uncertainty in its position?
A car feels a small force due to the light it sends out from its headlights,
66. A relatively long-lived excited state of an atom has a lifetime of 3.00
equal to the momentum of the light divided by the time in which it is
ms. What is the minimum uncertainty in its energy?
emitted. (a) Calculate the power of each headlight, if they exert a total
force of 2.00×10−2 N backward on the car. (b) What is unreasonable
67. (a) The lifetime of a highly unstable nucleus is 10−20 s . What is the
about this result? (c) Which assumptions are unreasonable or
smallest uncertainty in its decay energy? (b) Compare this with the rest
inconsistent?
energy of an electron.
1062 CHAPTER 29 | INTRODUCTION TO QUANTUM PHYSICS
68. The decay energy of a short-lived particle has an uncertainty of 1.0
(a) Calculate the velocity of electrons that form the same pattern as
MeV due to its short lifetime. What is the smallest lifetime it can have?
450-nm light when passed through a double slit. (b) Calculate the kinetic
69. The decay energy of a short-lived nuclear excited state has an
energy of each and compare them. (c) Would either be easier to
uncertainty of 2.0 eV due to its short lifetime. What is the smallest lifetime
generate than the other? Explain.
it can have?
81. Integrated Concepts
70. What is the approximate uncertainty in the mass of a muon, as
(a) What is the separation between double slits that produces a second-
determined from its decay lifetime?
order minimum at 45.0º for 650-nm light? (b) What slit separation is
71. Derive the approximate form of Heisenberg’s uncertainty principle for
needed to produce the same pattern for 1.00-keV protons.
energy and time, Δ EΔ t ≈ h , using the following arguments: Since the
82. Integrated Concepts
position of a particle is uncertain by Δ x ≈ λ , where λ is the wavelength A laser with a power output of 2.00 mW at a wavelength of 400 nm is
of the photon used to examine it, there is an uncertainty in the time the
projected onto calcium metal. (a) How many electrons per second are
photon takes to traverse Δ x . Furthermore, the photon has an energy
ejected? (b) What power is carried away by the electrons, given that the
related to its wavelength, and it can transfer some or all of this energy to
binding energy is 2.31 eV? (c) Calculate the current of ejected electrons.
the object being examined. Thus the uncertainty in the energy of the
(d) If the photoelectric material is electrically insulated and acts like a
object is also related to λ . Find Δ t and Δ E ; then multiply them to give 2.00-pF capacitor, how long will current flow before the capacitor voltage
stops it?
the approximate uncertainty principle.
83. Integrated Concepts
29.8 The Particle-Wave Duality Reviewed
One problem with x rays is that they are not sensed. Calculate the
72. Integrated Concepts
temperature increase of a researcher exposed in a few seconds to a
nearly fatal accidental dose of x rays under the following conditions. The
The 54.0-eV electron in Example 29.7 has a 0.167-nm wavelength. If
such electrons are passed through a double slit and have their first
energy of the x-ray photons is 200 keV, and 4.00×1013 of them are
maximum at an angle of 25.0º , what is the slit separation d ?
absorbed per kilogram of tissue, the specific heat of which is
0.830 kcal/kg ⋅ ºC . (Note that medical diagnostic x-ray machines
73. Integrated Concepts
cannot produce an intensity this great.)
An electron microscope produces electrons with a 2.00-pm wavelength. If
these are passed through a 1.00-nm single slit, at what angle will the first
84. Integrated Concepts
diffraction minimum be found?
A 1.00-fm photon has a wavelength short enough to detect some
74. Integrated Concepts
information about nuclei. (a) What is the photon momentum? (b) What is
its energy in joules and MeV? (c) What is the (relativistic) velocity of an
A certain heat lamp emits 200 W of mostly IR radiation averaging 1500
electron with the same momentum? (d) Calculate the electron’s kinetic
nm in wavelength. (a) What is the average photon energy in joules? (b)
energy.
How many of these photons are required to increase the temperature of a
person’s shoulder by 2.0ºC , assuming the a