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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 ( 5.0010
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
2x 
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
nx 

 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
2x 
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.