Content check — University Physics 3, Chapter 6: Photons and Matter Waves

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 6: Photons and Matter Waves

97 problems · 0 figures

Check Your Understanding 6.1 §6.1 source ↗

The flame of a peach-scented candle has a yellowish color and the flame of a Bunsen’s burner in a chemistry lab has a bluish color. Which flame has a higher temperature?

Answer

Bunsen’s burner

Check Your Understanding 6.2 §6.1 source ↗

An iron poker is being heated. As its temperature rises, the poker begins to glow—first dull red, then bright red, then orange, and then yellow. Use either the blackbody radiation curve or Wien’s law to explain these changes in the color of the glow.

Answer

The wavelength of the radiation maximum decreases with increasing temperature.

Check Your Understanding 6.3 §6.1 source ↗

Suppose that two stars,

α

and

β,

radiate exactly the same total power. If the radius of star

α

is three times that of star

β,

what is the ratio of the surface temperatures of these stars? Which one is hotter?

Answer

T_{α} / T_{β} =

1 / \sqrt{3} ≅ 0.58,

so the star

β

is hotter.

Check Your Understanding 6.4 §6.1 source ↗

A molecule is vibrating at a frequency of

5.0 \times 10^{14} Hz .

What is the smallest spacing between its vibrational energy levels?

Answer

3.3 \times 10^{−19} J

Check Your Understanding 6.5 §6.1 source ↗

Would the result in Example 6.4 be different if the mass were not 1.0 kg but a tiny mass of 1.0 µg, and the amplitude of vibrations were 0.10 µm?

Answer

No, because then

Δ E / E \approx 10^{−21}

Check Your Understanding 6.6 §6.2 source ↗

A yellow 589-nm light is incident on a surface whose work function is 1.20 eV. What is the stopping potential? What is the cut-off wavelength?

Answer

−0.91

V; 1040 nm

Check Your Understanding 6.7 §6.2 source ↗

Cut-off frequency for the photoelectric effect in some materials is

8.0 \times 10^{13} Hz .

When the incident light has a frequency of

1.2 \times 10^{14} Hz,

the stopping potential is measured as – 0.16 V. Estimate a value of Planck’s constant from these data (in units

J \cdot s

and

eV \cdot s

) and determine the percentage error of your estimation.

Answer

h =

6.40 \times 10^{−34} J \cdot s =

4.0 \times 10^{−15} eV \cdot s; − 3.5 %

Check Your Understanding 6.8 §6.3 source ↗

An incident 71-pm X-ray is incident on a calcite target. Find the wavelength of the X-ray scattered at a

60 °

angle. What is the smallest shift that can be expected in this experiment?

Answer

{(Δ λ)}_{min} = 0 m

at a

0 °

angle;

71.0 pm +

0.5 λ_{c} =

72. 215 pm

Check Your Understanding 6.9 §6.4 source ↗

What are the limits of the Lyman series? Can you see these spectral lines?

Answer

121.5 nm and 91.1 nm; no, these spectral bands are in the ultraviolet

Check Your Understanding 6.10 §6.4 source ↗

When an electron in a hydrogen atom is in the first excited state, what prediction does the Bohr model give about its orbital speed and kinetic energy? What is the magnitude of its orbital angular momentum?

Answer

v_{2} =

1.1 \times 10^{6} m / s ≅ 0.0036 c;

L_{2} =

2 ℏ K_{2} =

3.4 eV

Check Your Understanding 6.11 §6.5 source ↗

What is de Broglie’s wavelength of a nonrelativistic proton with a kinetic energy of 1.0 eV?

Answer

29 pm

Check Your Understanding 6.12 §6.5 source ↗

Find the de Broglie wavelength of an electron in the third excited state of hydrogen.

Answer

λ =

2 π n a_{0} =

2 (3.324 Å) =

6.648 Å

Check Your Understanding 6.13 §6.5 source ↗

Find the de Broglie wavelength and kinetic energy of a free electron that travels at a speed of 0.75c.

Answer

λ = 2.14 pm;

K = 261.56 keV

Check Your Understanding 6.14 §6.6 source ↗

For the situation described in Example 6.15, find the angular position of the fifth-order bright fringe on the viewing screen.

Answer

0.052 °

Check Your Understanding 6.15 §6.6 source ↗

Suppose that the diameter of the aperture in Example 6.16 is halved. How does it affect the resolving power?

Answer

doubles it

6-1 §6.1 source ↗

Which surface has a higher temperature – the surface of a yellow star or that of a red star?

Answer

yellow

6-3 §6.1 source ↗

Explain the color changes in a hot body as its temperature is increased.

Answer

goes from red to violet through the rainbow of colors

6-5 §6.1 source ↗

Two cavity radiators are constructed with walls made of different metals. At the same temperature, how would their radiation spectra differ?

Answer

would not differ

6-7 §6.1 source ↗

If everything radiates electromagnetic energy, why can we not see objects at room temperature in a dark room?

Answer

human eye does not see IR radiation

6-9 §6.2 source ↗

For the same monochromatic light source, would the photoelectric effect occur for all metals?

Answer

6-11 §6.2 source ↗

Explain how you can determine the work function from a plot of the stopping potential versus the frequency of the incident radiation in a photoelectric effect experiment. Can you determine the value of Planck’s constant from this plot?

Answer

from the slope

6-13 §6.2 source ↗

Speculate how increasing the temperature of a photoelectrode affects the outcomes of the photoelectric effect experiment.

Answer

Answers may vary

6-15 §6.2 source ↗

Is the photoelectric effect a consequence of the wave character of radiation or is it a consequence of the particle character of radiation? Explain briefly.

Answer

the particle character

6-17 §6.3 source ↗

Discuss any similarities and differences between the photoelectric and the Compton effects.

Answer

Answers may vary

6-19 §6.3 source ↗

Does changing the intensity of a monochromatic light beam affect the momentum of the individual photons in the beam? Does such a change affect the net momentum of the beam?

Answer

no; yes

6-21 §6.3 source ↗

Is it possible in the Compton experiment to observe scattered X-rays that have a shorter wavelength than the incident X-ray radiation?

Answer

6-23 §6.3 source ↗

At what scattering angle is the wavelength shift in the Compton effect equal to the Compton wavelength?

Answer

right angle

6-25 §6.4 source ↗

Do the various spectral lines of the hydrogen atom overlap?

Answer

6-27 §6.4 source ↗

When the absorption spectrum of hydrogen at room temperature is analyzed, absorption lines for the Lyman series are found, but none are found for the Balmer series. What does this tell us about the energy state of most hydrogen atoms at room temperature?

Answer

They are at ground state.

6-29 §6.4 source ↗

Discuss the similarities and differences between Thomson’s model of the hydrogen atom and Bohr’s model of the hydrogen atom.

Answer

Answers may vary

6-31 §6.4 source ↗

If, in a hydrogen atom, an electron moves to an orbit with a larger radius, does the energy of the hydrogen atom increase or decrease?

Answer

increase

6-33 §6.4 source ↗

Suppose an electron in a hydrogen atom makes a transition from the (n+1)th orbit to the nth orbit. Is the wavelength of the emitted photon longer for larger values of n, or for smaller values of n?

Answer

for larger n

6-35 §6.4 source ↗

Can a hydrogen atom absorb a photon whose energy is greater than 13.6 eV?

Answer

Yes, the excess of 13.6 eV will become kinetic energy of a free electron.

6-37 §6.4 source ↗

Do gravitational forces have a significant effect on atomic energy levels?

Answer

6-39 §6.5 source ↗

Which type of radiation is most suitable for the observation of diffraction patterns on crystalline solids; radio waves, visible light, or X-rays? Explain.

Answer

X-rays, best resolving power

6-41 §6.5 source ↗

If an electron and a proton are traveling at the same speed, which one has the shorter de Broglie wavelength?

Answer

proton

6-43 §6.5 source ↗

Why is the wave-like nature of matter not observed every day for macroscopic objects?

Answer

negligibly small de Broglie’s wavelengths

6-45 §6.5 source ↗

Why does the setup of Davisson–Germer experiment need to be enclosed in a vacuum chamber? Discuss what result you expect when the chamber is not evacuated.

Answer

to avoid collisions with air molecules

6-47 §6.6 source ↗

Discuss: How does the interference of water waves differ from the interference of electrons? How are they analogous?

Answer

Answers may vary

6-49 §6.6 source ↗

Give at least one argument in support of the particle-nature of radiation.

Answer

Answers may vary

6-51 §6.6 source ↗

Does the Heisenberg uncertainty principle allow a particle to be at rest in a designated region in space?

Answer

yes

6-53 §6.6 source ↗

Do the photons of red light produce better resolution in a microscope than blue light photons? Explain.

Answer

yes

6-55 §6.1 source ↗

A 200-W heater emits a 1.5-µm radiation. (a) What value of the energy quantum does it emit? (b) Assuming that the specific heat of a 4.0-kg body is

0.83 kcal / kg \cdot K,

how many of these photons must be absorbed by the body to increase its temperature by 2 K? (c) How long does the heating process in (b) take, assuming that all radiation emitted by the heater gets absorbed by the body?

Answer

a. 0.81 eV; b.

2.1 \times 10^{23};

c. 2 min 20 sec

6-57 §6.1 source ↗

(a) For what temperature is the peak of blackbody radiation spectrum at 400 nm? (b) If the temperature of a blackbody is 800 K, at what wavelength does it radiate the most energy?

Answer

a. 7245 K; b. 3.62 μm

6-59 §6.1 source ↗

Interstellar space is filled with radiation of wavelength

970 μ m.

This radiation is considered to be a remnant of the “big bang.” What is the corresponding blackbody temperature of this radiation?

Answer

about 3 K

6-61 §6.2 source ↗

A photon has energy 20 keV. What are its frequency and wavelength?

Answer

4.835 \times 10^{18}

Hz; 0.620 Å

6-63 §6.2 source ↗

What is the longest wavelength of radiation that can eject a photoelectron from silver? Is it in the visible range?

Answer

263 nm; no

6-65 §6.2 source ↗

Estimate the binding energy of electrons in magnesium, given that the wavelength of 337 nm is the longest wavelength that a photon may have to eject a photoelectron from magnesium photoelectrode.

Answer

3.68 eV

6-67 §6.2 source ↗

Estimate the work function of aluminum, given that the wavelength of 304 nm is the longest wavelength that a photon may have to eject a photoelectron from an aluminum photoelectrode.

Answer

4.09 eV

6-69 §6.2 source ↗

A 120-nm UV radiation illuminates a silver-plated electrode. What is the maximum kinetic energy of the ejected photoelectrons?

Answer

5.60 eV

6-71 §6.2 source ↗

A 600-nm light falls on a photoelectric surface and electrons with the maximum kinetic energy of 0.17 eV are emitted. Determine (a) the work function and (b) the cutoff frequency of the surface. (c) What is the stopping potential when the surface is illuminated with light of wavelength 400 nm?

Answer

a. 1.89 eV; b. 459 THz; c. 1.21 V

6-73 §6.2 source ↗

Find the wavelength of radiation that can eject 2.00-eV electrons from calcium electrode. The work function for calcium is 2.71 eV. In what range is this radiation?

Answer

264 nm; UV

6-75 §6.2 source ↗

Find the maximum velocity of photoelectrons ejected by an 80-nm radiation, if the work function of photoelectrode is 4.73 eV.

Answer

1.95 \times 10^{6} m/s

6-77 §6.3 source ↗

What is the momentum of a 4-cm microwave photon?

Answer

1.66 \times 10^{− 32} kg \cdot m/s

6-79 §6.3 source ↗

What is the energy of a photon whose momentum is

3.0 \times 10^{−24} kg \cdot m/s

Answer

56.21 eV

6-81 §6.3 source ↗

Find the momentum and energy of a 1.0-Å photon.

Answer

6.63 \times 10^{− 23} kg \cdot m/s;

124 keV

6-83 §6.3 source ↗

γ

-ray photon has a momentum of

8.00 \times 10^{−21} kg \cdot m/s .

Find its wavelength and energy.

Answer

82.9 fm; 15 MeV

6-85 §6.3 source ↗

Show that

p = h / λ

and

E_{f} = h f

are consistent with the relativistic formula

E^{2} =

p^{2} c^{2} +

m_{0}^{2} c^{2} .

Answer

(Proof)

6-87 §6.3 source ↗

For collisions with free electrons, compare the Compton shift of a photon scattered as an angle of

30 °

to that of a photon scattered at

45 ° .

Answer

Δ λ_{30} / Δ λ_{45} =

45.74 %

6-89 §6.4 source ↗

Calculate the wavelength of the first line in the Lyman series and show that this line lies in the ultraviolet part of the spectrum.

Answer

121.5 nm

6-91 §6.4 source ↗

Calculate the energy changes corresponding to the transitions of the hydrogen atom: (a) from

n = 3

n = 4;

(b) from

n = 2

n = 1;

and (c) from

n = 3

n = \infty .

Answer

a. 0.661 eV; b. –10.2 eV; c. 1.511 eV

6-93 §6.4 source ↗

What is the frequency of the photon absorbed when the hydrogen atom makes the transition from the ground state to the

n = 4

state?

Answer

3038 THz

6-95 §6.4 source ↗

When a hydrogen atom is in its third excided state, what are the shortest and longest wavelengths of the photons it can emit?

Answer

97.33 nm

6-97 §6.4 source ↗

For an electron in a hydrogen atom in the

n = 2

state, compute: (a) the angular momentum; (b) the kinetic energy; (c) the potential energy; and (d) the total energy.

Answer

a. h/

π;

b. 3.4 eV; c. – 6.8 eV; d. – 3.4 eV

6-99 §6.4 source ↗

It has been measured that it required 0.850 eV to remove an electron from the hydrogen atom. In what state was the atom before the ionization happened?

Answer

n = 4

6-101 §6.4 source ↗

Find the shortest wavelength in the Balmer series. In what part of the spectrum does this line lie?

Answer

365 nm; UV

6-103 §6.4 source ↗

Do the Balmer series and the Lyman series overlap? Why? Why not? (Hint: calculate the shortest Balmer line and the longest Lyman line.)

Answer

6-105 §6.4 source ↗

4.653 -μ m

emission line of atomic hydrogen corresponds to transition between the states

n_{f} = 5

and

n_{i} .

Find

n_{i} .

Answer

6-107 §6.5 source ↗

What is the de Broglie wavelength of an electron travelling at a speed of

5.0 \times 10^{6} m/s

Answer

145.5 pm

6-109 §6.5 source ↗

What is the de Broglie wavelength of a proton whose kinetic energy is 2.0 MeV? 10.0 MeV?

Answer

20 fm; 9 fm

6-111 §6.5 source ↗

(a) What is the energy of an electron whose de Broglie wavelength is that of a photon of yellow light with wavelength 590 nm? (b) What is the de Broglie wavelength of an electron whose energy is that of the photon of yellow light?

Answer

a. 2.103 eV; b. 0.846 nm

6-113 §6.5 source ↗

What is the wavelength of an electron that is moving at a 3% of the speed of light?

Answer

80.9 pm

6-115 §6.5 source ↗

What is the velocity of a 0.400-kg billiard ball if its wavelength is 7.50 fm?

Answer

2.21 \times 10^{− 19} m/s

6-117 §6.6 source ↗

An AM radio transmitter radiates 500 kW at a frequency of 760 kHz. How many photons per second does the emitter emit?

Answer

9.929 \times 10^{32}

6-119 §6.6 source ↗

Find the Lorentz factor

γ

and de Broglie’s wavelength for a 1.0-TeV proton in a particle accelerator.

Answer

γ = 1060;

0.00124 fm

6-121 §6.6 source ↗

If electron is to be diffracted significantly by a crystal, its wavelength must be about equal to the spacing, d, of crystalline planes. Assuming

d = 0.250 nm,

estimate the potential difference through which an electron must be accelerated from rest if it is to be diffracted by these planes.

Answer

24.11 V

6-123 §6.6 source ↗

Solar wind (radiation) that is incident on the top of Earth’s atmosphere has an average intensity of

1.3 kW / m^{2} .

Suppose that you are building a solar sail that is to propel a small toy spaceship with a mass of 0.1 kg in the space between the International Space Station and the moon. The sail is made from a very light material, which perfectly reflects the incident radiation. To assess whether such a project is feasible, answer the following questions, assuming that radiation photons are incident only in normal direction to the sail reflecting surface. (a) What is the radiation pressure (force per

m^{2}

) of the radiation falling on the mirror-like sail? (b) Given the radiation pressure computed in (a), what will be the acceleration of the spaceship when the sail has of an area of

10.0 m^{2}

? (c) Given the acceleration estimate in (b), how fast will the spaceship be moving after 24 hours when it starts from rest?

Answer

P =

2 I / c =

8.67 \times 10^{−6} N / m^{2};

a =

P A / m =

8.67 \times 10^{- 4} m / s^{2};

c. 74.91 m/s

6-125 §6.6 source ↗

Show that Wien’s displacement law results from Planck’s radiation law. (Hint: substitute

x = h c / λ k T

and write Planck’s law in the form

I (x, T) =

A x^{5} / (e^{x} - 1),

where

A =

2 π {(k T)}^{5} / (h^{4} c^{3}) .

Now, for fixed T, find the position of the maximum in I(x,T) by solving for x in the equation

d I (x, T) / d x =

0 .

)

Answer

x = 4.965

6-127 §6.AP source ↗

Determine the power intensity of radiation per unit wavelength emitted at a wavelength of 500.0 nm by a blackbody at a temperature of 10,000 K.

Answer

7.124 \times 10^{16} {W/m}^{3}

6-129 §6.AP source ↗

A quantum mechanical oscillator vibrates at a frequency of 250.0 THz. What is the minimum energy of radiation it can emit?

Answer

1.034 eV

6-131 §6.AP source ↗

A sodium lamp emits 2.0 W of radiant energy, most of which has a wavelength of about 589 nm. Estimate the number of photons emitted per second by the lamp.

Answer

5.93 \times 10^{18}

6-133 §6.AP source ↗

If the work function of a metal is 3.2 eV, what is the maximum wavelength that a photon can have to eject a photoelectron from this metal surface?

Answer

387.8 nm

6-135 §6.AP source ↗

A 400-nm laser beam is projected onto a calcium electrode. The power of the laser beam is 2.00 mW and the work function of calcium is 2.31 eV. (a) How many photoelectrons per second are ejected? (b) What net power is carried away by photoelectrons?

Answer

4.02 \times 10^{15};

b. 0.508 mW

6-137 §6.AP source ↗

A laser with a power output of 2.00 mW at a 400-nm wavelength is used to project a beam of light onto a calcium photoelectrode. (a) How many photoelectrons leave the calcium surface per second? (b) What power is carried away by ejected photoelectrons, given that the work function of calcium is 2.31 eV? (c) Calculate the photocurrent. (d) If the photoelectrode suddenly becomes electrically insulated and the setup of two electrodes in the circuit suddenly starts to act like a 2.00-pF capacitor, how long will current flow before the capacitor voltage stops it?

Answer

4.02 \times 10^{15};

b. 0.533 mW; c. 0.644 mA; d. 2.57 ns

6-139 §6.AP source ↗

(a) Calculate the wavelength of a photon that has the same momentum as a proton moving with 1% of the speed of light in a vacuum. (b) What is the energy of this photon in MeV? (c) What is the kinetic energy of the proton in MeV?

Answer

a. 0.132 pm; b. 9.39 MeV; c. 0.047 MeV

6-141 §6.AP source ↗

The momentum of light, as it is for particles, is exactly reversed when a photon is reflected straight back from a mirror, assuming negligible recoil of the mirror. The change in momentum is twice the photon’s incident momentum, as it is for the particles. Suppose that a beam of light has an intensity

1.0 kW / m^{2}

and falls on a

−2.0 {-m}^{2}

area of a mirror and reflects from it. (a) Calculate the energy reflected in 1.00 s. (b) What is the momentum imparted to the mirror? (c) Use Newton’s second law to find the force on the mirror. (d) Does the assumption of no-recoil for the mirror seem reasonable?

Answer

a. 2 kJ; b.

1.33 \times 10^{− 5} kg \cdot m/s;

1.33 \times 10^{− 5} N;

d. yes

6-143 §6.AP source ↗

A 0.75-nm photon is scattered by a stationary electron. The speed of the electron’s recoil is

1.5 \times 10^{6} m/s .

(a) Find the wavelength shift of the photon. (b) Find the scattering angle of the photon.

Answer

a. 0.003 nm; b.

105.56 °

6-145 §6.AP source ↗

A photon of wavelength 700 nm is incident on a hydrogen atom. When this photon is absorbed, the atom becomes ionized. What is the lowest possible orbit that the electron could have occupied before being ionized?

Answer

n = 3

6-147 §6.AP source ↗

Singly ionized atomic helium

{He}^{+ 1}

is a hydrogen-like ion. (a) What is its ground-state radius? (b) Calculate the energies of its four lowest energy states. (c) Repeat the calculations for the

{Li}^{2 +}

ion.

Answer

a_{0} / 2;

−54.4 eV / n^{2};

a_{0} / 3,

−122.4 eV / n^{2}

6-149 §6.AP source ↗

In extreme-temperature environments, such as those existing in a solar corona, atoms may be ionized by undergoing collisions with other atoms. One example of such ionization in the solar corona is the presence of

C^{5 +}

ions, detected in the Fraunhofer spectrum. (a) By what factor do the energies of the

C^{5 +}

ion scale compare to the energy spectrum of a hydrogen atom? (b) What is the wavelength of the first line in the Paschen series of

C^{5 +}

? (c) In what part of the spectrum are these lines located?

Answer

a. 36; b. 18.2 nm; c. UV

6-151 §6.AP source ↗

Experiments are performed with ultracold neutrons having velocities as small as 1.00 m/s. Find the wavelength of such an ultracold neutron and its kinetic energy.

Answer

396 nm; 5.23 neV

6-153 §6.AP source ↗

The spacing between crystalline planes in the NaCl crystal is 0.281 nm, as determined by X-ray diffraction with X-rays of wavelength 0.170 nm. What is the energy of neutrons in the neutron beam that produces diffraction peaks at the same locations as the peaks obtained with the X-rays?

Answer

7.3 keV

6-155 §6.AP source ↗

Calculate the velocity of a

1.0 -μ m

electron and a potential difference used to accelerate it from rest to this velocity.

Answer

728 m/s;

1.5 μ V

6-157 §6.AP source ↗

Find the de Broglie wavelength of an electron accelerated from rest in an X-ray tube in a potential difference of 100 keV. (Rest mass energy of an electron is

E_{0} = 511 keV .)

Answer

λ =

h c / \sqrt{K (2 E_{0} + K)} =

3.705 \times 10^{− 12} m, K =

100 keV

6-159 §6.AP source ↗

Compare the wavelength shift of a photon scattered by a free electron to that of a photon scattered at the same angle by a free proton.

Answer

Δ λ_{c}^{(electron)} / Δ λ_{c}^{(proton)} =

m_{p} / m_{e} =

1836

6-161 §6.AP source ↗

Consider a hydrogen-like ion where an electron is orbiting a nucleus that has charge

q = + Z e .

Derive the formulas for the energy

E_{n}

of the electron in nth orbit and the orbital radius

r_{n} .

Answer

(Proof)

6-163 §6.AP source ↗

An atom can be formed when a negative muon is captured by a proton. The muon has the same charge as the electron and a mass 207 times that of the electron. Calculate the frequency of the photon emitted when this atom makes the transition from

n = 2

to the

n = 1

state. Assume that the muon is orbiting a stationary proton.

Answer

5.1 \times 10^{17} Hz

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Problems and answers are adapted from University Physics Volume 3 by Samuel J. Ling, Jeff Sanny and William Moebs, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/university-physics-volume-3/pages/1-introduction. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.