College Physics

Jerry D. Wilson, Anthony J. Buffa, Bo Lou

Chapter 27

Quantum Physics - all with Video Answers

Educators

Chapter Questions

Problem 1

The walls of a blackbody cavity are at a temperature of $27^{\circ} \mathrm{C} .$ What is the wavelength of the radiation of maximum intensity?

Sheh Lit Chang

Sheh Lit Chang

University of Washington

Problem 2

Find the approximate temperature of a red star that emits light with a wavelength of maximum emission of $700 \mathrm{nm}$ (deep red).

Sheh Lit Chang

University of Washington

Problem 3

What are the wavelength and frequency of the most intense radiation component from a blackbody with a temperature of $0^{\circ} \mathrm{C} ?$

Sheh Lit Chang

University of Washington

Problem 4

If you have a fever, is the wavelength of the radiation component of maximum intensity emitted by your body (1) greater, (2) the same, or (3) smaller as compared with its value when your temperature is normal? Why?
(b) Assume that human skin has a temperature of $34^{\circ} \mathrm{C}$. What is the wavelength of the radiation component of maximum intensity emitted by our bodies? In what region of the EM spectrum is this wavelength?

Sheh Lit Chang

University of Washington

Problem 5

A "red-hot" object is measured to have a frequency of $1.0 \times 10^{14} \mathrm{~Hz}$. What is the Celsius temperature of the object?

Sheh Lit Chang

University of Washington

Problem 6

What is the minimum energy of a thermal oscillator in a blackbody producing radiation at $\lambda_{\max }$ at a temperature of $212^{\circ} \mathrm{F} ?$

Sheh Lit Chang

University of Washington

Problem 7

If the minimum energy of a thermal oscillator in a black body's most intense radiation is $3.5 \times 10^{-19} \mathrm{~J},$ what is the Celsius temperature of the black body?

Sheh Lit Chang

University of Washington

Problem 8

The temperature of a black body increases from $200^{\circ} \mathrm{C}$ to $400{ }^{\circ} \mathrm{C}$. (a) Will the frequency of the most intense spectral component emitted by this black body (1) increase, but not double; (2) double; (3) be reduced in half; or (4) decrease, but not by half? Why? (b) What is the change in the frequency of the most intense spectral component of this black body?

Sheh Lit Chang

University of Washington

Problem 9

The temperature of a blackbody is $500^{\circ} \mathrm{C}$. If the intensity of the emitted radiation, $2.0 \mathrm{~W} / \mathrm{m}^{2},$ were due entirely to the most intense frequency component, how many quanta of radiation would be emitted per second per square meter?

Sheh Lit Chang

University of Washington

Problem 10

The wavelength at which the Sun emits its most intense light is about $550 \mathrm{nm}$. Assuming the Sun radiates as a perfect black body, estimate (a) its surface tempera ture and (b) its total emitted power. [Hint: The Sun's data can be found in the inside back cover of the book.

Sheh Lit Chang

University of Washington

Problem 11

Each photon in a beam of light has an energy of $6.50 \times 10^{-19} \mathrm{~J} .$ What is the light's wavelength? What type of light is this?

Sheh Lit Chang

University of Washington

Problem 12

Compared with a quantum of red light $(\lambda=700 \mathrm{nm}),$ a quantum of violet light $(\lambda=400 \mathrm{nm})$ has (1) more, (2) the same amount of, (3) less energy. Why? (b) Determine the ratio of the photon energy associated with violet light to that related to red light.

Sheh Lit Chang

University of Washington

Problem 13

A source of UV light has a wavelength of $150 \mathrm{nm}$. How much energy does one of its photons have expressed in (a) joules and (b) electron-volts?

Sheh Lit Chang

University of Washington

Problem 14

The work function of a surface is $5.0 \times 10^{-19} \mathrm{~J}$. If light of wavelength of $300 \mathrm{nm}$ is incident on the surface, what is the maximum kinetic energy of the photoelectrons ejected from the surface?

Sheh Lit Chang

University of Washington

Problem 15

When light of wavelength of $200 \mathrm{nm}$ is incident on a surface, the maximum kinetic energy of the photoelectrons is measured to be $6.0 \times 10^{-19} \mathrm{~J}$. What is the work function of the surface?

Sheh Lit Chang

University of Washington

Problem 16

The photo electrons ejected from a surface require a stopping voltage of $5.0 \mathrm{~V}$. If the intensity of the light is tripled, what is the stopping voltage?

Sheh Lit Chang

University of Washington

Problem 17

What is the longest wavelength of light that can cause the release of electrons from a metal that has a work function of $3.50 \mathrm{eV} ?$

Sheh Lit Chang

University of Washington

Problem 18

Assume that a $100-\mathrm{W}$ light bulb gives off $2.50 \%$ of its energy as visible light. How many photons of visible light are given off in 1.00 min? (Use an average visible wavelength of $550 \mathrm{nm}$.)

Sheh Lit Chang

University of Washington

Problem 19

A metal with a work function of $2.40 \mathrm{eV}$ is illuminated by a beam of monochromatic light. If the stopping potential is $2.50 \mathrm{~V}$, what is the wavelength of the light?

Sheh Lit Chang

University of Washington

Problem 20

The work function of metal A is less than that of metal B. (a) The threshold wavelength for metal A is
(1) shorter than, (2) the same as, (3) longer than that of metal B. Why? (b) If the threshold wavelength for metal B is $620 \mathrm{nm}$ and the work function of metal $\mathrm{A}$ is twice that of metal B, what is the threshold wavelength for metal A?

Sheh Lit Chang

University of Washington

Problem 21

Figure 27.21 shows a graph of stopping potential versus frequency for a photoelectric material. Determine
(a) Planck's constant and (b) the work function of the material from the data contained in the graph.

Sheh Lit Chang

University of Washington

Problem 22

The photoelectric effect threshold wavelength for a certain metal is $400 \mathrm{nm}$. Calculate the maximum speed of photoelectrons if we use light having a wavelength of
(a) $300 \mathrm{nm}$
(b) $400 \mathrm{nm},$ and
(c) $500 \mathrm{nm}$

Sheh Lit Chang

University of Washington

Problem 23

When light of wavelength of $250 \mathrm{nm}$ is incident on a metal surface, the maximum speed of the photoelectrons is $4.0 \times 10^{5} \mathrm{~m} / \mathrm{s}$. What is the work function of the metal in eV?

Sheh Lit Chang

University of Washington

Problem 24

The work function of a material is $3.5 \mathrm{eV}$. If the material is illuminated with monochromatic light $(\lambda=300 \mathrm{nm}),$ what are (a) the stopping potential and (b) the cutoff frequency?

Sheh Lit Chang

University of Washington

Problem 25

Blue light with a wavelength of $420 \mathrm{nm}$ is incident on a certain material and causes the emission of photoelectrons with a maximum kinetic energy of $1.00 \times 10^{-19} \mathrm{~J}$. (a) What is the stopping voltage? (b) What is the material's work function? (c) What is the stopping voltage if red light $(\lambda=700 \mathrm{nm})$ is used instead? Explain.

Sheh Lit Chang

University of Washington

Problem 26

When the surface of a particular material is illuminated with monochromatic light of various frequencies, the stopping potentials for the photoelectrons are determined to be the following:Plot these data, and from the graph determine Planck's constant and the metal's work function.

Sheh Lit Chang

University of Washington

Problem 27

When a certain photoelectric material is illuminated with red light $(\lambda=700 \mathrm{nm})$ and then blue light $(\lambda=400 \mathrm{nm}),$ it is found that the maximum kinetic energy of the photoelectrons resulting from the blue light is twice that from red light. What is the work function of the material?

Sheh Lit Chang

University of Washington

Problem 28

What is half the maximum wavelength shift for Compton scattering from a free electron?

Sheh Lit Chang

University of Washington

Problem 29

When the wavelength shift for Compton scattering from a free electron is a maximum, what is the scattering angle?

Sheh Lit Chang

University of Washington

Problem 30

What is the change in wavelength when monochromatic X-rays are scattered by electrons through an angle of $30^{\circ} ?$

Sheh Lit Chang

University of Washington

Problem 31

A monochromatic beam of X-rays with a wavelength of $0.280 \mathrm{nm}$ is scattered by a metal foil. If the scattered beam has a wavelength of $0.281 \mathrm{nm},$ what is the observed scattering angle?

Sheh Lit Chang

University of Washington

Problem 32

$X$ -rays with a wavelength of $0.0045 \mathrm{nm}$ are used in a Compton scattering experiment. If the X-rays are scattered through an angle of $45^{\circ},$ what is the wavelength of the scattered radiation?

Sheh Lit Chang

University of Washington

Problem 33

A photon with an energy of $5.0 \mathrm{keV}$ is scattered by a free electron. (a) The recoiling electron could have an energy of (1) zero, (2) less than $5.0 \mathrm{keV}$, but not zero,
(3) $5.0 \mathrm{keV}$. Why? (b) If the wavelength of the scattered photon is $0.25 \mathrm{nm}$, what is the recoiling electron's kinetic energy?

Sheh Lit Chang

University of Washington

Problem 34

X-rays of wavelength $0.01520 \mathrm{nm}$ are scattered from a carbon atom. The wavelength shift is measured to be $0.000326 \mathrm{nm} .$ (a) What is the scattering angle? (b) How much energy, in $\mathrm{eV}$, does each photon impart to each electron?

Sheh Lit Chang

University of Washington

Problem 35

X-rays of frequency $1.210 \times 10^{18} \mathrm{~Hz}$ are scattered from electrons in an aluminum foil. The frequency of the scattered X-rays is $1.203 \times 10^{18} \mathrm{~Hz}$. (a) What is the scattering angle?
(b) What is the recoiling speed of the electrons?

Sheh Lit Chang

University of Washington

Problem 36

The Compton effect can occur for scattering from any particle-for example, from a proton. (a) Compared with the Compton wavelength for an electron, the Compton wavelength for a proton is (1) longer, (2) the same,
(3) shorter. Why? (b) What is the value of the Compton wavelength for a proton? (c) Determine the ratio of the maximum Compton wavelength shift for scattering by an electron to that for scattering by a proton.

Sheh Lit Chang

University of Washington

Problem 37

Find the energy of a hydrogen atom whose electron is in the (a) $n=2$ state and (b) $n=3$ state.

Sheh Lit Chang

University of Washington

Problem 38

Find the radius of the electron orbit in a hydrogen atom for states with the following principal quantum numbers: (a) $n=2,$ (b) $n=4,$ (c) $n=5$.

Sheh Lit Chang

University of Washington

Problem 39

Scientists are now beginning to study "large" atoms, that is, atoms with orbits that are almost large enough to be measured in our everyday units of measurement. For what excited state (give an approximate principal quantum number) of a hydrogen atom would the diameter of the orbit be in the order of $\left(10^{-5} \mathrm{~m}\right)$ - that is, close to the diameter of a human hair?

Sheh Lit Chang

University of Washington

Problem 40

Find the binding energy of the hydrogen electron for states with the following principal quantum numbers:
(a) $n=3,($ b) $n=5$, (c) $n=7$.

Sheh Lit Chang

University of Washington

Problem 41

Find the energy required to excite a hydrogen electron from (a) the ground state to the first excited state and (b) the first excited state to the second excited state. (c) Classify the type of light needed to create each of the transitions.

Sheh Lit Chang

University of Washington

Problem 42

What is the frequency of light that would excite the electron of a hydrogen atom (a) from a state with a principal quantum number of $n=2$ to that with a principal (b) What about from $n=2$ quantum number of $n=5 ?$ to $n=\infty$ ?

Sheh Lit Chang

University of Washington

Problem 43

A hydrogen atom has an ionization energy of $13.6 \mathrm{eV}$. When it absorbs a photon with an energy greater than this energy, the electron will be emitted with some kinetic energy. (a) If the energy of such a photon is doubled, the kinetic energy of the emitted electron will (1) more than double, (2) remain the same, (3) exactly double, (4) increase, but less than double. Why? (b) Photons associated with light of a frequency of $7.00 \times 10^{15} \mathrm{~Hz}$ and $1.40 \times 10^{16} \mathrm{~Hz}$ are absorbed by a hydrogen atom. What is the kinetic energy of the emitted electron?

Sheh Lit Chang

University of Washington

Problem 44

A hydrogen atom in its ground state is excited to the $n=5$ level. It then makes a transition directly to the $n=2$ level before returning to the ground state. (a) What are the wavelengths of the emitted photons? (b) Would any of the emitted light be in the visible region?

Sheh Lit Chang

University of Washington

Problem 45

For which of the following transitions in a hydrogen atom is the photon of longest wavelength emitted:
(1) $n=5$ to $n=3,$ (2) $n=6$ to $n=2,$ or $($ 3) $n=2$ to $n=1 ?(\mathrm{~b})$ Justify your answer mathematically.

Sheh Lit Chang

University of Washington

Problem 46

The hydrogen spectrum has a series of lines called the Lyman series, which results from transitions to the ground state. What is the longest wavelength in this series, and in what region of the EM spectrum does it lie?

Sheh Lit Chang

University of Washington

Problem 47

A hydrogen atom absorbs light of wavelength $486 \mathrm{nm}$. (a) How much energy did the atom absorb?
(b) What are the values of the principal quantum numbers of the initial and final states of this transition?

Sheh Lit Chang

University of Washington

Problem 48

If the electron in a hydrogen atom is to make a transition from the first excited state to the fourth excited state, what frequency of photon is needed? What type of light is this?

Sheh Lit Chang

University of Washington

Problem 49

How many transitions in a hydrogen atom result in the absorption of red light: (1) one, (2) two, (3) three, or (4) four? (b) What are the principal quantum numbers of the initial and final states for this process? (c) What are the energy of the required photon and the wavelength of the light associated with it?

Sheh Lit Chang

University of Washington

Problem 50

What is the binding energy for an electron in the ground state in the following hydrogen-like ions:
(a) $\mathrm{He}^{+}(Z=2)$ and
(b) $\mathrm{Li}^{2+}(Z=3) ?$

Sheh Lit Chang

University of Washington

Problem 51

Show that the orbital speeds of an electron in the Bohr orbits are given (to two significant figures) by $v_{n}=\left(2.2 \times 10^{6} \mathrm{~m} / \mathrm{s}\right) / n$

Sheh Lit Chang

University of Washington

Problem 52

For an electron in the ground state of a hydrogen atom, calculate its (a) potential energy, (b) kinetic energy, and (c) total energy. [Hint: You will need to use the orbital radius. $]$

Sheh Lit Chang

University of Washington

Problem 53

Suppose a hypothetical atom had two metastable excited states, one $2.0 \mathrm{eV}$ above the ground state and one $4.0 \mathrm{eV}$ above the ground state. If used in a laser with transitions only to the ground state, (a) what is the wavelength for each excited state? (b) Which transition is in the visible range?

Sheh Lit Chang

University of Washington

Problem 54

In order to achieve population inversion between two states that are separated by an energy difference of $3.5 \mathrm{eV}$ what wavelength of pumping light should be used?

Sheh Lit Chang

University of Washington

Problem 55

Light of wavelength $340 \mathrm{nm}$ is incident on a metal surface and ejects electrons that have a maximum speed of $3.5 \times 10^{5} \mathrm{~m} / \mathrm{s}$. (a) What is the work function of the metal? (b) What is its stopping voltage?
(c) What is its threshold wavelength?

Sheh Lit Chang

University of Washington

Problem 56

A $10.0-\mathrm{keV}$ X-ray photon is successively scattered by two free electrons initially at rest. In the first case, it is scattered through an angle of $41^{\circ}$; the second scattering is through an angle of $72^{\circ} .$ (a) What is the final photon energy?
(b) How much kinetic energy does each electron receive?

Sheh Lit Chang

University of Washington

Problem 57

Under the right circumstances, if a photon's energy is above a minimum energy level, that energy can be completely converted into creating an electron-positron pair (a positron is identical to an electron except that it has a positive charge). Recall that the energy equivalent of the electron mass is $0.511 \mathrm{MeV}$. Determine (a) the minimum-energy photon required to create such a pair, and (b) the wavelength of light associated with photons of this energy. (c) If a photon with twice the minimum energy were used, what would be the total kinetic energy (electron plus positron) of the two particles after their creation?

Sheh Lit Chang

University of Washington

Problem 58

Consider an electron in its first excited state in a hydrogen atom. Determine its (a) orbital speed, (b) angular speed, (c) linear momentum, and (d) angular momentum.

Sheh Lit Chang

University of Washington

Problem 59

A gamma-ray photon scatters off a free proton (initially at rest) at an angle of $45^{\circ} .$ The wavelength of the scattered light is measured to be $6.20 \times 10^{-13} \mathrm{~m}$. (a) What was the energy of the incoming photon? (b) How much kinetic energy did the proton receive? (You may need to carry an extra figure or two in your intermediate answers.)

Sheh Lit Chang

University of Washington

Problem 60

In the nuclear version of the photoelectric effect (called the photonuclear effect), a high-energy photon is absorbed by an atomic nucleus and a proton is freed from that nucleus. If the minimum energy needed to free a proton from a particular nucleus is $5.00 \mathrm{MeV},$ (a) determine the maximum (threshold) wavelength of light that can cause this. (b) If a photon of half this wavelength is used instead, determine the kinetic energy of the ejected proton.

Sheh Lit Chang

University of Washington

Problem 61

A photon of wavelength $320 \mathrm{nm}$ is absorbed by a hydrogen atom when the electron is in the second excited state. What is the speed of the ionized electron?

Sheh Lit Chang

University of Washington