University Physics with Modern Physics

Roger A. Freedman, Hugh D. Young

Chapter 39

Particles Behaving as Waves - all with Video Answers

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Chapter Questions

Problem 1

(a) An electron moves with a speed of $4.70 \times 10^{6} \mathrm{~m} / \mathrm{s}$. What is its de Broglic wavelength? (b) A proton moves with the same speed. Determine its de Broglie wavelength.

Narendra Kumar

Narendra Kumar

Numerade Educator

Problem 2

For crystal diffraction experiments (discussed in Section 39.1 ) wavelengths on the order of $0.20 \mathrm{nm}$ are often appropriate. Find the energy in electron volts for a particle with this wavelength if the particle is (a) a photon; (b) an electron; (c) an alpha particle $\left(m=6.64 \times 10^{-27} \mathrm{~kg}\right)$

Nishant Kumar

Numerade Educator

Problem 3

An electron has a de Broglie wavelength of $2.80 \times 10^{-10} \mathrm{~m}$ Determine (a) the magnitude of its momentum and (b) its kinetic energy (in joules and in electron volts).

Salamat Ali

Numerade Educator

Problem 4

Wavelength of an Alpha Particle. An alpha particle $\left(m=6.64 \times 10^{-27} \mathrm{~kg}\right)$ emitted in the radioactive decay of uranium- 238 has an energy of $4.20 \mathrm{MeV}$. What is its de Broglie wavelength?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 5

An electron is moving with a speed of $8.00 \times 10^{6} \mathrm{~m} / \mathrm{s}$. What is the speed of a proton that has the same de Broglie wavelength as this electron?

Salamat Ali

Numerade Educator

Problem 6

(a) A nonrelativistic free particle with mass $m$ has kinetic energy $K$. Derive an expression for the de Broglie wavelength of the particle in terms of $m$ and $K$. (b) What is the de Broglie wavelength of an $800 \mathrm{eV}$ electron?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 7

Calculate the de Broglie wavelength of (a) a $50.0 \mathrm{~kg}$ woman jogging leisurely at $2.0 \mathrm{~m} / \mathrm{s},$ (b) a free electron with kinetic energy $2.0 \mathrm{MeV}$. and (c) a free electron with kinetic energy $20 \mathrm{eV}$. Use the proper relativistic expression when necessary.

Narendra Kumar

Numerade Educator

Problem 8

What is the de Broglie wavelength for an electron with speed
(a) $v=0.480 c$ and $(b) v=0.960 c ?$ (Hint: Use the correct relativistic expression for linear momentum if necessary.)

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 9

Wavelength of a Bullet. Calculate the de Broglie wavelength of a $5.00 \mathrm{~g}$ bullet that is moving at $340 \mathrm{~m} / \mathrm{s}$. Will the bullet exhibit wavelike properties?

Narendra Kumar

Numerade Educator

Problem 10

Through what potential difference must electrons be accelerated if they are to have (a) the same wavelength as an x ray of waveIength $0.220 \mathrm{nm}$ and (b) the same energy as the x ray in part (a)?

Narendra Kumar

Numerade Educator

Problem 11

(a) What accelerating potential is needed to produce electrons of wavelength $5.00 \mathrm{nm} ?$ (b) What would be the energy of photons having the same wavelength as these electrons? (c) What would be the wavelength of photons having the same energy as the electrons in part (a)?

Elan Stopnitzky

Elan Stopnitzky

Numerade Educator

Problem 12

A beam of electrons is accelerated from rest through a potential difference of $0.100 \mathrm{kV}$ and then passes through a thin slit. When viewed far from the slit, the diffracted beam shows its first diffraction minima at $\pm 14.6^{\circ}$ from the original direction of the beam.
(a) Do we need to use relativity formulas? How do you know? (b) How wide is the slit?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 13

A beam of neutrons that all have the same energy scatters from atoms that have a spacing of $0.0910 \mathrm{nm}$ in the surface plane of a crystal. The $m=1$ intensity maximum occurs when the angle $\theta$ in Fig. 39.2 is $28.6^{\circ} .$ What is the kinctic energy (in electron volts) of each neutron in the beam?

Narendra Kumar

Numerade Educator

Problem 14

(a) In an electron microscope, what accelerating voltage is needed to produce electrons with wavelength $0.0600 \mathrm{nm} ?$ (b) If protons are used instead of electrons, what accelerating voltage is needed to produce protons with wavelength $0.0600 \mathrm{nm} ?$ (Hint: In each case the initial kinetic energy is negligible.)

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 15

A photon and a free electron each have an energy of $6.00 \mathrm{eV}$.
(a) What is the wavelength of the photon if it is traveling in air? (b) What is the de Broglie wavelength of the electron? (c) Which wavelength is longer?

Narendra Kumar

Numerade Educator

Problem 16

A photon traveling in air has a wavelength of $500 \mathrm{nm},$ and a free electron has a de Broglie wavelength of $500 \mathrm{nm}$. (a) What is the energy of each, in eV? (b) Which energy is greater?

Narendra Kumar

Numerade Educator

Problem 17

A beam of alpha particles is incident on a target of lead. A particular alpha particle comes in "head-on" to a particular lead nucleus and stops $6.50 \times 10^{-14} \mathrm{~m}$ away from the center of the nucleus. (This point is well outside the nucleus.) Assume that the lead nucleus, which has 82 protons, remains at rest. The mass of the alpha particle is $6.64 \times 10^{-27} \mathrm{~kg} .$ (a) Calculate the electrostatic potential energy at the instant that the alpha particle stops. Express your result in joules and in $\mathrm{MeV}$. (b) What initial kinetic energy (in joules and in MeV) did the alpha particle have? (c) What was the initial speed of the alpha particle?

Ren Jie Tuieng

Numerade Educator

Problem 18

A 4.78 MeV alpha particle from a ${ }^{226}$ Ra decay makes a head-on collision with a uranium nucleus. A uranium nucleus has 92 protons. (a) What is the distance of closest approach of the alpha particle to the center of the nucleus? Assume that the uranium nucleus remains at rest and that the distance of closest approach is much greater than the radius of the uranium nucleus. (b) What is the force on the alpha particle at the instant when it is at the distance of closest approach?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 19

A hydrogen atom is in a state with energy $-1.51 \mathrm{eV}$. In the Bohr model, what is the angular momentum of the electron in the atom, with respect to an axis at the nucleus?

Ren Jie Tuieng

Numerade Educator

Problem 20

A hydrogen atom initially in its ground level absorbs a photon, which excites the atom to the $n=3$ level. Determine the wavelength and frequency of the photon.

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 21

A triply ionized beryllium ion, $\mathrm{Be}^{3+}$ (a beryllium atom with three electrons removed), behaves very much like a hydrogen atom except that the nuclear charge is four times as great. (a) What is the ground-level energy of $\mathrm{Be}^{3+}$ ? How does this compare to the groundlevel energy of the hydrogen atom? (b) What is the ionization energy of $\mathrm{Be}^{3+} ?$ How does this compare to the ionization energy of the hydrogen atom? (c) For the hydrogen atom, the wavelength of the photon emitted in the $n=2$ to $n=1$ transition is $122 \mathrm{nm}$ (see Example 39.6 ). What is the wavelength of the photon emitted when a $\mathrm{Be}^{3+}$ ion undergoes this transition? (d) For a given value of $n$, how does the radius of an orbit in $\mathrm{Be}^{3+}$ compare to that for hydrogen?

Ren Jie Tuieng

Numerade Educator

Problem 22

Consider the Bohr-model description of a hydrogen atom.
(a) Calculate $E_{2}-E_{1}$ and $E_{10}-E_{9} .$ As $n$ increases, does the energy separation between adjacent energy levels increase, decrease, or stay the same? (b) Show that $E_{n+1}-E_{n}$ approaches $(27.2 \mathrm{eV}) / n^{3}$ as $n$ becomes large. (c) How does $r_{n+1}-r_{n}$ depend on $n ?$ Does the radial distance between adjacent orbits increase, decrease, or stay the same as $n$ increases?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 23

(a) Using the Bohr model, calculate the speed of the electron in a hydrogen atom in the $n=1,2,$ and 3 levels. (b) Calculate the orbital period in each of these levels.
(c) The average lifetime of the first excited level of a hydrogen atom is $1.0 \times 10^{-8} \mathrm{~s}$. In the Bohr model, how many orbits does an electron in the $n=2$ level complete before returning to the ground level?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 24

An electron is in a bound state of a hydrogen atom. The energy state of the atom is labeled with principal quantum number $n .$ In the Bohr model description of this bound state, the electron has linear momentum $p=6.65 \times 10^{-25} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s} .$ In the Bohr model description. what are (a) the kinetic energy of the electron, (b) the angular momentum of the electron, and (c) the quantum number $n ?$

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 25

$\mathrm{CP} \quad$ The energy-level scheme for the hypothetical oneelectron element Searsium is shown in Fig. E39.25. The potential energy is taken to be zero for an electron at an infinite distance from the nucleus. (a) How much energy (in electron volts) does it take to ionize an electron from the ground level?
(b) An $18 \mathrm{eV}$ photon is absorbed by a Searsium atom in its ground level. As the atom returns to its ground level, what possible energies can the emitted photons have? Assume that there can be transitions between all pairs of levels. (c) What will happen if a photon with an energy of $8 \mathrm{eV}$ strikes a Searsium atom in its ground level? Why? (d) Photons emitted in the Searsium transitions $n=3 \rightarrow n=2$ and $n=3 \rightarrow n=1$ will eject photoelectrons from an unknown metal, but the photon emitted from the transition $n=4 \rightarrow n=3$ will not. What are the limits (maximum and minimum possible values) of the work function of the metal?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 26

A positronium atom consists of a positron and an electron. In Bohr-like model, the two particles rotate in circles about their common center of mass. (a) Calculate the reduced mass of a positronium atom in terms of the mass of an electron.
(b) Determine the orbital radius of its ground-state electron. (c) Find its ground-state energy. (d) The longest visible-light emission wavelength for ordinary hydrogen is $656.3 \mathrm{nm}$ in air and is for the $n=3$ to $n=2$ transition. Calculate the wavelength for the same transition in positronium.

Eduard Sanchez

Numerade Educator

Problem 27

In a set of experiments on a hypothetical one-electron atom, you measure the wavelengths of the photons emitted from transitions ending in the ground level $(n=1),$ as shown in the energylevel diagram in Fig. E39.27. You also observe that it takes $17.50 \mathrm{eV}$ to ionize this atom. (a) What is the energy of the atom in each of the levels $(n=1, n=2,$ etc. $)$ shown in the figure?
(b) If an electron made a transition from the $n=4$ to the $n=2$ level, what wavelength of light would it emit?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 28

Find the longest and shortest wavelengths in the Lyman and Paschen series for hydrogen. In what region of the electromagnetic spectrum does each series lie?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 29

(a) An atom initially in an energy level with $E=-6.52 \mathrm{eV}$ absorbs a photon that has wavelength $860 \mathrm{nm}$. What is the internal energy of the atom after it absorbs the photon? (b) An atom initially in an cnergy level with $E=-2.68 \mathrm{eV}$ emits a photon that has wavelength $420 \mathrm{nm}$. What is the internal energy of the atom after it emits the photon?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 30

Use Balmer's formula to calculate (a) the wavelength,
(b) the frequency, and (c) the photon energy for the $\mathrm{H}_{\gamma}$ line of the Balmer series for hydrogen.

Katie Mcalpine

Numerade Educator

Problem 31

BIO Laser Surgery. Using a mixture of $\mathrm{CO}_{2}, \mathrm{~N}_{2},$ and sometimes He, $\mathrm{CO}_{2}$ lasers emit a wavelength of $10.6 \mu \mathrm{m}$. At power outputs of $0.100 \mathrm{~kW},$ such lasers are used for surgery. How many photons per second does a $\mathrm{CO}_{2}$ laser deliver to the tissue during its use in an operation?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 32

BIO Removing Birthmarks. Pulsed dye lasers emit light of wavelength $585 \mathrm{nm}$ in $0.45 \mathrm{~ms}$ pulses to remove skin blemishes such as birthmarks. The beam is usually focused onto a circular spot $5.0 \mathrm{~mm}$ in diameter. Suppose that the output of one such laser is $20.0 \mathrm{~W}$. (a) What is the energy of each photon, in eV? (b) How many photons per square millimeter are delivered to the blemish during each pulse?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 33

How many photons per second are emitted by a $7.50 \mathrm{~mW}$ $\mathrm{CO}_{2}$ laser that has a wavelength of $10.6 \mu \mathrm{m} ?$

Ren Jie Tuieng

Numerade Educator

Problem 34

BIO PRK Surgery. Photorefractive keratectomy (PRK) is a laser-based surgical procedure that corrects near- and farsightedness by removing part of the lens of the eye to change its curvature and hence focal length. This procedure can remove layers $0.25 \mu \mathrm{m}$ thick using pulses lasting 12.0 ns from a laser beam of wavelength 193 nm. Low-intensity beams can be used because each individual photon has enough energy to break the covalent bonds of the tissue. (a) In what part of the electromagnetic spectrum does this light lic? (b) What is the energy of a single photon? (c) If a $1.50 \mathrm{~mW}$ beam is used, how many photons are delivered to the lens in each pulse?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 35

Figure $39.19 \mathrm{a}$ shows the energy levels of the sodium atom. The two lowest excited levels are shown in columns labeled ${ }^{2} P_{3 / 2}$ and ${ }^{2} P_{1 / 2}$. Find the ratio of the number of atoms in a ${ }^{2} P_{3 / 2}$ state to the number in a ${ }^{2} P_{1 / 2}$ state for a sodium gas in thermal equilibrium at $500 \mathrm{~K}$. In which state are more atoms found?

Ren Jie Tuieng

Numerade Educator

Problem 36

The temperature of a blackbody is changed so that the intensity $I$ of radiation from the blackbody increases by a factor of $16 . \mathrm{By}$ what factor does the peak wavelength $\lambda_{m}$ change?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 37

$\mathrm{A} 100 \mathrm{~W}$ incandescent light bulb has a cylindrical tungsten filament $30.0 \mathrm{~cm}$ long. $0.40 \mathrm{~mm}$ in diameter, and with an emissivity of $0.26 .$ (a) What is the temperature of the filament? (b) For what wavelength does the spectral emittance of the bulb peak?
(c) Incandescent light bulbs are not very efficient sources of visible light. Explain why this is so.

Ren Jie Tuieng

Numerade Educator

Problem 38

Determine $\lambda_{\mathrm{m}},$ the wavelength at the peak of the Planck distribution, and the corresponding frequency $f,$ at these temperatures:
(a) $3.00 \mathrm{~K} ;$ (b) $300 \mathrm{~K} ;$ (c) $3000 \mathrm{~K}$.

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 39

Radiation has been detected from space that is characteristic of an ideal radiator at $T=2.728 \mathrm{~K}$. (This radiation is a relic of the Big Bang at the beginning of the universe.) For this temperature, at what wavelength does the Planck distribution peak? In what part of the electromagnetic spectrum is this wavelength?

Salamat Ali

Numerade Educator

Problem 40

The wavelength $10.0 \mu \mathrm{m}$ is in the infrared region of the electromagnetic spectrum, whereas $600 \mathrm{nm}$ is in the visible region and $100 \mathrm{nm}$ is in the ultraviolet. What is the temperature of an ideal blackbody for which the peak wavelength $\lambda_{\mathrm{m}}$ is equal to each of these wavelengths?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 41

Two stars, both of which behave like ideal blackbodies, radiate the same total energy per second. The cooler one has a surface temperature $T$ and a diameter 3.0 times that of the hotter star. (a) What is the temperature of the hotter star in terms of $T ?$ (b) What is the ratio of the peak-intensity wavelength of the hot star to the peak-intensity wavelength of the cool star?

Salamat Ali

Numerade Educator

Problem 42

Sirius $\mathrm{B}$. The brightest star in the sky is Sirius, the Dog Star. It is actually a binary system of two stars, the smaller one (Sirius B) being a white dwarf. Spectral analysis of Sirius B indicates that its surface temperature is $24,000 \mathrm{~K}$ and that it radiates energy at a total rate of $1.0 \times 10^{25} \mathrm{~W}$. Assume that it behaves like an ideal blackbody.
(a) What is the total radiated intensity of Sirius $\mathrm{B}$ ? (b) What is the peak-intensity wavelength? Is this wavelength visible to humans? (c) What is the radius of Sirius B? Express your answer in kilometers and as a fraction of our sun's radius.
(d) Which star radiates more total energy per second, the hot Sirius $\mathrm{B}$ or the (relatively) cool sun with a surface temperature of $5800 \mathrm{~K}$ ? To find out, calculate the ratio of the total power radiated by our sun to the power radiated by Sirius B.

Salamat Ali

Numerade Educator

Problem 43

(a) The $x$ -coordinate of an electron is measured with an uncertainty of $0.30 \mathrm{~mm}$. What is the $x$ -component of the electron's velocity, $v_{x},$ if the minimum percent uncertainty in a simultaneous measurement of $v_{x}$ is $1.0 \% ?$ (b) Repeat part (a) for a proton.

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 44

A pesky $1.5 \mathrm{mg}$ mosquito is annoying you as you attempt to study physics in your room, which is $5.0 \mathrm{~m}$ wide and $2.5 \mathrm{~m}$ high. You decide to swat the bothersome insect as it flies toward you, but you need to estimate its speed to make a successful hit.
(a) What is the maximum uncertainty in the horizontal position of the mosquito? (b) What limit does the Heisenberg uncertainty principle place on your ability to know the horizontal velocity of this mosquito? Is this limitation a serious impediment to your attempt to swat it?

Elan Stopnitzky

Elan Stopnitzky

Numerade Educator

Problem 45

(a) The uncertainty in the $y$ -component of a proton's position is $2.0 \times 10^{-12} \mathrm{~m} .$ What is the minimum uncertainty in a simultaneous measurement of the $y$ -component of the proton's velocity? (b) The uncertainty in the $z$ -component of an electron's velocity is $0.250 \mathrm{~m} / \mathrm{s}$. What is the minimum uncertainty in a simultaneous measurement of the $z$ -coordinate of the electron?

Salamat Ali

Numerade Educator

Problem 46

A $10.0 \mathrm{~g}$ marble is gently placed on a horizontal tabletop that is $1,75 \mathrm{~m}$ wide. (a) What is the maximum uncertainty in the horizontal position of the marble? (b) According to the Heisenberg uncertainty principle, what is the minimum uncertainty in the horizontal velocity of the marble? (c) In light of your answer to part (b), what is the longest time the marble could remain on the table? Compare this time to the age of the universe, which is approximately 14 billion years. (Hint: Can you know that the horizontal velocity of the marble is exactly zero?)

Andrew Duncan

Numerade Educator

Problem 47

A scientist has devised a new method of isolating individual particles. He claims that this method enables him to detect simultaneously the position of a particle along an axis with a standard deviation of $0.12 \mathrm{nm}$ and its momentum component along this axis with a standard deviation of $3.0 \times 10^{-25} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}$. Use the Heisenberg uncertainty principle to evaluate the validity of this claim.

Ren Jie Tuieng

Numerade Educator

Problem 48

An atom with mass $m$ emits a photon of wavelength $\lambda$.
(a) What is the recoil speed of the atom? (b) What is the kinetic energy $K$ of the recoiling atom? (c) Find the ratio $K / E$, where $E$ is the energy of the emitted photon. If this ratio is much less than unity, the recoil of the atom can be neglected in the emission process. Is the recoil of the atom more important for small or large atomic masses? For long or short wavelengths?
(d) Calculate $K$ (in electron volts) and $K / E$ for a hydrogen atom (mass $1.67 \times 10^{-27} \mathrm{~kg}$ ) that emits an ultraviolet photon of energy $10.2 \mathrm{eV}$. Is recoil an important consideration in this emission process?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 49

The negative muon has a charge equal to that of an electron but a mass that is 207 times as great. Consider a hydrogenlike atom consisting of a proton and a muon. (a) What is the reduced mass of the atom? (b) What is the ground-level energy (in electron volts)? (c) What is the wavelength of the radiation emitted in the transition from the $n=2$ level to the $n=1$ level?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 50

A hydrogen atom in an excited bound state labeled with principal quantum number $n=3$ absorbs a photon that has wavelength $\lambda$. The atom is ionized and the electron has kinetic energy $8.00 \mathrm{eV}$ after it has left the atom. What was the wavelength $\lambda$ of the photon?

Andrew Duncan

Numerade Educator

Problem 51

The wavelengths $\lambda$ in the Pickering emission series are given by $\frac{1}{\lambda}=\left(1.097 \times 10^{7} \mathrm{~m}^{-1}\right)\left[\frac{1}{4}-\frac{1}{(n / 2)^{2}}\right]$ for $n=5,6,7, \ldots$ and were at-
tributed to hydrogen by some scientists. However, Bohr realized that this was not a hydrogen series, but rather belonged to another element, ionized so that it has only one electron. (a) What are the shortest and longest wavelengths in the Pickering series? (b) Which element gives rise to this series, and what is the common final-state quantum number $n_{\mathrm{L}}$ for each transition in the series?

Andrew Duncan

Numerade Educator

Problem 52

In the Bohr model of the hydrogen atom, what is the de Broglie wavelength of the clectron when it is in (a) the $n=1$ level and
(b) the $n=4$ level? In both cases, compare the de Broglie wavelength to the circumference $2 \pi r_{n}$ of the orbit.

Andrew Duncan

Numerade Educator

Problem 53

A sample of hydrogen atoms is irradiated with light with wavelength $85.5 \mathrm{nm},$ and electrons are observed leaving the gas. (a) If each hydrogen atom were initially in its ground level, what would be the maximum kinetic energy in electron volts of these photo-clectrons?
(b) A few electrons are detected with energies as much as $10.2 \mathrm{eV}$ greater than the maximum kinetic energy calculated in part (a). How can this be?

Andrew Duncan

Numerade Educator

Problem 54

CALC Consider two energy levels, 1 and $2,$ of an atom that have nearly the same energies, so that their energies $E_{1}$ and $E_{2}$ relative to the ground state are close in value. When an atom undergoes a transition from one of these states to the ground state, the wavelength of the photon emitted is $\lambda_{1}$ or $\lambda_{2},$ respectively. since $\Delta E=\left|E_{1}-E_{2}\right|$ is small, $\Delta \lambda=\left|\lambda_{1}-\lambda_{2}\right|$ is small. (a) Use the fact that $\Delta \lambda$ is small to derive an expression for $\Delta E$ in terms of $\Delta \lambda$ and $\lambda \approx \lambda_{1}=\lambda_{2}$. (b) The transitions that produce the $589.0 \mathrm{nm}$ and $589.6 \mathrm{nm}$ wavelengths in the atomic emission spectrum of sodium are shown in Fig. $39.19 .$ Use the result from part (a) to calculate the energy difference, in eV, between the two initial states for these two transitions.

Andrew Duncan

Numerade Educator

Problem 55

The Red Super giant Betelgeuse. The star Betelgeuse has a surface temperature of $3000 \mathrm{~K}$ and is 600 times the diameter of our sun. (If our sun were that large, we would be inside it!) Assume that it radiates like an ideal black body. (a) If Betelgeuse were to radiate all of its energy at the peak-intensity wavelength, how many photons per second would it radiate? (b) Find the ratio of the power radiated by Betelgeuse to the power radiated by our sun (at $5800 \mathrm{~K}$ ).

Andrew Duncan

Numerade Educator

Problem 56

CP Light from an ideal spherical black body $15.0 \mathrm{~cm}$ in diameter is analyzed by using a diffraction grating that has 3850 lines/cm. When you shine this light through the grating. you observe that the peak-intensity wavelength forms a first-order bright fringe at $\pm 14.4^{\circ}$ from the central bright fringe. (a) What is the temperature of the black body? (b) How long will it take this sphere to radiate $12.0 \mathrm{MJ}$ of energy at constant temperature?

Andrew Duncan

Numerade Educator

Problem 57

$\mathrm{CP}$ The moon has a mass of $7.35 \times 10^{22} \mathrm{~kg}$, and the length of a sidereal day is 27.3 days. (a) Estimate the de Broglie wavelength of the moon in its orbit around the earth. (b) Using $M_{\text {earth }}$ for the mass of the earth and $M_{\text {moon }}$ for the mass of the moon, we can use Newton's law of gravitation to determine the radius of the moon's orbit in terms of an integer-valued quantum number $m$ as $R_{m}=m^{2} a_{\text {moon }},$ where $a_{\text {moon }}$ is the analog of the Bohr radius for the earth-moon gravitational system. Determine $a_{\text {moon }}$ in terms of Newton's constant $G,$ Planck's constant $h,$ and the masses $M_{\text {earth }}$ and $M_{\text {moon }}$ (c) The mass of the earth is $M_{\text {earth }}=5.97 \times 10^{24} \mathrm{~kg} .$ Estimate the numerical value of $a_{\mathrm{moon}}$
(d) The radius of the moon's orbit is $3.84 \times 10^{8} \mathrm{~m}$. Estimate the moon's quantum number $m$. (e) The quantized energy levels of the moon are given by $E=-E_{0} / m^{2}$. Estimate the quantum ground-state energy $E_{0}$ of the moon.

Andrew Duncan

Numerade Educator

Problem 58

An Ideal Blackbody. A large cavity that has a very small hole and is maintained at a temperature $T$ is a good approximation to an ideal radiator or blackbody. Radiation can pass into or out of the cavity only through the hole. The cavity is a perfect absorber, since any radiation incident on the hole becomes trapped inside the cavity. Such a cavity at $400^{\circ} \mathrm{C}$ has a hole with area $4.00 \mathrm{~mm}^{2}$. How long does it take for the cavity to radiate $100 \mathrm{~J}$ of energy through the hole?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 59

CALC (a) Write the Planck distribution law in terms of the frequency $f,$ rather than the wavelength $\lambda,$ to obtain $I(f) .(\mathrm{b})$ Show that $\int_{0}^{\infty} I(\lambda) d \lambda=\frac{2 \pi^{5} k^{4}}{15 c^{2} h^{3}} T^{4}$ where $I(\lambda)$ is the Planck distribution formula of Eq. (39.24). Hint:
Change the integration variable from $\lambda$ to $f$. You'll need to use the following tabulated integral:
$$\int_{0}^{\infty} \frac{x^{3}}{e^{a x}-1} d x=\frac{1}{240}\left(\frac{2 \pi}{\alpha}\right)^{4}$$ (c) The result of part (b) is $I$ and has the form of the Stefan-Boltzmann law, $I=\sigma T^{4}$ (Eq. 39.19 ). Evaluate the constants in part (b) to show that $\sigma$ has the value given in Section $39.5 .$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 60

CP A beam of $40 \mathrm{eV}$ electrons traveling in the $+x$ -direction passes through a slit that is parallel to the $y$ -axis and $5.0 \mu \mathrm{m}$ wide. The diffraction pattern is recorded on a screen $2.5 \mathrm{~m}$ from the slit. (a) What is the de Broglie wavelength of the electrons?
(b) How much time does it take the electrons to travel from the slit to the screen? (c) Use the width of the central diffraction pattern to calculate the uncertainty in the $y$ -component of momentum of an electron just after it has passed through the slit. (d) Use the result of part
(c) and the Heisenberg uncertainty principle [(Eq. 39.29 ) for $y$ ] to estimate the minimum uncertainty in the $y$ -coordinate of an electron just after it has passed through the slit. Compare your result to the width of the slit.

Andrew Duncan

Numerade Educator

Problem 61

If you could keep utterly motionless, your de Broglie wavelength would be infinite. As soon as you make the slightest motion, however, your wavelength collapses. (a) Estimate the lowest speed you can perceive.
(b) Estimate your wavelength if you moved with that slowest perceptible speed. (c) A grain of sand has a mass of about $0.5 \mathrm{mg}$. Estimate the wavelength of a grain of sand moving at your slowest perceptible speed. (It should be clear that the wave aspects of macroscopic material things are hidden from us by our size.) (d) If nature were to alter her laws so that Planck's constant became $h=1 \mathrm{~J} \cdot \mathrm{s},$ then what would be the wavelength of a grain of sand moving at $1 \mathrm{~m} / \mathrm{s} ?$ (e) Under these same circumstances, estimate your own wavelength if you ran at $2.5 \mathrm{~m} / \mathrm{s}$.
(f) A baseball has a mass of 145 g. Estimate the speed that a baseball would need to have a perceptible diffraction, meaning a central maximum subtending $10^{\circ}$, when thrown through a doorway, if $h$ were $1 \mathrm{~J} \cdot \mathrm{s}$.

Andrew Duncan

Numerade Educator

Problem 62

CP Electrons go through a single slit $300 \mathrm{nm}$ wide and strike a screen $24.0 \mathrm{~cm}$ away. At angles of $\pm 20.0^{\circ}$ from the center of the diffraction pattern, no electrons hit the screen, but electrons hit at all points closer to the center. (a) How fast were these electrons moving when they went through the slit? (b) What will be the next pair of larger angles at which no electrons hit the screen?

Andrew Duncan

Numerade Educator

Problem 63

CP A beam of electrons is accelerated from rest and then passes through a pair of identical thin slits that are $1.25 \mathrm{nm}$ apart. You observe that the first double-slit interference dark fringe occurs at $\pm 18.0^{\circ}$ from the original direction of the beam when viewed on a distant screen. (a) Are these electrons relativistic? How do you know? (b) Through what potential difference were the electrons accelerated?

Andrew Duncan

Numerade Educator

Problem 64

CP Coherent light is passed through two narrow slits whose separation is $20.0 \mu \mathrm{m}$. The second-order bright fringe in the interference pattern is located at an angle of 0.0300 rad. If electrons are used instead of light, what must the kinetic energy (in electron volts) of the electrons be if they are to produce an interference pattern for which the second-order maximum is also at 0.0300 rad?

Andrew Duncan

Numerade Educator

Problem 65

CP An electron beam and a photon beam pass through identical slits. On a distant screen, the first dark fringe occurs at the same angle for both of the beams. The electron speeds are much slower than that of light. (a) Express the energy of a photon in terms of the kinetic energy $K$ of one of the electrons. (b) Which is greater, the energy of a photon or the kinetic energy of an electron?

Andrew Duncan

Numerade Educator

Problem 66

BIO What is the de Broglie wavelength of a red blood cell, with mass $1.00 \times 10^{-11} \mathrm{~g}$, that is moving with a speed of $0.400 \mathrm{~cm} / \mathrm{s} ?$ Do we need to be concerned with the wave nature of the blood cells when we describe the flow of blood in the body?

Andrew Duncan

Numerade Educator

Problem 67

High-speed electrons are used to probe the interior structure of the atomic nucleus. For such electrons the expression $\lambda=h / p$ still holds, but we must use the relativistic expression for momentum, $p=m v / \sqrt{1-v^{2} / c^{2}}$. (a) Show that the speed of an electron that has de Broglie wavelength $\lambda$ is $v=\frac{c}{\sqrt{1+(m c \lambda / h)^{2}}}$ (b) The quantity $h / m c$ equals $2.426 \times 10^{-12} \mathrm{~m}$. (As we saw in Section 38.3 , this same quantity appears in Eq. $(38.7),$ the expression for Compton scattering of photons by electrons.) If $\lambda$ is small compared to $h / m c,$ the denominator in the expression found in part (a) is close to unity and the speed $v$ is very close to $c .$ In this case it is convenient to write $v=(1-\Delta) c$ and express the speed of the electron in terms of $\Delta$ rather than $v .$ Find an expression for $\Delta$ valid when $\lambda \ll h / m c .[$Hint: Use the binomial expansion $(1+z)^{n}=1+n z+\left[n(n-1) z^{2} / 2\right]+\cdots,$ valid for the case $|z|<1 .]$ (c) How fast must an electron move for its de Broglie wavelength to be $1.00 \times 10^{-15} \mathrm{~m}$, comparable to the size of a proton? Express your answer in the form $v=(1-\Delta) c$, and state the value of $\Delta$.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 68

Suppose that the uncertainty of position of an electron is equal to the radius of the $n=1$ Bohr orbit for hydrogen. Calculate the simultaneous minimum uncertainty of the corresponding momentum component, and compare this with the magnitude of the momentum of the electron in the $n=1$ Bohr orbit. Discuss your results.

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 69

$\mathrm{CP}$ (a) A particle with mass $m$ has kinetic energy equal to three times its rest energy.What is the de Broglie wavelength of this particle? (Hint: You must use the relativistic expressions for momentum and kinetic energy: $E^{2}=(p c)^{2}+\left(m c^{2}\right)^{2}$ and $\left.K=E-m c^{2} .\right)$
(b) Determine the numerical value of the kinetic energy (in MeV) and the wavelength (in meters) if the particle in part (a) is (i) an electron and (ii) a proton.

Andrew Duncan

Numerade Educator

Problem 70

Proton Energy in a Nucleus. The radii of atomic nuclei are of the order of $5.0 \times 10^{-15} \mathrm{~m}$. (a) Estimate the minimum uncertainty in the momentum of a proton if it is confined within a nucleus.
(b) Take this uncertainty in momentum to be an estimate of the magnitude of the momentum. Use the relativistic relationship between energy and momentum, Eq. $(37.39),$ to obtain an estimate of the kinetic energy of a proton confined within a nucleus. (c) For a proton to remain bound within a nucleus, what must the magnitude of the (negative) potential energy for a proton be within the nucleus? Give your answer in eV and in MeV. Compare to the potential energy for an electron in a hydrogen atom, which has a magnitude of a few tens of eV. (This shows why the interaction that binds the nucleus together is called the "strong nuclear force.")

Ren Jie Tuieng

Numerade Educator

Problem 71

Electron Energy in a Nucleus. The radii of atomic nuclei are of the order of $5.0 \times 10^{-15} \mathrm{~m}$. (a) Estimate the minimum uncertainty in the momentum of an electron if it is confined within a nucleus.
(b) Take this uncertainty in momentum to be an estimate of the magnitude of the momentum. Use the relativistic relationship between energy and momentum, Eq. (37.39), to obtain an estimate of the kinetic energy of an electron confined within a nucleus
(c) Compare the energy calculated in part (b) to the magnitude of the Coulomb potential energy of a proton and an electron separated by $5.0 \times 10^{-15} \mathrm{~m} .$ On the basis of your result, could there be electron:
within the nucleus? (Note: It is interesting to compare this result to that of Problem $39.70 .$ )

Andrew Duncan

Numerade Educator

Problem 72

The neutral pion $\left(\pi^{0}\right)$ is an unstable particle produced in high-energy particle collisions. Its mass is about 264 times that of the electron, and it exists for an average lifetime of $8.4 \times 10^{-17} \mathrm{~s}$ before decaying into two gamma-ray photons. Using the relationship $E=m c^{2}$ between rest mass and energy, find the uncertainty in the mass of the particle and express it as a fraction of the mass.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 73

Doorway Diffraction. If your wavelength were $1.0 \mathrm{~m},$ you would undergo considerable diffraction in moving through a doorway.
(a) What must your speed be for you to have this wavelength? (Assume that your mass is $60.0 \mathrm{~kg} .$ ) (b) At the speed calculated in part (a), how many years would it take you to move $0.80 \mathrm{~m}$ (one step)? Will you notice diffraction effects as you walk through doorways?

Elan Stopnitzky

Elan Stopnitzky

Numerade Educator

Problem 74

Atomic Spectra Uncertainties. A certain atom has an energy level $2.58 \mathrm{eV}$ above the ground level. Once excited to this level, the atom remains in this level for $1.64 \times 10^{-7} \mathrm{~s}$ (on average) before emitting a photon and returning to the ground level. (a) What is the energy of the photon (in electron volts)? What is its wavelength (in nanometers)? (b) What is the smallest possible uncertainty in energy of the photon? Give your answer in electron volts. (c) Show that $|\Delta E / E|=|\Delta \lambda / \lambda| $
if $|\Delta \lambda / \lambda| \ll 1 .$ Use this to calculate the magnitude of the smallest possible uncertainty in the wavelength of the photon. Give your answer in nanometers.

Andrew Duncan

Numerade Educator

Problem 75

For x rays with wavelength $0.0300 \mathrm{nm},$ the $m=1$ intensity maximum for a crystal occurs when the angle $\theta$ in Fig. 39.2 is $35.8^{\circ} .$ At what angle $\theta$ does the $m=1$ maximum occur when a beam of $4.50 \mathrm{keV}$ electrons is used instead? Assume that the electrons also scatter from the atoms in the surface plane of this same crystal.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 76

A certain atom has an energy state $3.50 \mathrm{eV}$ above the ground state. When excited to this state, the atom remains for $2.0 \mu \mathrm{s},$ on average, before it emits a photon and returns to the ground state. (a) What are the energy and wavelength of the photon? (b) What is the smallest possible uncertainty in energy of the photon?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 77

BIO Structure of a Virus. To investigate the structure of extremely small objects, such as viruses, the wavelength of the probing wave should be about one-tenth the size of the object for sharp images. But as the wavelength gets shorter, the energy of a photon of light gets greater and could damage or destroy the object being studied. One alternative is to use electron matter waves instead of light. Viruses vary considerably in size, but $50 \mathrm{nm}$ is not unusual. Suppose you want to study such a virus, using a wave of wavelength $5.00 \mathrm{nm}$. (a) If you use light of this wavelength, what would be the energy (in eV) of a single photon? (b) If you use an electron of this wavelength, what would be its kinetic energy (in eV)? Is it now clear why matter waves (such as in the electron microscope) are often preferable to electromagnetic waves for studying microscopic objects?

Andrew Duncan

Numerade Educator

Problem 78

CALC Zero-Point Energy. Consider a particle with mass $m$ moving in a potential $U=\frac{1}{2} k x^{2},$ as in a mass-spring system. The total energy of the particle is $E=\left(p^{2} / 2 m\right)+\frac{1}{2} k x^{2}$. Assume that $p$ and $x$ are approximately related by the Heisenberg uncertainty principle, so $p x=h$ (a) Calculate the minimum possible value of the energy $E,$ and the value of $x$ that gives this minimum $E .$ This lowest possible energy, which is not zero, is called the zero-point energy. (b) For the $x$ calculated in part (a), what is the ratio of the kinetic to the potential energy of the particle?

Andrew Duncan

Numerade Educator

Problem 79

CALC A particle with mass $m$ moves in a potential energy$U(x)=A|x|,$ where $A$ is a positive constant. In a simplified picture, quarks (the constituents of protons, neutrons, and other particles, as will be described in Chapter 44 ) have a potential energy of interaction of approximately this form, where $x$ represents the separation between a pair of quarks. Because $U(x) \rightarrow \infty$ as $x \rightarrow \infty,$ it's not possible to separate quarks from each other (a phenomenon called quark confinement).
(a) Classically, what is the force acting on this particle as a function of $x ?$
(b) Using the uncertainty principle as in Problem 39.78 , determine approximately the zero-point energy of the particle.

Andrew Duncan

Numerade Educator

Problem 80

Imagine another universe in which the value of Planck's constant is $0.0663 \mathrm{~J} \cdot \mathrm{s},$ but in which the physical laws and all other physical constants are the same as in our universe. In this universe, two physics students are playing catch. They are $12 \mathrm{~m}$ apart, and one throws a $0.25 \mathrm{~kg}$ ball directly toward the other with a speed of $6.0 \mathrm{~m} / \mathrm{s}$. (a) What is the uncertainty in the ball's horizontal momentum, in a direction perpendicular to that in which it is being thrown, if the student throwing the ball knows that it is located within a cube with volume $125 \mathrm{~cm}^{3}$ at the time she throws it? (b) By what horizontal distance could the ball miss the second student?

Jonathan Everett

Jonathan Everett

Numerade Educator

Problem 81

DATA For your work in a mass spectrometry lab, you are investigating the absorption spectrum of one-electron ions. To maintain the atoms in an ionized state, you hold them at low density in an ion trap, a device that uses a configuration of electric fields to confine ions. The majority of the ions are in their ground state, so that is the initial state for the absorption transitions that you observe. (a) If the longest wavelength that you observe in the absorption spectrum is $13.56 \mathrm{nm},$ what is the atomic number $Z$ for the ions? (b) What is the next shorter wavelength that the ions will absorb? (c) When one of the ions absorbs a photon of wavelength $6.78 \mathrm{nm},$ a free electron is produced. What is the kinetic energy (in electron volts) of the electron?

Andrew Duncan

Numerade Educator

Problem 82

DATA In the crystallography lab where you work, you are given a single crystal of an unknown substance to identify. To obtain one piece of information about the substance, you repeat the DavissonGermer experiment to determine the spacing of the atoms in the surface planes of the crystal. You start with electrons that are essentially stationary and accelerate them through a potential difference of magnitude $V_{a c}$ The electrons then scatter off the atoms on the surface of the crystal (as in Fig. $39.3 \mathrm{~b}$ ). Next you measure the angle $\theta$ that locates the first-order diffraction peak. Finally, you repeat the measurement for different values of $V_{\mathrm{ac}}$. Your results are given in the table. $$\begin{array}{l|rrrrrr}
\boldsymbol{V}_{\mathbf{a c}}(\mathbf{V}) & 106.3 & 69.1 & 49.9 & 25.2 & 16.9 & 13.6 \\
\hline \boldsymbol{\theta}\left({ }^{\circ} \mathbf{)}\right. & 20.4 & 24.8 & 30.2 & 45.5 & 59.1 & 73.1
\end{array}$$ (a) Graph your data in the form $\sin \theta$ versus $1 / \sqrt{V_{\mathrm{ac}}} .$ What is the slope of the straight line that best fits the data points when plotted in this way?
(b) Use your results from part (a) to calculate the value of $d$ for this crystal.

Declan Nell

Numerade Educator

Problem 83

DATA As an amateur astronomer, you are studying the apparent brightness of stars. You know that a star's apparent brightness depends on its distance from the earth and also on the fraction of its radiated energy that is in the visible region of the electromagnetic spectrum. But, as a first step, you search the Internet for information on the surface temperatures and radii of some selected stars so that you can calculate their total radiated power. You find the data given in the table. $$ \begin{array}{l|cccc}
\text { Star } & \text { Polaris } & \text { Vega } & \text { Antares } & \alpha \text { Centauri B } \\
\hline \text { Surface temperature (K) } & 6015 & 9602 & 3400 & 5260 \\
\hline \begin{array}{l}
\text { Radius relative to that of the } \\
\text { sun }\left(R_{\text {sun }}\right)
\end{array} & 46 & 2.73 & 883 & 0.865 \\
& & & &
\end{array}$$ The radius is given in units of the radius of the sun, $R_{\text {sun }}=6.96 \times 10^{8} \mathrm{~m}$ The surface temperature is the effective temperature that gives the measured photon luminosity of the star if the star is assumed to radiate as an ideal blackbody. The photon luminosity is the power emitted in the form of photons. (a) Which star in the table has the greatest radiated power? (b) For which of these stars, if any, is the peak wavelength $\lambda_{\mathrm{m}}$ in the visible range $(380-750 \mathrm{nm}) ?$ (c) The sun has a total radiated power of $3.85 \times 10^{26} \mathrm{~W} .$ Which of these stars, if any, have a total radiated power less than that of our sun?

Umar Sohail Qureshi

Umar Sohail Qureshi

Numerade Educator

Problem 84

CP CALC An advanced civilization uses a fusion reactor to maintain a high temperature on a spherical blackbody whose radiation supplies energy for use on a spacecraft. The blackbody has radius $r=1.23 \mathrm{~m}$ and is surrounded by a larger sphere whose inside surface is densely lined with photocells (Fig. $\mathbf{P} 39.84$ ). These capture all photons that have energy within $1.00 \%$ of a central value $E_{0}$. The photocells are designed so that $E_{0}$ corresponds to the peak cnergy radiated from the blackbody. The range $\Delta E$ is sufficiently small that we can approximate the spectral emittance as constant over the range $E_{0} \pm \Delta E .$ This device generates an emf across a motor of resistance $1.00 \mathrm{k} \Omega$. (a) The spectral emittance $I=\int I(\lambda) d \lambda$ can be structured as $\int I(E) d E$ by parameterizing the integral using $E=h c / \lambda$ Use Eq. (39.24) to determine $I(E) .$ Be careful with the differential factor.
(b) Determine the energy $E_{0}$ that maximizes $I(E),$ and determine $I_{\max }=I\left(E_{0}\right)$ in terms of the temperature $T$. (Hint: You have to solve the equation $3-x=3 e^{-x} .$ Use the solution $x=2.821,$ which is accurate to four significant figures.) (c) If the device is designed for $T=1000 \mathrm{~K}$, then how much power is supplied by the motor, and what is the current $I ?$ (Hint: Integrate the spectral emittance around the blackbody.) (d) If the device is designed for $T=5000 \mathrm{~K}$, then how much power is supplied and what is the current?
(figure cant copy)

Khaled Yasein

Numerade Educator

Problem 85

CP An alpha particle is incident with kinetic energy $K$ on a
gold nucleus at rest. The aim is direct. (a) If $m$ is the mass of an alpha particle and $M$ is the mass of a gold nucleus, solve the classical conditions for energy and momentum conservation to determine the recoil speed $V$ of the nucleus after the collision.
(b) Determine an expression for the fractional energy lost to the nucleus. (c) Is your result independent of the initial kinetic energy? (d) An alpha particle has mass $m=6.64 \times 10^{-27} \mathrm{~kg},$ and a gold nucleus has mass $M=1.32 \times 10^{-25} \mathrm{~kg}$
If $K=5.00 \mathrm{MeV},$ then what is the speed $V$ as a fraction of $c,$ and what proportion of the original energy is transferred to the gold nucleus?
(e) According to the classical analysis, what speed of the incident alpha particle would result in a nuclear speed $V$ of $0.10 \mathrm{c} ?$ (f) Is that possible?

Eduard Sanchez

Numerade Educator

Problem 86

CP CALC You have entered a contest in which the contestants drop a marble with mass $20.0 \mathrm{~g}$ from the roof of a building onto a small target $25.0 \mathrm{~m}$ below. From uncertainty considerations, what is the typical distance by which you'll miss the target, given that you aim with the highest possible precision? (Hint: The uncertainty $\Delta x_{f}$ in the $x$ -coordinate of the marble when it reaches the ground comes in part from the uncertainty $\Delta x_{i}$ in the $x$ -coordinate initially and in part from the initial uncertainty in $v_{x}$. The latter gives rise to an uncertainty $\Delta v_{x}$ in the horizontal motion of the marble as it falls. The values of $\Delta x_{i}$ and $\Delta v_{x}$ are related by the uncertainty principle. A small $\Delta x_{i}$ gives rise to a large $\Delta v_{x}$, and vice versa. Find the value of $\Delta x_{i}$ that gives the smallest total uncertainty in $x$ at the ground. Ignore any effects of air resistance.

Eduard Sanchez

Numerade Educator

Problem 87

(a) Show that in the Bohr model, the frequency of revolution of an electron in its circular orbit around a stationary hydrogen nucleus is $f=m e^{4} / 4 \epsilon_{0}^{2} n^{3} h^{3} .(b)$ In classical physics, the frequency of revolution of the electron is equal to the frequency of the radiation that it emits. Show that when $n$ is very large, the frequency of revolution does indeed equal the radiated frequency calculated from Eq. (39.5) for a transition from $n_{1}=n+1$ to $n_{2}=n .$ (This illustrates Bohr's correspondence principle, which is often used as a check on quantum calculations. When $n$ is small, quantum physics gives results that are very different from those of classical physics. When $n$ is large, the differences are not significant, and the two methods then "correspond." In fact, when Bohr first tackled the hydrogen atom problem, he sought to determine $f$ as a function of $n$ such that it would correspond to classical results for large $n .$ )

Ren Jie Tuieng

Numerade Educator

Problem 88

How does the wavelength of a helium ion compare to that of an electron accelerated through the same potential difference? (a) The helium ion has a longer wavelength, because it has greater mass.
(b) The helium ion has a shorter wavelength, because it has greater mass. (c) The wavelengths are the same, because the kinetic energy is the same. (d) The wavelengths are the same, because the electric charge is the same.

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 89

Can the first type of helium-ion microscope, used for surface imaging. produce helium ions with a wavelength of $0.1 \mathrm{pm} ?$ (a) Yes; the voltage required is $21 \mathrm{kV}$. (b) Yes; the voltage required is $42 \mathrm{kV}$. (c) No; a voltage higher than $50 \mathrm{kV}$ is required. (d) No; a voltage lower than $10 \mathrm{kV}$ is required.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 90

Why is it easier to use helium ions rather than neutral helium atoms in such a microscope? (a) Helium atoms are not electrically charged, and only electrically charged particles have wave properties.
(b) Helium atoms form molecules, which are too large to have wave properties. (c) Neutral helium atoms are more difficult to focus with electric and magnetic fields. (d) Helium atoms have much larger mass than helium ions do and thus are more difficult to accelerate.

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 91

In the second type of helium-ion microscope, a $1.2 \mathrm{MeV}$ ion passing through a cell loses $0.2 \mathrm{MeV}$ per $\mu \mathrm{m}$ of cell thickness. If the energy of the ion can be measured to $6 \mathrm{keV},$ what is the smallest difference in thickness that can be discemed?
(a) $0.03 \mu \mathrm{m}$
(b) $0.06 \mu \mathrm{m}$
(c) $3 \mu \mathrm{m} ;$ (d) $6 \mu \mathrm{m}$

Andrew Duncan

Numerade Educator