# Chapter 29

Chapter thumbnail # Chapter 29 : Introduction to Quantum Physics - all with Video Solutions

### Problem 1

A LiBr molecule oscillates with a frequency of $1.7 \times 10^{13} \textrm{ Hz}$. (a) What is the difference in energy in eV between allowed oscillator states? (b) What is the approximate value of n for a state having an energy of 1.0 eV?

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### Problem 3

A physicist is watching a 15-kg orangutan at a zoo swing lazily in a tire at the end of a rope. He (the physicist) notices that each oscillation takes 3.00 s and hypothesizes that the energy is quantized. (a) What is the difference in energy in joules between allowed oscillator states? (b) What is the value of n for a state where the energy is 5.00 J? (c) Can the quantization be observed?

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### Problem 15

Photoelectrons from a material with a binding energy of 2.71 eV are ejected by 420-nm photons. Once ejected, how long does it take these electrons to travel 2.50 cm to a detection device?

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### Problem 16

A laser with a power output of 2.00 mW at a wavelength of 400 nm is projected onto calcium metal. (a) How many electrons per second are ejected? (b) What power is carried away by the electrons, given that the binding energy is 2.71 eV?

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### Problem 17

(a) Calculate the number of photoelectrons per second ejected from a $1.00 \textrm{ mm}^2$ area of sodium metal by 500-nm EM radiation having an intensity of $1.30 \textrm{ kW/m}^2$ (the intensity of sunlight above the Earth’s atmosphere). (b) Given that the binding energy is 2.28 eV, what power is carried away by the electrons? (c) The electrons carry away less power than brought in by the photons. Where does the other power go? How can it be recovered?

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### Problem 18

Red light having a wavelength of 700 nm is projected onto magnesium metal to which electrons are bound by 3.68 eV. (a) Use $\textrm{KE}_e = hf - \textrm{BE}$ to calculate the kinetic energy of e the ejected electrons. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

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### Problem 19

(a) What is the binding energy of electrons to a material from which 4.00-eV electrons are ejected by 400-nm EM radiation? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

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### Problem 21

(a) Find the energy in joules and eV of photons in radio waves from an FM station that has a 90.0-MHz broadcast frequency. (b) What does this imply about the number of photons per second that the radio station must broadcast?

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### Problem 26

(a) Calculate the energy in eV of an IR photon of frequency $2.00\times 10^{13}\textrm{ Hz}$. (b) How many of these photons would need to be absorbed simultaneously by a tightly bound molecule to break it apart? (c) What is the energy in eV of a $\gamma$ ray of frequency $3.00\times 10^{20}\textrm{ Hz}$? (d) How many tightly bound molecules could a single such γ ray break apart?

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### Problem 30

(a) What is the ratio of power outputs by two microwave ovens having frequencies of 950 and 2560 MHz, if they emit the same number of photons per second? (b) What is the ratio of photons per second if they have the same power output?

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### Problem 32

Some satellites use nuclear power. (a) If such a satellite emits a 1.00-W flux of $\gamma$ rays having an average energy of 0.500 MeV, how many are emitted per second? (b) These γ rays affect other satellites. How far away must another satellite be to only receive one γ ray per second per square meter?

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### Problem 33

(a) If the power output of a 650-kHz radio station is 50.0 kW, how many photons per second are produced? (b) If the radio waves are broadcast uniformly in all directions, find the number of photons per second per square meter at a distance of 100 km. Assume no reflection from the ground or absorption by the air.

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### Problem 35

(a) How far away must you be from a 650-kHz radio station with power 50.0 kW for there to be only one photon per second per square meter? Assume no reflections or absorption, as if you were in deep outer space. (b) Discuss the implications for detecting intelligent life in other solar systems by detecting their radio broadcasts.

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### Problem 36

Assuming that 10.0% of a 100-W light bulb’s energy output is in the visible range (typical for incandescent bulbs) with an average wavelength of 580 nm, and that the photons spread out uniformly and are not absorbed by the atmosphere, how far away would you be if 500 photons per second enter the 3.00-mm diameter pupil of your eye? (This number easily stimulates the retina.)

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### Problem 41

(a) A $\gamma$-ray photon has a momentum of $8.00\times 10^{-21} \textrm{ kg}\cdot \textrm{m/s}$. What is its wavelength? (b) Calculate its energy in MeV.

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### Problem 42

(a) Calculate the momentum of a photon having a wavelength of $2.50\textrm{ }\mu\textrm{m}$. (b) Find the velocity of an electron having the same momentum. (c) What is the kinetic energy of the electron, and how does it compare with that of the photon?

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### Problem 43

(a) Calculate the momentum of a photon having a wavelength of $10.0 \textrm{ nm}$ (b) Find the velocity of an electron having the same momentum. (c) What is the kinetic energy of the electron, and how does it compare with that of the photon?

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### Problem 44

(a) Calculate the wavelength of a photon that has the same momentum as a proton moving at 1.00% of the speed of light. (b) What is the energy of the photon in MeV? (c) What is the kinetic energy of the proton in MeV?

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### Problem 46

Take a ratio of relativistic rest energy, $E = \gamma mc^2$, to relativistic momentum, $p = \gamma mu$, and show that in the limit that mass approaches zero, you find $\dfrac{E}{p} = c$.

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### Problem 48

A car feels a small force due to the light it sends out from its headlights, equal to the momentum of the light divided by the time in which it is emitted. (a) Calculate the power of each headlight, if they exert a total force of $2.00\times 10^{-2}\textrm{ N}$ backward on the car. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

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### Problem 51

At what velocity does a proton have a 6.00-fm wavelength (about the size of a nucleus)? Assume the proton is nonrelativistic. (1 femtometer = $10^{-15} \textrm{ m}$. )

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### Problem 55

(a) Find the velocity of a neutron that has a 6.00-fm wavelength (about the size of a nucleus). Assume the neutron is nonrelativistic. (b) What is the neutron’s kinetic energy in MeV?

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### Problem 58

(a) Calculate the velocity of an electron that has a wavelength of $1.00\textrm{ }\mu\textrm{m}$. (b) Through what voltage must the electron be accelerated to have this velocity?

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### Problem 59

The velocity of a proton emerging from a Van de Graaff accelerator is 25.0% of the speed of light. (a) What is the proton’s wavelength? (b) What is its kinetic energy, assuming it is nonrelativistic? (c) What was the equivalent voltage through which it was accelerated?

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### Problem 61

(a) Assuming it is nonrelativistic, calculate the velocity of an electron with a 0.100-fm wavelength (small enough to detect details of a nucleus). (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

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### Problem 62

(a) If the position of an electron in a membrane is measured to an accuracy of $1.00\textrm{ }\mu\textrm{m}$, what is the electron’s minimum uncertainty in velocity? (b) If the electron has this velocity, what is its kinetic energy in eV? (c) What are the implications of this energy, comparing it to typical molecular binding energies?

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### Problem 63

(a) If the position of a chlorine ion in a membrane is measured to an accuracy of 1.00 μm , what is its minimum uncertainty in velocity, given its mass is $5.86 \times 10^{-26} \textrm{ kg}$ (b) If the ion has this velocity, what is its kinetic energy in eV, and how does this compare with typical molecular binding energies?

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### Problem 64

Suppose the velocity of an electron in an atom is known to an accuracy of $2.0\times 10^{3}\textrm{ m/s}$ (reasonably accurate compared with orbital velocities). What is the electron’s minimum uncertainty in position, and how does this compare with the approximate 0.1-nm size of the atom?

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### Problem 65

The velocity of a proton in an accelerator is known to an accuracy of 0.250% of the speed of light. (This could be small compared with its velocity.) What is the smallest possible uncertainty in its position?

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### Problem 71

Derive the approximate form of Heisenberg’s uncertainty principle for energy and time, $\Delta E \Delta t \approx h$, using the following arguments: Since the position of a particle is uncertain by $\Delta x \approx \lambda$, where $\lambda$ is the wavelength of the photon used to examine it, there is an uncertainty in the time the photon takes to traverse $\Delta x$ . Furthermore, the photon has an energy related to its wavelength, and it can transfer some or all of this energy to the object being examined. Thus the uncertainty in the energy of the object is also related to $\lambda$. Find $\Delta t$ and $\Delta E$ ; then multiply them to give the approximate uncertainty principle.

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### Problem 74

A certain heat lamp emits 200 W of mostly IR radiation averaging 1500 nm in wavelength. (a) What is the average photon energy in joules? (b) How many of these photons are required to increase the temperature of a person’s shoulder by $2.0\textrm{C}^\circ$ , assuming the affected mass is 4.0 kg with a specific heat of $0.83\textrm{ kcal/kg}\cdot\textrm{C}^\circ$. Also assume no other significant heat transfer. (c) How long does this take?

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### Problem 75

On its high power setting, a microwave oven produces 900 W of 2560 MHz microwaves. (a) How many photons per second is this? (b) How many photons are required to increase the temperature of a 0.500-kg mass of pasta by $45.0 \textrm{ C}^\circ$ , assuming a specific heat of $0.900 \textrm{ kcal/(kg}\cdot \textrm{C}^\circ\textrm{)}$ ? Neglect all other heat transfer. (c) How long must the microwave operator wait for their pasta to be ready?

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### Problem 76

(a) Calculate the amount of microwave energy in joules needed to raise the temperature of 1.00 kg of soup from $20^\circ\textrm{C}$ to $100^\circ\textrm{C}$ . (b) What is the total momentum of all the microwave photons it takes to do this? (c) Calculate the velocity of a 1.00-kg mass with the same momentum. (d) What is the kinetic energy of this mass?

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### Problem 77

(a) What is $\lambda$ for an electron emerging from the Stanford Linear Accelerator with a total energy of 50.0 GeV? (b) Find its momentum. (c) What is the electron’s wavelength?

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### Problem 78

(a) What is $\gamma$ for a proton having an energy of 1.00 TeV, produced by the Fermilab accelerator? (b) Find its momentum. (c) What is the proton’s wavelength?

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### Problem 79

An electron microscope passes 1.00-pm-wavelength electrons through a circular aperture $2.00 \textrm{ }\mu\textrm{m}$ in diameter. What is the angle between two just-resolvable point sources for this microscope?

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### Problem 80

(a) Calculate the velocity of electrons that form the same pattern as 450-nm light when passed through a double slit. (b) Calculate the kinetic energy of each and compare them. (c) Would either be easier to generate than the other? Explain.

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### Problem 81

(a) What is the separation between double slits that produces a second-order minimum at $45.0^\circ$ for 650-nm light? (b) What slit separation is needed to produce the same pattern for 1.00-keV protons.

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### Problem 82

A laser with a power output of 2.00 mW at a wavelength of 400 nm is projected onto calcium metal. (a) How many electrons per second are ejected? (b) What power is carried away by the electrons, given that the binding energy is 2.71 eV? (c) Calculate the current of ejected electrons. (d) If the photoelectric material is electrically insulated and acts like a 2.00-pF capacitor, how long will current flow before the capacitor voltage stops it?

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### Problem 83

One problem with x-rays is that they are not sensed. Calculate the temperature increase of a researcher exposed in a few seconds to a nearly fatal accidental dose of x-rays under the following conditions. The energy of the x-ray photons is 200 keV, and $4.00 \times 10^{13}$ of them are absorbed per kilogram of tissue, the specific heat of which is $0.830 \textrm{ kcal/(kg}\cdot\textrm{C}^\circ\textrm{)}$ . (Note that medical diagnostic x-ray machines cannot produce an intensity this great.)

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### Problem 84

A 1.00-fm photon has a wavelength short enough to detect some information about nuclei. (a) What is the photon momentum? (b) What is its energy in joules and MeV? (c) What is the (relativistic) velocity of an electron with the same momentum? (d) Calculate the electron’s kinetic energy.

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### Problem 85

The momentum of light is exactly reversed when reflected straight back from a mirror, assuming negligible recoil of the mirror. Thus the change in momentum is twice the photon momentum. Suppose light of intensity $1.00 \textrm{ kW/m}^2$ reflects from a mirror of area $2.00 \textrm{ m}^2$ . (a) Calculate the energy reflected in 1.00 s. (b) What is the momentum imparted to the mirror? (c) Using the most general form of Newton’s second law, what is the force on the mirror? (d) Does the assumption of no mirror recoil seem reasonable?

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### Problem 86

Sunlight above the Earth’s atmosphere has an intensity of $1.30\times 10 \textrm{ kW/m}^2$ . If this is reflected straight back from a mirror that has only a small recoil, the light’s momentum is exactly reversed, giving the mirror twice the incident momentum. (a) Calculate the force per square meter of mirror. (b) Very low mass mirrors can be constructed in the near weightlessness of space, and attached to a spaceship to sail it. Once done, the average mass per square meter of the spaceship is 0.100 kg. Find the acceleration of the spaceship if all other forces are balanced. (c) How fast is it moving 24 hours later?

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### Problem 1 (AP)

The visible spectrum of sunlight shows a range of colors from red to violet. This spectrum has numerous dark lines spread throughout it. Noting that the surface of the Sun is much cooler than the interior, so that the surface is comparable to a cool gas through which light passes, which of the following statements correctly explains the dark lines?
1. The cooler, denser surface material scatters certain wavelengths of light, forming dark lines.
2. The atoms at the surface absorb certain wavelengths of light, causing the dark lines at those wavelengths.
3. The atoms in the Sun’s interior emit light of specific wavelength, so that parts of the spectrum are dark.
4. The atoms at the surface are excited by the high interior temperatures, so that the dark lines are merely wavelengths at which those atoms don’t emit energy.

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### Problem 2 (AP)

A log in a fireplace burns for nearly an hour, at which point it consists mostly of small, hot embers. These embers glow a bright orange and whitish-yellow color. Describe the characteristics of the energy of this system, both in terms of energy transfer and the quantum behavior of blackbodies.

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### Problem 3 (AP)

A metal exposed to a beam of light with a wavelength equal to or shorter than a specific wavelength emits electrons. What property of light, as described in the quantum explanation of blackbody radiation, accounts for this photoelectric process?
1. The energy of light increases as its speed increases.
2. The energy of light increases as its intensity increases.
3. The energy of light increases as its frequency increases.
4. The energy of light increases as its wavelength increases.

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### Problem 4 (AP)

During his experiments that confirmed the existence of electromagnetic waves, Heinrich Hertz used a spark across a gap between two electrodes to provide the rapidly changing electric current that produced electromagnetic waves. He noticed, however, that production of the spark required a lower voltage in a well-lighted laboratory than when the room was dark. Describe how this curious event can be explained in terms of the quantum interpretation of the photoelectric effect.

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### Problem 5 (AP)

A microwave oven produces electromagnetic radiation in the radio portion of the spectrum. These microwave photons are absorbed by water molecules, resulting in an increase in the molecules’ rotational energies. This added energy is transferred by heat to the surrounding food, which as a result becomes hot very quickly. If the energy absorbed by a water molecule is $1.0\times 10^{-5}\textrm{eV}$, what is the corresponding wavelength of the microwave photons?
1. 1.22 GHz
2. 2.45 GHz
3. 4.90 GHz
4. 9.80 GHz

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### Problem 6 (AP)

In the intensity versus frequency curve for x rays (Figure 29.14), the intensity is mostly a smooth curve associated with bremsstrahlung (“breaking radiation”). However, there are two spikes (characteristic x rays) that exhibit high-intensity output. Explain how the smooth curve can be described by classical electrodynamics, whereas the peaks require a quantum mechanical interpretation. (Recall that the acceleration or deceleration of electric charges causes the emission of electromagnetic radiation.)

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### Problem 7 (AP)

The mass of a proton is $1.67\times 10^{-27}\textrm{ kg}$. If a proton has the same momentum as a photon with a wavelength of 325 nm, what is its speed?
1. $2.73\times 10^{-3}\textrm{ m/s}$
2. $0.819\textrm{ m/s}$
3. $1.22 \textrm{ m/s}$
4. $2.71\times 10^{4}\textrm{ m/s}$

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### Problem 8 (AP)

A strip of metal foil with a mass of $5.00\times 10^{-7}\textrm{ kg}$ suspended in a vacuum and exposed to a pulse of light. The velocity of the foil changes from zero to $1.00\times 10^{-3}\textrm{ m/s}$ in the same direction as the initial light pulse, and the light pulse is entirely reflected from the surface of the foil. Given that the wavelength of the light is 450 nm, and assuming that this wavelength is the same before and after the collision, how many photons in the pulse collide with the foil?

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### Problem 9 (AP)

In an experiment in which the Compton effect is observed, a “gamma ray” photon with a wavelength of $5.00 \times 10^{-13}\textrm{ m}$ scatters from an electron. If the change in the electron energy is $1.60 \times 10^{-15} \textrm{ J}$, what is the wavelength of the photon after the collision with the electron?
1. $4.95\times 10^{-13}\textrm{ m}$
2. $4.98 \times 10^{-13}\textrm{ m}$
3. $5.02 \times 10^{-13}\textrm{ m}$
4. $5.05\times 10^{-13}\textrm{ m}$

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### Problem 10 (AP)

Consider two experiments involving a metal sphere with a radius of $2.00\textrm{ }\mu\textrm{m}$ that is suspended in a vacuum. In one experiment, a pulse of $N$ photons reflects from the surface of the sphere, causing the sphere to acquire momentum. In a second experiment, an identical pulse of photons is completely absorbed by the sphere, so that the sphere acquires momentum. Identify each type of collision as either elastic or inelastic, and, assuming that the change in the photon wavelength can be ignored, use linear momentum conservation to derive the expression for the momentum of the sphere in each experiment.

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### Problem 11 (AP)

The ground state of a certain type of atom has energy of $-E_o$. What is the wavelength of a photon with enough energy to ionize the atom when it is in the ground state, so that the ejected electron has kinetic energy equal to $2E_o$?
1. $\lambda = \dfrac{hc}{3E_o}$
2. $\lambda = \dfrac{hc}{2E_o}$
3. $\lambda = \dfrac{hc}{E_o}$
4. $\lambda = \dfrac{2hc}{E_o}$

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### Problem 12 (AP)

While the quantum model explains many physical processes that the classical model cannot, it must be consistent with those processes that the classical model does explain. Energy and momentum conservation are fundamental principles of classical physics. Use the Compton and photoelectric effects to explain how these conservation principles carry over to the quantum model of light.

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### Problem 13 (AP)

The least massive particle known to exist is the electron neutrino. Though scientists once believed that it had no mass, like the photon, they have now determined that this particle has an extremely low mass, equivalent to a few electron volts. Assuming a mass of $2.2 \textrm{ eV/c}^2$ (or $3.9\times 10^{-36}\textrm{ kg}$) and a speed of $4.4\times 10^{6}\textrm{ m/s}$, which of the following values equals the neutrino’s de Broglie wavelength?
1. $3.8\times 10^{-5}\textrm{ m}$
2. $4.7\times 10^{-7}\textrm{ m}$
3. $1.7\times 10^{-10}\textrm{ m}$
4. $8.9\times 10^{-14}\textrm{ m}$

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### Problem 14 (AP)

Using the definition of the de Broglie wavelength, explain how wavelike properties of matter increase with a decrease in mass or decrease in speed. Use as examples an electron (mass = $9.11\times 10^{-31} \textrm{ kg}$) with a speed of $5.0\times 10^{6}\textrm{ m/s}$ and a proton (mass = $1.67\times 10^{-27}\textrm{ kg}$) with a speed of $8.0\times 10^{6}\textrm{ m/s}$.

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### Problem 15 (AP)

In a Davisson-Germer type of experiment, a crystal with a parallel-plane separation, $d$, of $9.1\times 10^{-2}\textrm{ nm}$ produces constructive interference with an electron beam at an angle of $\theta = 50^\circ$. Which of the following is the maximum de Broglie wavelength for these electrons?
1. $0.07\textrm{ nm}$
2. $0.09\textrm{ nm}$
3. $0.14\textrm{ nm}$
4. $0.21\textrm{ nm}$

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### Problem 16 (AP)

In a Davisson-Germer experiment, electrons with a speed of $6.5\times 10^{6}\textrm{ m/s}$ exhibit third-order (n = 3) constructive interference for a crystal with unknown plane separation, $d$. Given an angle of incidence of $\theta = 45^\circ$, compute the value for $d$. Compare the de Broglie wavelength to electromagnetic radiation with the same wavelength. (Recall that the mass of the electron is $9.11\times 10^{-31}\textrm{ kg}$.)

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### Problem 17 (AP)

Which of the following describes one of the main features of wave-particle duality?
1. As speed increases, the wave nature of matter becomes more evident.
2. As momentum decreases, the particle nature of matter becomes more evident.
3. As energy increases, the wave nature of matter becomes easier to observe.
4. As mass increases, the wave nature of matter is less easy to observe.

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### Problem 18 (AP)

Explain why Heisenberg’s uncertainty principle limits the precision with which either momentum or position of a subatomic particle can be known, but becomes less applicable for matter at the macroscopic level.

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