Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 38

Photons and Matter Waves - all with Video Answers

Educators

Chapter Questions

Problem 1

Monochromatic light (that is, light of a single wavelength) is to be absorbed by a sheet of photographic film and thus recorded on the film. Photon absorption will occur if the photon energy equals or exceeds $0.6 \mathrm{eV}$, the smallest amount of energy needed to dissociate an AgBr molecule in the film. (a) What is the greatest wavelength of light that can be recorded by the film? (b) In what region of the electromagnetic spectrum is this wavelength located?

Salamat Ali

Salamat Ali

Numerade Educator

Problem 2

How fast must an electron move to have a kinetic energy equal to the photon energy of sodium light at wavelength $590 \mathrm{~nm}$ ?

Averell Hause

Carnegie Mellon University

Problem 3

At what rate does the Sun emit photons? For simplicity, assume that the Sun's entire emission at the rate of $3.9 \times 10^{26} \mathrm{~W}$ is at the single wavelength of $550 \mathrm{~nm}$.

Salamat Ali

Numerade Educator

Problem 4

A helium-neon laser emits red light at wavelength $\lambda=633 \mathrm{~nm}$ in a beam of diameter $3.5 \mathrm{~mm}$ and at an energy-emission rate of $5.0 \mathrm{~mW}$. A detector in the beam's path totally absorbs the beam. At what rate per unit area does the detector absorb photons?

Averell Hause

Carnegie Mellon University

Problem 5

The meter was once defined as $1650763.73$ wavelengths of the orange light emitted by a source containing krypton-86 atoms. What is the photon energy of that light?

Salamat Ali

Numerade Educator

Problem 6

What is the photon energy for yellow light from a highway sodium lamp at a wavelength of $589 \mathrm{~nm}$ ?

Averell Hause

Carnegie Mellon University

Problem 7

A light detector (your eye) has an area of $2.00 \times 10^{-6} \mathrm{~m}^{2}$ and absorbs $80 \%$ of the incident light, which is at wavelength $500 \mathrm{~nm}$. The detector faces an isotropic source, $3.00 \mathrm{~m}$ from the source. If the detector absorbs photons at the rate of exactly $4.000 \mathrm{~s}^{-1}$, at what power does the emitter emit light?

Salamat Ali

Numerade Educator

Problem 8

The beam emerging from a $1.5 \mathrm{~W}$ argon laser $(\lambda=515 \mathrm{~nm})$ has a diameter $d$ of $3.0 \mathrm{~mm}$. The beam is focused by a lens system with an effective focal length $f_{\mathrm{L}}$ of $2.5 \mathrm{~mm}$. The focused beam strikes a totally absorbing screen, where it forms a circular diffraction pattern whose central disk has a radius $R$ given by $1.22 f_{\mathrm{L}} \lambda / d$. It can be shown that $84 \%$ of the incident energy ends up within this central disk. At what rate are photons absorbed by the screen in the central disk of the diffraction pattern?

Averell Hause

Carnegie Mellon University

Problem 9

A $100 \mathrm{~W}$ sodium lamp $(\lambda=589 \mathrm{~nm})$ radiates energy uniformly in all directions. (a) At what rate are photons emitted by the lamp? (b) At what distance from the lamp will a totally absorbing screen absorb photons at the rate of $1.00$ photon $/ \mathrm{cm}^{2} \cdot \mathrm{s} ?(\mathrm{c})$ What is the photon flux (photons per unit area per unit time) on a small screen $2.00 \mathrm{~m}$ from the lamp?

Salamat Ali

Numerade Educator

Problem 10

A satellite in Earth orbit maintains a panel of solar cells of area $2.60 \mathrm{~m}^{2}$ perpendicular to the direction of the Sun's light rays. The intensity of the light at the panel is $1.39 \mathrm{~kW} / \mathrm{m}^{2}$. (a) At what rate does solar energy arrive at the panel? (b) At what rate are solar photons absorbed by the panel? Assume that the solar radiation is monochromatic, with a wavelength of $550 \mathrm{~nm}$, and that all the solar radiation striking the panel is absorbed. (c) How long would it take for a "mole of photons" to be absorbed by the panel?

Averell Hause

Carnegie Mellon University

Problem 11

An ultraviolet lamp emits light of wavelength $400 \mathrm{~nm}$ at the rate of $400 \mathrm{~W}$. An infrared lamp emits light of wavelength $700 \mathrm{~nm}$, also at the rate of $400 \mathrm{~W}$. (a) Which lamp emits photons at the greater rate and (b) what is that greater rate?

Salamat Ali

Numerade Educator

Problem 12

Under ideal conditions, a visual sensation can occur in the human visual system if light of wavelength $550 \mathrm{~nm}$ is absorbed by the eye's retina at a rate as low as 100 photons per second. What is the corresponding rate at which energy is absorbed by the retina?

Averell Hause

Carnegie Mellon University

Problem 13

A special kind of lightbulb emits monochromatic light of wavelength $630 \mathrm{~nm}$. Electrical energy is supplied to it at the rate of $60 \mathrm{~W}$, and the bulb is $93 \%$ efficient at converting that energy to light energy. How many photons are emitted by the bulb during its lifetime of $730 \mathrm{~h}$ ?

Salamat Ali

Numerade Educator

Problem 14

A light detector has an absorbing area of $2.00 \times 10^{-6} \mathrm{~m}^{2}$ and absorbs $50 \%$ of the incident light, which is at wavelength $600 \mathrm{~nm}$. The detector faces an isotropic source, $12.0 \mathrm{~m}$ from the source. The energy $E$ emitted by the source versus time $t$ is given in Fig. $38-26\left(E_{s}=7.2 \mathrm{~nJ},\right.$, $t_{s}=2.0 \mathrm{~s}$ ). At what rate are photons absorbed by the detector?

Averell Hause

Carnegie Mellon University

Problem 15

Light strikes a sodium surface, causing photoelectric emission. The stopping potential for the ejected electrons is $5.0 \mathrm{~V}$, and the work function of sodium is $2.2 \mathrm{eV}$. What is the wavelength of the incident light?

Salamat Ali

Numerade Educator

Problem 16

Find the maximum kinetic energy of electrons ejected from a certain material if the material's work function is $2.3 \mathrm{eV}$ and the frequency of the incident radiation is $3.0 \times 10^{15} \mathrm{~Hz}$.

Averell Hause

Carnegie Mellon University

Problem 17

The work function of tungsten is $4.50 \mathrm{eV}$. Calculate the speed of the fastest electrons ejected from a tungsten surface when light whose photon energy is $5.80 \mathrm{eV}$ shines on the surface.

Salamat Ali

Numerade Educator

Problem 18

You wish to pick an element for a photocell that will operate via the photoelectric effect with visible light. Which of the following are suitable (work functions are in parentheses): tantalum $(4.2$ $\mathrm{eV}$ ), tungsten $(4.5 \mathrm{eV})$, aluminum $(4.2 \mathrm{eV})$, barium $(2.5 \mathrm{eV})$, lithium $(2.3 \mathrm{eV}) ?$

Averell Hause

Carnegie Mellon University

Problem 19

(a) If the work function for a certain metal is $1.8 \mathrm{eV}$, what is the stopping potential for electrons ejected from the metal when light of wavelength $400 \mathrm{~nm}$ shines on the metal? (b) What is the maximum speed of the ejected electrons?

Salamat Ali

Numerade Educator

Problem 20

Wuppose the fractional efficiency of a cesium surface (with work function $1.80 \mathrm{eV}$ ) is $1.0 \times 10^{-16}$; that is, on average one electron is ejected for every $10^{16}$ photons that reach the surface. What would be the current of electrons ejected from such a surface if it were illuminated with $600 \mathrm{~nm}$ light from a $2.00 \mathrm{~mW}$ laser and all the ejected electrons took part in the charge flow?

Keshav Singh

Numerade Educator

Problem 21

X rays with a wavelength of $71 \mathrm{pm}$ are directed onto a gold foil and eject tightly bound electrons from the gold atoms. The ejected electrons then move in circular paths of radius $r$ in a region of uniform magnetic field $\vec{B}$. For the fastest of the ejected electrons, the product $B r$ is equal to $1.88 \times 10^{-4} \mathrm{~T} \cdot \mathrm{m}$. Find (a) the maximum kinetic energy of those electrons and (b) the work done in removing them from the gold atoms.

Salamat Ali

Numerade Educator

Problem 22

The wavelength associated with the cutoff frequency for silver is $325 \mathrm{~nm}$. Find the maximum kinetic energy of electrons ejected from a silver surface by ultraviolet light of wavelength $254 \mathrm{~nm}$.

Averell Hause

Carnegie Mellon University

Problem 23

Light of wavelength $200 \mathrm{~nm}$ shines on an aluminum surface; $4.20 \mathrm{eV}$ is required to eject an electron. What is the kinetic energy of (a) the fastest and (b) the slowest ejected electrons? (c) What is the stopping potential for this situation? (d) What is the cutoff wavelength for aluminum?

Salamat Ali

Numerade Educator

Problem 24

In a photoelectric experiment using a sodium surface, you find a stopping potential of $1.85 \mathrm{~V}$ for a wavelength of $300 \mathrm{~nm}$ and a stopping potential of $0.820 \mathrm{~V}$ for a wavelength of $400 \mathrm{~nm}$. From these data find (a) a value for the Planck constant, (b) the work function $\Phi$ for sodium, and (c) the cutoff wavelength $\lambda_{0}$ for sodium.

Averell Hause

Carnegie Mellon University

Problem 25

The stopping potential for electrons emitted from a surface illuminated by light of wavelength $491 \mathrm{~nm}$ is $0.710 \mathrm{~V}$. When the incident wavelength is changed to a new value, the stopping potential is $1.43 \mathrm{~V}$. (a) What is this new wavelength? (b) What is the work function for the surface?

Salamat Ali

Numerade Educator

Problem 26

An orbiting satellite can become charged by the photoelectric effect when sunlight ejects electrons from its outer surface. Satellites must be designed to minimize such charging because it can ruin the sensitive microelectronics. Suppose a satellite is coated with platinum, a metal with a very large work function $(\Phi=5.32 \mathrm{eV})$. Find the longest wavelength of incident sunlight that can eject an electron from the platinum.

Averell Hause

Carnegie Mellon University

Problem 27

Light of wavelength $2.40 \mathrm{pm}$ is directed onto a target containing free electrons. (a) Find the wavelength of light scattered at $30.0^{\circ}$ from the incident direction. (b) Do the same for a scattering angle of $120^{\circ}$.

Salamat Ali

Numerade Educator

Problem 28

(a) In MeV/c, what is the magnitude of the momentum associated with a photon having an energy equal to the electron rest energy? What are the (b) wavelength and (c) frequency of the corresponding radiation?

Averell Hause

Carnegie Mellon University

Problem 29

What (a) frequency, (b) photon energy, and (c) photon momentum magnitude (in $\mathrm{keV} / \mathrm{c}$ ) are associated with $\mathrm{x}$ rays having wavelength $35.0 \mathrm{pm}$ ?

Salamat Ali

Numerade Educator

Problem 30

What is the maximum wavelength shift for a Compton collision between a photon and a free proton?

Averell Hause

Carnegie Mellon University

Problem 31

What percentage increase in wavelength leads to a $75 \%$ loss of photon energy in a photon-free electron collision?

Salamat Ali

Numerade Educator

Problem 32

X rays of wavelength $0.0100 \mathrm{~nm}$ are directed in the positive direction of an $x$ axis onto a target containing loosely bound electrons. For Compton scattering from one of those electrons, at an angle of $180^{\circ}$, what are (a) the Compton shift, (b) the corresponding change in photon energy, (c) the kinetic energy of the recoiling electron, and (d) the angle between the positive direction of the $x$ axis and the electron's direction of motion?

Averell Hause

Carnegie Mellon University

Problem 33

Calculate the percentage change in photon energy during a collision like that in Fig. $38-5$ for $\phi=90^{\circ}$ and for radiation in (a) the microwave range, with $\lambda=3.0 \mathrm{~cm} ;$ (b) the visible range, with $\lambda=500 \mathrm{~nm} ;(\mathrm{c})$ the $\mathrm{x}$ -ray range, with $\lambda=25 \mathrm{pm} ;$ and $(\mathrm{d})$ the gamma-ray range, with a gamma photon energy of $1.0 \mathrm{MeV}$. (e) What are your conclusions about the feasibility of detecting the Compton shift in these various regions of the electromagnetic spectrum, judging solely by the criterion of energy loss in a single photon-electron encounter?

Salamat Ali

Numerade Educator

Problem 34

A photon undergoes Compton scattering off a stationary free electron. The photon scatters at $90.0^{\circ}$ from its initial direction; its initial wavelength is $3.00 \times 10^{-12} \mathrm{~m}$. What is the electron's kinetic energy?

Averell Hause

Carnegie Mellon University

Problem 35

Calculate the Compton wavelength for (a) an electron and (b) a proton. What is the photon energy for an electromagnetic wave with a wavelength equal to the Compton wavelength of (c) the electron and (d) the proton?

Salamat Ali

Numerade Educator

Problem 36

Gamma rays of photon energy $0.511 \mathrm{MeV}$ are directed onto an aluminum target and are scattered in various directions by loosely bound electrons there. (a) What is the wavelength of the incident gamma rays? (b) What is the wavelength of gamma rays scattered at $90.0^{\circ}$ to the incident beam? (c) What is the photon energy of the rays scattered in this direction?

Averell Hause

Carnegie Mellon University

Problem 37

Consider a collision between an x-ray photon of initial energy $50.0 \mathrm{keV}$ and an electron at rest, in which the photon is scattered backward and the electron is knocked forward. (a) What is the energy of the backscattered photon? (b) What is the kinetic energy of the electron?

Salamat Ali

Numerade Educator

Problem 38

Show that when a photon of energy $E$ is scattered from a free electron at rest, the maximum kinetic energy of the recoiling electron is given by
$$
K_{\max }=\frac{E^{2}}{E+m c^{2} / 2} .
$$

Averell Hause

Carnegie Mellon University

Problem 39

Through what angle must a $200 \mathrm{keV}$ photon be scattered by a free electron so that the photon loses $10 \%$ of its energy?-o39 Through what angle must a $200 \mathrm{keV}$ photon be scattered by a free electron so that the photon loses $10 \%$ of its energy?

Keshav Singh

Numerade Educator

Problem 40

what is the maximum kinetic energy of electrons knocked out of a thin copper foil by Compton scattering of an incident beam of $17.5 \mathrm{keV}$ x rays? Assume the work function is negligible.

Averell Hause

Carnegie Mellon University

Problem 41

What are (a) the Compton shift $\Delta \lambda$, (b) the fractional Compton shift $\Delta \lambda / \lambda$, and $(\mathrm{c})$ the change $\Delta E$ in photon energy for light of wavelength $\lambda=590 \mathrm{~nm}$ scattering from a free, initially stationary electron if the scattering is at $90^{\circ}$ to the direction of the incident beam? What are (d) $\Delta \lambda,(\mathrm{e}) \Delta \lambda / \lambda$, and $(\mathrm{f}) \Delta E$ for $90^{\circ}$ scattering for photon energy $50.0 \mathrm{keV}$ (x-ray range)?

Salamat Ali

Numerade Educator

Problem 42

The Sun is approximately an ideal blackbody radiator with a surface temperature of $5800 \mathrm{~K} .$ (a) Find the wavelength at which its spectral radiancy is maximum and (b) identify the type of electromagnetic wave corresponding to that wavelength. (See Fig. 33-1.)
(c) As we shall discuss in Chapter 44, the universe is approximately an ideal blackbody radiator with radiation emitted when atoms first formed. Today the spectral radiancy of that radiation peaks at a wavelength of $1.06 \mathrm{~mm}$ (in the microwave region). What is the corresponding temperature of the universe?

Averell Hause

Carnegie Mellon University

Problem 43

Just after detonation, the fireball in a nuclear blast is approximately an ideal blackbody radiator with a surface temperature of about $1.0 \times 10^{7} \mathrm{~K} .$ (a) Find the wavelength at which the thermal radiation is maximum and (b) identify the type of electromagnetic wave corresponding to that wavelength. (See Fig. 33-1.) This radiation is almost immediately absorbed by the surrounding air molecules, which produces another ideal blackbody radiator with a surface temperature of about $1.0 \times 10^{5} \mathrm{~K}$. (c) Find the wavelength at which the thermal radiation is maximum and (d) identify the type of electromagnetic wave corresponding to that wavelength.

Keshav Singh

Numerade Educator

Problem 44

For the thermal radiation from an ideal blackbody radiator with a surface temperature of $2000 \mathrm{~K}$, let $I_{c}$ represent the intensity per unit wavelength according to the classical expression for the spectral radiancy and $I_{P}$ represent the corresponding intensity per unit wavelength according to the Planck expression. What is the ratio $I_{c} / I_{P}$ for a wavelength of (a) $400 \mathrm{~nm}$ (at the blue end of the visible spectrum) and (b) $200 \mu \mathrm{m}$ (in the far infrared)? (c) Does the classical expression agree with the Planck expression in the shorter wavelength range or the longer wavelength range?

Averell Hause

Carnegie Mellon University

Problem 45

Assuming that your surface temperature is $98.6^{\circ} \mathrm{F}$ and that you are an ideal blackbody radiator (you are close), find (a) the wavelength at which your spectral radiancy is maximum, (b) the power at which you emit thermal radiation in a wavelength range of $1.00 \mathrm{~nm}$ at that wavelength, from a surface area of $4.00 \mathrm{~cm}^{2}$, and (c) the corresponding rate at which you emit photons from that area. Using a wavelength of $500 \mathrm{~nm}$ (in the visible range), $(\mathrm{d})$ recalculate the power and (e) the rate of photon emission. (As you have noticed, you do not visibly glow in the dark.)

Keshav Singh

Numerade Educator

Problem 46

Calculate the de Broglie wavelength of (a) a $1.00 \mathrm{keV}$ electron, (b) a $1.00 \mathrm{keV}$ photon, and (c) a $1.00 \mathrm{keV}$ neutron.

Keshav Singh

Numerade Educator

Problem 47

In an old-fashioned television set, electrons are accelerated through a potential difference of $25.0 \mathrm{kV}$. What is the de Broglie wavelength of such electrons? (Relativity is not needed.)

Salamat Ali

Numerade Educator

Problem 48

The smallest dimension (resolving power) that can be resolved by an electron microscope is equal to the de Broglie wavelength of its electrons. What accelerating voltage would be required for the electrons to have the same resolving power as could be obtained using 100 keV gamma rays?

Averell Hause

Carnegie Mellon University

Problem 49

Singly charged sodium ions are accelerated through a potential difference of $300 \mathrm{~V}$. (a) What is the momentum acquired by such an ion? (b) What is its de Broglie wavelength?

Salamat Ali

Numerade Educator

Problem 50

Electrons accelerated to an energy of $50 \mathrm{GeV}$ have a de Broglie wavelength $\lambda$ small enough for them to probe the structure within a target nucleus by scattering from the structure. Assume that the energy is so large that the extreme relativistic relation $p=$ $E / c$ between momentum magnitude $p$ and energy $E$ applies. (In this extreme situation, the kinetic energy of an electron is much greater than its rest energy.) (a) What is $\lambda ?$ (b) If the target nucleus has radius $R=5.0 \mathrm{fm}$, what is the ratio $R / \lambda ?$

Averell Hause

Carnegie Mellon University

Problem 51

The wavelength of the yellow spectral emission line of sodium is $590 \mathrm{~nm}$. At what kinetic energy would an electron have that wavelength as its de Broglie wavelength?

Salamat Ali

Numerade Educator

Problem 52

A stream of protons, each with a speed of $0.9900 c$, are directed into a two-slit experiment where the slit separation is $4.00 \times$ $10^{-9} \mathrm{~m}$. A two-slit interference pattern is built up on the viewing screen. What is the angle between the center of the pattern and the second minimum (to either side of the center)?

Keshav Singh

Numerade Educator

Problem 53

What is the wavelength of (a) a photon with energy $1.00 \mathrm{eV}$, (b) an electron with energy $1.00 \mathrm{eV},(\mathrm{c})$ a photon of energy $1.00 \mathrm{GeV}$, and (d) an electron with energy $1.00 \mathrm{GeV}$ ?

Salamat Ali

Numerade Educator

Problem 54

An electron and a photon each have a wavelength of $0.20 \mathrm{~nm}$. What is the momentum (in $\mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$ ) of the (a) electron and (b) photon? What is the energy (in $\mathrm{eV}$ ) of the (c) electron and (d) photon?

Averell Hause

Carnegie Mellon University

Problem 55

The highest achievable resolving power of a microscope is limited only by the wavelength used; that is, the smallest item that can be distinguished has dimensions about equal to the wavelength. Suppose one wishes to "see" inside an atom. Assuming the atom to have a diameter of $100 \mathrm{pm}$, this means that one must be able to resolve a width of, say, $10 \mathrm{pm} .$ (a) If an electron microscope is used, what minimum electron energy is required? (b) If a light microscope is used, what minimum photon energy is required? (c) Which microscope seems more practical? Why?

Salamat Ali

Numerade Educator

Problem 56

The existence of the atomic nucleus was discovered in 1911 by Ernest Rutherford, who properly interpreted some experiments in which a beam of alpha particles was scattered from a metal foil of atoms such as gold. (a) If the alpha particles had a kinetic energy of $7.5 \mathrm{MeV}$, what was their de Broglie wavelength? (b) Explain whether the wave nature of the incident alpha particles should have been taken into account in interpreting these experiments. The mass of an alpha particle is $4.00 \mathrm{u}$ (atomic mass units), and its distance of closest approach to the nuclear center in these experiments was about $30 \mathrm{fm}$. (The wave nature of matter was not postulated until more than a decade after these crucial experiments were first performed.)

Averell Hause

Carnegie Mellon University

Problem 57

A nonrelativistic particle is moving three times as fast as an electron. The ratio of the de Broglie wavelength of the particle to that of the electron is $1.813 \times 10^{-4} .$ By calculating its mass, identify the particle.

Salamat Ali

Numerade Educator

Problem 58

What are (a) the energy of a photon corresponding to wavelength $1.00 \mathrm{~nm}$, (b) the kinetic energy of an electron with de Broglie wavelength $1.00 \mathrm{~nm},(\mathrm{c})$ the energy of a photon corresponding to wavelength $1.00 \mathrm{fm}$, and (d) the kinetic energy of an electron with de Broglie wavelength $1.00 \mathrm{fm}$ ?

Averell Hause

Carnegie Mellon University

Problem 59

If the de Broglie wavelength of a proton is $100 \mathrm{fm}$, (a) what is the speed of the proton and (b) through what electric potential would the proton have to be accelerated to acquire this speed?

Salamat Ali

Numerade Educator

Problem 60

Suppose we put $A=0$ in Eq. $38-24$ and relabeled $B$ as $\psi_{0}$. (a) What would the resulting wave function then describe? (b) How, if at all, would Fig. $38-13$ be altered?

Averell Hause

Carnegie Mellon University

Problem 61

The function $\psi(x)$ displayed in Eq. $38-27$ can describe a free particle, for which the potential energy is $U(x)=0$ in Schrödinger's equation (Eq. $38-19$ ). Assume now that $U(x)=U_{0}=$ a constant in that equation. Show that Eq. $38-27$ is a solution of Schrödinger's equation, with
$$
k=\frac{2 \pi}{h} \sqrt{2 m\left(E-U_{0}\right)}
$$
giving the angular wave number $k$ of the particle.

Keshav Singh

Numerade Educator

Problem 62

Show that Eq. $38-24$ is indeed a solution of Eq. $38-22$ by substituting $\psi(x)$ and its second derivative into Eq. $38-22$ and noting that an identity results.

Averell Hause

Carnegie Mellon University

Problem 63

(a) Write the wave function $\psi(x)$ displayed in Eq. $38-27$ in the form $\psi(x)=a+i b$, where $a$ and $b$ are real quantities. (Assume that $\psi_{0}$ is real.) (b) Write the time-dependent wave function $\Psi(x, t)$ that corresponds to $\psi(x)$ written in this form.

Salamat Ali

Numerade Educator

Problem 64

Show that the angular wave number $k$ for a nonrelativistic free particle of mass $m$ can be written as
$$
k=\frac{2 \pi \sqrt{2 m K}}{h},
$$
in which $K$ is the particle's kinetic energy.

Averell Hause

Carnegie Mellon University

Problem 65

(a) Let $n=a+i b$ be a complex number, where $a$ and $b$ are real (positive or negative) numbers. Show that the product $n n^{*}$ is always a positive real number. (b) Let $m=c+i d$ be another complex number. Show that $|n m|=|n||m|$.

Salamat Ali

Numerade Educator

Problem 66

In Eq. $38-25$ keep both terms, putting $A=B=\psi_{0}$. The equation then describes the superposition of two matter waves of equal amplitude, traveling in opposite directions. (Recall that this is the condition for a standing wave.) (a) Show that $|\Psi(x, t)|^{2}$ is then given by
$$
|\Psi(x, t)|^{2}=2 \psi_{0}^{2}[1+\cos 2 k x] .
$$
(b) Plot this function, and demonstrate that it describes the square of the amplitude of a standing matter wave. (c) Show that the nodes of this standing wave are located at
$$
x=(2 n+1)\left(\frac{1}{4} \lambda\right), \quad \text { where } n=0,1,2,3, \ldots
$$
and $\lambda$ is the de Broglie wavelength of the particle. (d) Write a similar expression for the most probable locations of the particle.

Averell Hause

Carnegie Mellon University

Problem 67

The uncertainty in the position of an electron along an $x$ axis is given as $50 \mathrm{pm}$, which is about equal to the radius of a hydrogen atom. What is the least uncertainty in any simultaneous measurement of the momentum component $p_{x}$ of this electron?

Salamat Ali

Numerade Educator

Problem 68

You will find in Chapter 39 that electrons cannot move in definite orbits within atoms, like the planets in our solar system. To see why, let us try to "observe" such an orbiting electron by using a light microscope to measure the electron's presumed orbital position with a precision of, say, $10 \mathrm{pm}$ (a typical atom has a radius of about $100 \mathrm{pm}$ ). The wavelength of the light used in the microscope must then be about $10 \mathrm{pm}$. (a) What would be the photon energy of this light? (b) How much energy would such a photon impart to an electron in a head-on collision? (c) What do these results tell you about the possibility of "viewing" an atomic electron at two or more points along its presumed orbital path? (Hint: The outer electrons of atoms are bound to the atom by energies of only a few electron-volts.)

Averell Hause

Carnegie Mellon University

Problem 69

Figure $38-13$ shows a case in which the momentum component $p_{x}$ of a particle is fixed so that $\Delta p_{x}=0 ;$ then, from Heisenberg's uncertainty principle (Eq. $38-28)$, the position $x$ of the particle is completely unknown. From the same principle it follows that the opposite is also true; that is, if the position of a particle is exactly known $(\Delta x=0)$, the uncertainty in its momentum is infinite.
Consider an intermediate case, in which the position of a particle is measured, not to infinite precision, but to within a distance of $\lambda / 2 \pi$, where $\lambda$ is the particle's de Broglie wavelength. Show that the uncertainty in the (simultaneously measured) momentum component is then equal to the component itself; that is, $\Delta p_{x}=p$. Under these circumstances, would a measured momentum of zero surprise you? What about a measured momentum of $0.5 p ?$ Of $2 p ?$ Of $12 p ?$

Keshav Singh

Numerade Educator

Problem 70

An electron moves through a region of uniform electric potential of $-200 \mathrm{~V}$ with a (total) energy of $500 \mathrm{eV}$. What are its (a) kinetic energy (in electron-volts), (b) momentum, (c) speed, (d) de Broglie wavelength, and (e) angular wave number?

Averell Hause

Carnegie Mellon University

Problem 71

For the arrangement of Figs. $38-14$ and $38-15$, electrons in the incident beam in region 1 have energy $E=800 \mathrm{eV}$ and the potential step has a height of $U_{1}=600 \mathrm{eV}$. What is the angular wave number in (a) region 1 and (b) region $2 ?$ (c) What is the reflection coefficient? (d) If the incident beam sends $5.00 \times 10^{5}$ electrons against the potential step, approximately how many will be reflected?

Salamat Ali

Numerade Educator

Problem 72

For the arrangement of Figs. $38-14$ and $38-15$, electrons in the incident beam in region 1 have a speed of $1.60 \times 10^{7} \mathrm{~m} / \mathrm{s}$ and region 2 has an electric potential of $V_{2}=-500 \mathrm{~V}$. What is the angular wave number in (a) region 1 and (b) region $2 ?$ (c) What is the reflection coefficient? (d) If the incident beam sends $3.00 \times 10^{9}$ electrons against the potential step, approximately how many will be reflected?

Averell Hause

Carnegie Mellon University

Problem 73

The current of a beam of electrons, each with a speed of $900 \mathrm{~m} / \mathrm{s}$, is $5.00 \mathrm{~mA}$. At one point along its path, the beam encounters a potential step of height $-1.25 \mu \mathrm{V}$. What is the current on the other side of the step boundary?

Salamat Ali

Numerade Educator

Problem 74

Consider a potential energy barrier like that of Fig. $38-17$ but whose height $U_{b}$ is $6.0 \mathrm{eV}$ and whose thickness $L$ is $0.70 \mathrm{~nm}$. What is the energy of an incident electron whose transmission coefficient is $0.0010$ ?

Averell Hause

Carnegie Mellon University

Problem 75

A 3.0 MeV proton is incident on a potential energy barrier of thickness $10 \mathrm{fm}$ and height $10 \mathrm{MeV}$. What are (a) the transmission coefficient $T,(\mathrm{~b})$ the kinetic energy $K_{t}$ the proton will have on the other side of the barrier if it tunnels through the barrier, and (c) the kinetic energy $K_{r}$ it will have if it reflects from the barrier? A $3.0 \mathrm{MeV}$ deuteron (the same charge but twice the mass as a proton) is incident on the same barrier. What are (d) $T$, (e) $K_{t}$, and (f) $K_{r}$ ?

Salamat Ali

Numerade Educator

Problem 76

(a) Suppose a beam of $5.0$ eV protons strikes a potential energy barrier of height $6.0 \mathrm{eV}$ and thickness $0.70 \mathrm{~nm}$, at a rate equivalent to a current of $1000 \mathrm{~A}$. How long would you have to wait $-$ on average $-$ for one proton to be transmitted? (b) How long would you have to wait if the beam consisted of electrons rather than protons?

Keshav Singh

Numerade Educator

Problem 77

An electron with total energy $E=5.1 \mathrm{eV}$ approaches a barrier of height $U_{b}=6.8 \mathrm{eV}$ and thickness $L=$ $750 \mathrm{pm}$. What percentage change in the transmission coefficient $T$ occurs for a $1.0 \%$ change in (a) the barrier height, (b) the barrier thickness, and (c) the kinetic energy of the incident electron?

Salamat Ali

Numerade Educator

Problem 78

The current of a beam of electrons, each with a speed of $1.200 \times 10^{3} \mathrm{~m} / \mathrm{s}$, is $9.000 \mathrm{~mA}$. At one point along its path, the beam encounters a potential barrier of height $-4.719 \mu V$ and thickness $200.0 \mathrm{~nm}$. What is the transmitted current?

Averell Hause

Carnegie Mellon University

Problem 79

Figure $38-13$ shows that because of Heisenberg's uncertainty principle, it is not possible to assign an $x$ coordinate to the position of a free electron moving along an $x$ axis. (a) Can you assign a $y$ or a $z$ coordinate? (Hint: The momentum of the electron has no $y$ or $z$ component.) (b) Describe the extent of the matter wave in three dimensions.

Salamat Ali

Numerade Educator

Problem 80

A spectral emission line is electromagnetic radiation that is emitted in a wavelength range narrow enough to be taken as a single wavelength. One such emission line that is important in astronomy has a wavelength of $21 \mathrm{~cm}$. What is the photon energy in the electromagnetic wave at that wavelength?

Averell Hause

Carnegie Mellon University

Problem 81

Using the classical equations for momentum and kinetic energy, show that an electron's de Broglie wavelength in nanometers can be written as $\lambda=1.226 / \sqrt{K}$, in which $K$ is the electron's kinetic energy in electron-volts.

Salamat Ali

Numerade Educator

Problem 82

Derive Eq. 38-11, the equation for the Compton shift, from Eqs. $38-8,38-9$, and $38-10$ by eliminating $v$ and $\theta$.

Averell Hause

Carnegie Mellon University

Problem 83

Neutrons in thermal equilibrium with matter have an average kinetic energy of $(3 / 2) k T$, where $k$ is the Boltzmann constant and $T$, which may be taken to be $300 \mathrm{~K}$, is the temperature of the environment of the neutrons. (a) What is the average kinetic energy of such a neutron? (b) What is the corresponding de Broglie wavelength?

Salamat Ali

Numerade Educator

Problem 84

Consider a balloon filled with helium gas at room temperature and atmospheric pressure. Calculate (a) the average de Broglie wavelength of the helium atoms and (b) the average distance between atoms under these conditions. The average kinetic energy of an atom is equal to $(3 / 2) k T$, where $k$ is the Boltzmann constant. (c) Can the atoms be treated as particles under these conditions? Fxplain.

Keshav Singh

Numerade Educator

Problem 85

In about 1916, R. A. Millikan found the following stoppingpotential data for lithium in his photoelectric experiments:
$$
\begin{array}{llllll}
\hline \text { Wavelength (nm) } & 433.9 & 404.7 & 365.0 & 312.5 & 253.5 \\
\text { Stopping } & & & & & \\
\text { potential (V) } & 0.55 & 0.73 & 1.09 & 1.67 & 2.57 \\
\hline
\end{array}
$$
Use these data to make a plot like Fig. $38-2$ (which is for sodium) and then use the plot to find (a) the Planck constant and (b) the work function for lithium.

Narayan Hari

Numerade Educator

Problem 86

Show that $|\psi|^{2}=|\Psi|^{2}$, with $\psi$ and $\Psi$ related as in Eq. $38-14$. That is, show that the probability density does not depend on the time variable.

Averell Hause

Carnegie Mellon University

Problem 87

Show that $\Delta E / E$, the fractional loss of energy of a photon during a collision with a particle of mass $m$, is given by
$$
\frac{\Delta E}{E}=\frac{h f^{\prime}}{m c^{2}}(1-\cos \phi),
$$
where $E$ is the energy of the incident photon, $f^{\prime}$ is the frequency of the scattered photon, and $\phi$ is defined as in Fig. $38-5$.

Salamat Ali

Numerade Educator

Problem 88

A bullet of mass $40 \mathrm{~g}$ travels at $1000 \mathrm{~m} / \mathrm{s}$. Although the bullet is clearly too large to be treated as a matter wave, determine what Eq. $38-17$ predicts for the de Broglie wavelength of the bullet at that speed.

Averell Hause

Carnegie Mellon University

Problem 89

(a) The smallest amount of energy needed to eject an electron from metallic sodium is $2.28 \mathrm{eV}$. Does sodium show a photoelectric effect for red light, with $\lambda=680 \mathrm{~nm} ?$ (That is, does the light cause electron emission?) (b) What is the cutoff wavelength for photoelectric emission from sodium? (c) To what color does that wavelength correspond?

Salamat Ali

Numerade Educator

Problem 90

Imagine playing baseball in a universe (not ours!) where the Planck constant is $0.60 \mathrm{~J} \cdot \mathrm{s}$ and thus quantum physics affects macroscopic objects. What would be the uncertainty in the position of a $0.50 \mathrm{~kg}$ baseball that is moving at $20 \mathrm{~m} / \mathrm{s}$ along an axis if the uncertainty in the speed is $1.0 \mathrm{~m} / \mathrm{s}$ ?

Bettina Hanlon

Numerade Educator