Answer: (a) A nonrelativistic free particle with mass m

If a proton and an electron have the same speed, which has the longer de Broglie wavelength? Explain.

If a proton and an electron have the same kinetic energy, which has the longer de Broglie wavelength? Explain.

Does a photon have a de Broglie wavelength? If so, how is it related to the wavelength of the associated electromagnetic wave? Explain

When an electron beam goes through a very small hole, it produces a diffraction pattern on a screen, just like that of light. Does this mean that an electron spreads out as it goes through the hole? What does this pattern mean?

Galaxies tend to be strong emitters of Lyman@a photons (from the n = 2 to n = 1 transition in atomic hydrogen). But the intergalactic mediumthe very thin gas between the galaxies tends to absorb Lyman@a photons. What can you infer from these observations about the temperature in these two environments? Explain

A doubly ionized lithium atom 1Li++2 is one that has had two of its three electrons removed. The energy levels of the remaining single-electron ion are closely related to those of the hydrogen atom. The nuclear charge for lithium is +3e instead of just +e. How are the energy levels related to those of hydrogen? How is the radius of the ion in the ground level related to that of the hydrogen atom? Explain

The emission of a photon by an isolated atom is a recoil process in which momentum is conserved. Thus Eq. (39.5) should include a recoil kinetic energy \(K_r\) for the atom. Why is this energy negligible in that equation?

How might the energy levels of an atom be measured directlythat is, without recourse to analysis of spectra?

Elements in the gaseous state emit line spectra with welldefined wavelengths. But hot solid bodies always emit a continuous spectrumthat is, a continuous smear of wavelengths. Can you account for this difference?

As a body is heated to a very high temperature and becomes self-luminous, the apparent color of the emitted radiation shifts from red to yellow and finally to blue as the temperature increases. Why does the color shift? What other changes in the character of the radiation occur?

Do the planets of the solar system obey a distance law 1rn = n2 r12 as the electrons of the Bohr atom do? Should they? Why (or why not)? (Consult Appendix F for the appropriate distances.)

You have been asked to design a magnet system to steer a beam of 54-eV electrons like those described in Example 39.1 (Section 39.1). The goal is to be able to direct the electron beam to a specific target location with an accuracy of 1.0 mm. In your design, do you need to take the wave nature of electrons into account? Explain

Why go through the expense of building an electron microscope for studying very small objects such as organic molecules? Why not just use extremely short electromagnetic waves, which are much cheaper to generate?

Which has more total energy: a hydrogen atom with an electron in a high shell (large n) or in a low shell (small n)? Which is moving faster: the high-shell electron or the low-shell electron? Is there a contradiction here? Explain

Does the uncertainty principle have anything to do with marksmanship? That is, is the accuracy with which a bullet can be aimed at a target limited by the uncertainty principle? Explain.

Suppose a two-slit interference experiment is carried out using an electron beam. Would the same interference pattern result if one slit at a time is uncovered instead of both at once? If not, why not? Doesn’t each electron go through one slit or the other? Or does every electron go through both slits? Discuss the latter possibility in light of the principle of complementarity.

Equation (39.30) states that the energy of a system can have uncertainty. Does this mean that the principle of conservation of energy is no longer valid? Explain.

Laser light results from transitions from long-lived metastable states. Why is it more monochromatic than ordinary light?

Could an electron-diffraction experiment be carried out using three or four slits? Using a grating with many slits? What sort of results would you expect with a grating? Would the uncertainty principle be violated? Explain

As the lower half of Fig. 39.4 shows, the diffraction pattern made by electrons that pass through aluminum foil is a series of concentric rings. But if the aluminum foil is replaced by a single crystal of aluminum, only certain points on these rings appear in the pattern. Explain.

Why can an electron microscope have greater magnification than an ordinary microscope?

When you check the air pressure in a tire, a little air always escapes; the process of making the measurement changes the quantity being measured. Think of other examples of measurements that change or disturb the quantity being measured.

(a) An electron moves with a speed of 4.70 * 106 m>s. What is its de Broglie wavelength? (b) A proton moves with the same speed. Determine its de Broglie wavelength.

For crystal diffraction experiments (discussed in Section 39.1), wavelengths on the order of 0.20 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 1m = 6.64 * 10-27 kg2.

An electron has a de Broglie wavelength of 2.80 * 10-10 m. Determine (a) the magnitude of its momentum and (b) its kinetic energy (in joules and in electron volts).

Wavelength of an Alpha Particle. An alpha particle 1m = 6.64 * 10-27 kg2 emitted in the radioactive decay of uranium-238 has an energy of 4.20 MeV. What is its de Broglie wavelength?

An electron is moving with a speed of 8.00 * 106 m>s. What is the speed of a proton that has the same de Broglie wavelength as this electron?

(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-eV electron?

(a) If a photon and an electron each have the same energy of 20.0 eV, find the wavelength of each. (b) If a photon and an electron each have the same wavelength of 250 nm, find the energy of each. (c) You want to study an organic molecule that is about 250 nm long using either a photon or an electron microscope. Approximately what wavelength should you use, and which probe, the electron or the photon, is likely to damage the molecule the least?

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

Wavelength of a Bullet. Calculate the de Broglie wavelength of a 5.00-g bullet that is moving at 340 m>s. Will the bullet exhibit wavelike properties?

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

(a) What accelerating potential is needed to produce electrons of wavelength 5.00 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)?

A beam of electrons is accelerated from rest through a potential difference of 0.100 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?

A beam of neutrons that all have the same energy scatters from atoms that have a spacing of 0.0910 nm in the surface plane of a crystal. The m = 1 intensity maximum occurs when the angle u in Fig. 39.2 is 28.6. What is the kinetic energy (in electron volts) of each neutron in the beam?

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

A CD-ROM is used instead of a crystal in an electrondiffraction experiment. The surface of the CD-ROM has tracks of tiny pits with a uniform spacing of 1.60 mm. (a) If the speed of the electrons is 1.26 * 104 m>s, at which values of u will the m = 1 and m = 2 intensity maxima appear? (b) The scattered electrons in these maxima strike at normal incidence a piece of photographic film that is 50.0 cm from the CD-ROM. What is the spacing on the film between these maxima?

A 4.78-MeV alpha particle from a 226Ra 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?

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 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?

The siliconsilicon single bond that forms the basis of the mythical silicon-based creature the Horta has a bond strength of 3.80 eV. What wavelength of photon would you need in a (mythical) phasor disintegration gun to destroy the Horta?

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

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 photo

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 ground-level 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 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?

Consider the Bohr-model description of a hydrogen atom. (a) Calculate E2 - E1 and E10 - E9. As n increases, does the energy separation between adjacent energy levels increase, decrease, or stay the same? (b) Show that En+1 - En approaches 127.2 eV2>n3 as n becomes large. (c) How does rn+1 - rn depend on n? Does the radial distance between adjacent orbits increase, decrease, or stay the same as n increases?

(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 * 10-8 s. In the Bohr model, how many orbits does an electron in the n = 2 level complete before returning to the ground level?

Consider the Bohr-model description of a hydrogen atom. (a) Calculate \(K_1, U_1\), and \(E_1\) for the n = 1 energy level. How are \(K_1\) and \(U_1\) related? (b) Show that for any value of n, both \(U_n = -2K_n\) and \(K_n = -E_n\).

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-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 eV strikes a Searsium atom in its ground level? Why? (d) Photons emitted in the Searsium transitions n = 3 S n = 2 and n = 3 S n = 1 will eject photoelectrons from an unknown metal, but the photon emitted from the transition n = 4 S n = 3 will not. What are the limits (maximum and minimum possible values) of the work function of the metal?

(a) For one-electron ions with nuclear charge Z, what is the speed of the electron in a Bohr-model orbit labeled with n? Give your answer in terms of v1, the orbital speed for the n = 1 Bohr orbit in hydrogen. (b) What is the largest value of Z for which the n = 1 orbital speed is less than 10% of the speed of light in vacuum?

In a set of experiments on a hypothetical oneelectron atom, you measure the wavelengths of the photons emitted from transitions ending in the ground level 1n = 12, as shown in the energy-level diagram in Fig. E39.27. You also observe that it takes 17.50 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?

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?

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

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.

Laser Surgery. Using a mixture of CO2, N2, and sometimes He, CO2 lasers emit a wavelength of 10.6 mm. At power outputs of 0.100 kW, such lasers are used for surgery. How many photons per second does a CO2 laser deliver to the tissue during its use in an operation?

Removing Birthmarks. Pulsed dye lasers emit light of wavelength 585 nm in 0.45-ms pulses to remove skin blemishes such as birthmarks. The beam is usually focused onto a circular spot 5.0 mm in diameter. Suppose that the output of one such laser is 20.0 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?

How many photons per second are emitted by a \(7.50-\mathrm{mW} \ \mathrm{CO}_2\) laser that has a wavelength of \(10.6 \ \mu \mathrm{m}\)?

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 mm 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 lie? (b) What is the energy of a single photon? (c) If a 1.50-mW beam is used, how many photons are delivered to the lens in each pulse?

A large number of neon atoms are in thermal equilibrium. What is the ratio of the number of atoms in a 5s state to the number in a 3p state at (a) 300 K; (b) 600 K; (c) 1200 K? The energies of these states, relative to the ground state, are E5s = 20.66 eV and E3p = 18.70 eV. (d) At any of these temperatures, the rate at which a neon gas will spontaneously emit 632.8-nm radiation is quite low. Explain why.

Figure 39.19a shows the energy levels of the sodium atom. The two lowest excited levels are shown in columns labeled 2 P3>2 and 2 P1>2. Find the ratio of the number of atoms in a 2 P3>2 state to the number in a 2 P1>2 state for a sodium gas in thermal equilibrium at 500 K. In which state are more atoms found?

A 100-W incandescent light bulb has a cylindrical tungsten filament 30.0 cm long, 0.40 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.

Determine lm, the wavelength at the peak of the Planck distribution, and the corresponding frequency , at these temperatures: (a) 3.00 K; (b) 300 K; (c) 3000 K.

Radiation has been detected from space that is characteristic of an ideal radiator at T = 2.728 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?

The shortest visible wavelength is about 400 nm. What is the temperature of an ideal radiator whose spectral emittance peaks at this wavelength?

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?

The wavelength 10.0 mm is in the infrared region of the electromagnetic spectrum, whereas 600 nm is in the visible region and 100 nm is in the ultraviolet. What is the temperature of an ideal blackbody for which the peak wavelength lm is equal to each of these wavelengths?

Sirius 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 K and that it radiates energy at a total rate of 1.0 * 1025 W. Assume that it behaves like an ideal blackbody. (a) What is the total radiated intensity of Sirius 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 suns radius. (d) Which star radiates more total energy per second, the hot Sirius B or the (relatively) cool sun with a surface temperature of 5800 K? To find out, calculate the ratio of the total power radiated by our sun to the power radiated by Sirius B.

A pesky 1.5-mg mosquito is annoying you as you attempt to study physics in your room, which is 5.0 m wide and 2.5 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?

(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 m/s. What is the minimum uncertainty in a simultaneous measurement of the z-coordinate of the electron?

A 10.0-g marble is gently placed on a horizontal tabletop that is 1.75 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?)

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 nm and its momentum component along this axis with a standard deviation of 3.0 * 10-25 kg # m>s. Use the Heisenberg uncertainty principle to evaluate the validity of this claim.

(a) The x-coordinate of an electron is measured with an uncertainty of 0.30 mm. What is the x-component of the electrons velocity, vx, if the minimum percent uncertainty in a simultaneous measurement of vx is 1.0%? (b) Repeat part (a) for a proton

An atom in a metastable state has a lifetime of 5.2 ms. What is the uncertainty in energy of the metastable state?

An atom with mass m emits a photon of wavelength l. (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 * 10-27 kg) that emits an ultraviolet photon of energy 10.2 eV. Is recoil an important consideration in this emission process?

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?

A large number of hydrogen atoms are in thermal equilibrium. Let n2>n1 be the ratio of the number of atoms in an n = 2 excited state to the number of atoms in an n = 1 ground state. At what temperature is n2>n1 equal to (a) 10-12; (b) 10-8 ; (c) 10-4 ? (d) Like the sun, other stars have continuous spectra with dark absorption lines (see Fig. 39.9). The absorption takes place in the stars atmosphere, which in all stars is composed primarily of hydrogen. Explain why the Balmer absorption lines are relatively weak in stars with low atmospheric temperatures such as the sun (atmosphere temperature 5800 K) but strong in stars with higher atmospheric temperatures.

(a) What is the smallest amount of energy in electron volts that must be given to a hydrogen atom initially in its ground level so that it can emit the Ha line in the Balmer series? (b) How many different possibilities of spectral-line emissions are there for this atom when the electron starts in the n = 3 level and eventually ends up in the ground level? Calculate the wavelength of the emitted photon in each case.

In the Bohr model of the hydrogen atom, what is the de Broglie wavelength of the electron 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 2prn of the orbit.

A sample of hydrogen atoms is irradiated with light with wavelength 85.5 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 photoelectrons? (b) A few electrons are detected with energies as much as 10.2 eV greater than the maximum kinetic energy calculated in part (a). How can this be?

Take 380750 nm to be the wavelength range of the visible spectrum. (a) What are the largest and smallest photon energies for visible light? (b) The lowest six energy levels of the one-electron He+ ion are given in Fig. 39.27. For these levels, what transitions give absorption or emission of visible-light photons?

The Red Supergiant Betelgeuse. The star Betelgeuse has a surface temperature of 3000 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 blackbody. (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 K).

Light from an ideal spherical blackbody 15.0 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 blackbody? (b) How long will it take this sphere to radiate 12.0 MJ of energy at constant temperature?

What must be the temperature of an ideal blackbody so that photons of its radiated light having the peak-intensity wavelength can excite the electron in the Bohr-model hydrogen atom from the ground level to the n = 4 energy level?

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 J of energy through the hole?

(a) Write the Planck distribution law in terms of the frequency f, rather than the wavelength l, to obtain I1f2. (b) Show that L q 0 I1l2 dl = 2p5 k4 15c2 h3 T4 where I1l2 is the Planck distribution formula of Eq. (39.24). Hint: Change the integration variable from l to f. You will need to use the following tabulated integral: L q 0 x3 eax - 1 dx = 1 240 a 2p a b 4 (c) The result of part (b) is I and has the form of the Stefan Boltzmann law, I = sT4 (Eq. 39.19). Evaluate the constants in part (b) to show that s has the value given in Section 39.5.

A beam of 40-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 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.

(a) What is the energy of a photon that has wavelength 0.10 mm? (b) Through approximately what potential difference must electrons be accelerated so that they will exhibit wave nature in passing through a pinhole 0.10 mm in diameter? What is the speed of these electrons? (c) If protons rather than electrons were used, through what potential difference would protons have to be accelerated so they would exhibit wave nature in passing through this pinhole? What would be the speed of these protons?

Electrons go through a single slit 300 nm wide and strike a screen 24.0 cm away. At angles of 20.0 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?

A beam of electrons is accelerated from rest and then passes through a pair of identical thin slits that are 1.25 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?

Coherent light is passed through two narrow slits whose separation is 20.0 mm. 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?

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?

What is the de Broglie wavelength of a red blood cell, with mass 1.00 * 10-11 g, that is moving with a speed of 0.400 cm>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?

High-speed electrons are used to probe the interior structure of the atomic nucleus. For such electrons the expression l = h>p still holds, but we must use the relativistic expression for momentum, p = mv>21 - v2>c2 . (a) Show that the speed of an electron that has de Broglie wavelength l is v = c 21 + 1mcl>h22 (b) The quantity h>mc equals 2.426 * 10-12 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 l is small compared to h>mc, 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 = 11 - 2c and express the speed of the electron in terms of rather than v. Find an expression for valid when l V h>mc. [Hint: Use the binomial expansion 11 + z2n = 1 + nz + 3n1n - 12z 2>24 + g, valid for the case 0z0 6 1.4 (c) How fast must an electron move for its de Broglie wavelength to be 1.00 * 10-15 m, comparable to the size of a proton? Express your answer in the form v = 11 - 2c, and state the value of .

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

(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: E2 = 1pc22 + 1mc2 22 and K = E - mc2 .) (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.

Proton Energy in a Nucleus. The radii of atomic nuclei are of the order of 5.0 * 10-15 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.)

Electron Energy in a Nucleus. The radii of atomic nuclei are of the order of 5.0 * 10-15 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 * 10-15 m. On the basis of your result, could there be electrons within the nucleus? (Note: It is interesting to compare this result to that of Problem 39.72

The neutral pion 1p0 2 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 * 10-17 s before decaying into two gamma-ray photons. Using the relationship E = mc2 between rest mass and energy, find the uncertainty in the mass of the particle and express it as a fraction of the mass.

Doorway Diffraction. If your wavelength were 1.0 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 kg.) (b) At the speed calculated in part (a), how many years would it take you to move 0.80 m (one step)? Will you notice diffraction effects as you walk through doorways?

Atomic Spectra Uncertainties. A certain atom has an energy level 2.58 eV above the ground level. Once excited to this level, the atom remains in this level for 1.64 * 10-7 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 0 E>E0 = 0 l>l 0 if 0 l>l 0 V 1. Use this to calculate the magnitude of the smallest possible uncertainty in the wavelength of the photon. Give your answer in nanometers.

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

A certain atom has an energy state 3.50 eV above the ground state. When excited to this state, the atom remains for 2.0 ms, 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?

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 nm is not unusual. Suppose you want to study such a virus, using a wave of wavelength 5.00 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?

Zero-Point Energy. Consider a particle with mass m moving in a potential U = 1 2 kx2 , as in a massspring system. The total energy of the particle is E = 1p2>2m2 + 1 2 kx2 . Assume that p and x are approximately related by the Heisenberg uncertainty principle, so px 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?

A particle with mass m moves in a potential energy U1x2 = A0 x 0 , 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 U1x2 S q as x S q, its 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 39.80, determine approximately the zero-point energy of the particle.

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 m apart, and one throws a 0.25-kg ball directly toward the other with a speed of 6.0 m/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?

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 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 nm, a free electron is produced. What is the kinetic energy (in electron volts) of the electron?

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 Vac. The electrons then scatter off the atoms on the surface of the crystal (as in Fig. 39.3b). Next you measure the angle u that locates the first-order diffraction peak. Finally, you repeat the measurement for different values of Vac. Your results are given in the table. Vac 1V2 106.3 69.1 49.9 25.2 16.9 13.6 U 12 20.4 24.8 30.2 45.5 59.1 73.1 (a) Graph your data in the form sin u versus 1>2Vac. 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.

As an amateur astronomer, you are studying the apparent brightness of stars. You know that a stars 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. Star Polaris Vega Antares a Centauri B Surface temperature (K) 6015 9602 3400 5260 Radius relative to that of the sun 1Rsun2 46 2.73 883 0.865 The radius is given in units of the radius of the sun, Rsun = 6.96 * 108 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 lm in the visible range (380750 nm)? (c) The sun has a total radiated power of 3.85 * 1026 W. Which of these stars, if any, have a total radiated power less than that of our sun?

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

(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 = me4>4P 2 0 n3 h3 . (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 n1 = n + 1 to n2 = n. (This illustrates Bohrs 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 as a function of n such that it would correspond to classical results for large n.)

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.

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

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.

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