Determine the wavelength at the peak of the Planck

(a) An electron moves with a speed of 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 \(\left(m=6.64 \times 10^{-27} \mathrm{~kg}\right)\).

An electron has a de Broglie wavelength of 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 emitted in the radioactive decay of uranium-238 has an energy of 4.20 MeV. What is its de Broglie wavelength?

In the Bohr model of the hydrogen atom, what is the de Broglie wavelength for the electron when it is in (a) the n = 1 level and (b) the n = 4 level? In each case, compare the de Broglie wavelength to the circumference \(2 \pi r_{n}\) of the orbit. Text Transcription: 2pi r_n

(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?

Why Dont We Diffract? (a) Calculate the de Broglie wavelength of a typical person walking through a doorway. Make reasonable approximations for the necessary quantities. (b) Will the person in part (a) exhibit wavelike behavior when walking through the single slit of a doorway? Why?

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

(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?

How fast would an electron have to move so that its de Broglie wavelength is 1.00 mm?

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?

Find the wavelengths of a photon and an electron that have the same energy of 25 eV. (Note: The energy of the electron is its kinetic energy.)

(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)?

Through what potential difference must electrons be accelerated so they will have (a) the same wavelength as an x ray of wavelength 0.150 nm and (b) the same energy as the x ray in part (a)?

(a) Approximately how fast should an electron move so it has a wavelength that makes it useful to measure the distance between adjacent atoms in typical crystals (about 0.10 nm)? (b) What is the kinetic energy of the electron in part (a)? (c) What would be the energy of a photon of the same wavelength as the electron in part (b)? (d) Which would make a more effective probe of smallscale structures: electrons or photons? Why?

A beam of electrons is accelerated from rest through a potential difference of 0.100 kV and then passes through a thin slit. The diffracted beam shows its first diffraction minima at \(\pm 11.5^{\circ}\) from the original direction of the beam when viewed far from the slit. (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 intensity maximum occurs when the angle in Fig. 39.2 is What is the kinetic energy (in electron volts) of each neutron in the beam?

A beam of 188-eV electrons is directed at normal incidence onto a crystal surface as shown in Fig. 39.3b. The intensity maximum occurs at an angle (a) What is the spacing between adjacent atoms on the surface? (b) At what other angle or angles is there an intensity maximum? (c) For what electron energy (in electron volts) would the intensity maximum occur at For this energy, is there an intensity maximum? Explain.

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 (a) If the speed of the electrons is at which values of will the and 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) 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.)

You want to study a biological specimen by means of a wavelength of 10.0 nm, and you have a choice of using electromagnetic waves or an electron microscope. (a) Calculate the ratio of the energy of a 10.0-nm-wavelength photon to the kinetic energy of a 10.0-nm-wavelength electron. (b) In view of your answer to part (a), which would be less damaging to the specimen you are studying: photons or electrons?

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

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 \(6.50\times10^{-14}\) 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 \(6.64\times10^{-27}\) 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 silicon–silicon 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 the ground level absorbs a photon, which excites it to the n = 4 level. Determine the wavelength and frequency of the photon.

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 \(\mathrm{Be}^{3+}\)? of 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?

(a) Show that, as n gets very large, the energy levels of the hydrogen atom get closer and closer together in energy. (b) Do the radii of these energy levels also get closer together?

hydrogen atom in the , 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 In the Bohr model, how many orbits does an electron in the level complete before returning to the ground level?

The energy-level scheme for the hypothetical oneelectron element Searsium is shown in Fig. E39.30. 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 and will eject photoelectrons from an unknown metal, but the photon emitted from the transition will not. What are the limits (maximum and minimum possible values) of the work function of the metal?

In a set of experiments on a hypothetical one-electron atom, you measure the wavelengths of the photons emitted from transitions ending in the ground state as shown in the energy-level diagram in Fig. E39.31. 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 etc.) shown in the figure? (b) If an electron made a transition from the to the 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.

BIO Laser Surgery. Using a mixture of CO2, N2, and sometimes He, CO2 lasers emit a wavelength of 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?

BIO 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-mW \(CO_2\) laser that has a wavelength of \(10.6 \mu \mathrm{m}\)?

BIO PRK Surgery. Photorefractive keratectomy (PRK) is a laser-based surgical procedure that corrects near- and farsightedness by removing part of the lens of the eye to change its curvature and hence focal length. This procedure can remove layers 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=20.66eV and E3p=18.70eV. (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 \({ }^{2} P_{3 / 2} \text { and }{ }^{2} P_{1 / 2}\).Find the ratio of the number of atoms in a \({ }^{2} P_{3 / 2}\) state to the number in a \({ }^{2} P_{1 / 2}\) state for a sodium gas in thermal equilibrium at 500 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 \(\lambda_{\mathrm{m}}\), the wavelength at the peak of the Planck distribution, and the corresponding frequency f, 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 (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 (b) What is the ratio of the peak-intensity wavelength of the hot star to the peak-intensity wavelength of the cool star?

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 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.

Blue Supergiants. A typical blue supergiant star (the type that explodes and leaves behind a black hole) has a surface temperature of 30,000 K and a visual luminosity 100,000 times that of our sun. Our sun radiates at the rate of (Visual luminosity is the total power radiated at visible wavelengths.) (a) Assuming that this star behaves like an ideal blackbody, what is the principal wavelength it radiates? Is this light visible? Use your answer to explain why these stars are blue. (b) If we assume that the power radiated by the star is also 100,000 times that of our sun, what is the radius of this star? Compare its size to that of our sun, which has a radius of (c) Is it really correct to say that the visual luminosity is proportional to the total power radiated? Explain.

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?

By extremely careful measurement, you determine the x-coordinate of a car’s center of mass with an uncertainty of only \(1.00\ \mu\mathrm{m}\).. The car has a mass of 1200 kg. (a) What is the minimum uncertainty in the x-component of the velocity of the car’s center of mass as prescribed by the Heisenberg uncertainty principle? (b) Does the uncertainty principle impose a practical limit on our ability to make simultaneous measurements of the positions and velocities of ordinary objects like cars, books, and people? Explain.

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 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 What is the x-component of the electrons velocity, if the minimum percentage uncertainty in a simultaneous measurement of 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?

(a) The uncertainty in the y-component of a protons position is What is the minimum uncertainty in a simultaneous measurement of the y-component of the protons velocity? (b) The uncertainty in the z- component of an electrons velocity is What is the minimum uncertainty in a simultaneous measurement of the z-coordinate of the electron?

The negative muon has a charge equal to that of an elec-tron 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?

An atom with mass m emits a photon of wavelength (a) What is the recoil speed of the atom? (b) What is the kinetic energy K of the recoiling atom? (c) Find the ratio 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 for a hydrogen atom (mass ) that emits an ultraviolet photon of energy 10.2 eV. Is recoil an important consideration in this emission process?

(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 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 level and eventually ends up in the ground level? Calculate the wavelength of the emitted photon in each case.

A large number of hydrogen atoms are in thermal equilibrium. Let be the ratio of the number of atoms in an excited state to the number of atoms in an ground state. At what temperature is equal to (a) (b) (c) (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 ) but strong in stars with higher atmospheric temperatures.

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?

Bohr Orbits of a Satellite. A 20.0-kg satellite circles the earth once every 2.00 h in an orbit having a radius of 8060 km. (a) Assuming that Bohrs angular-momentum result applies to satellites just as it does to an electron in the hydrogen atom, find the quantum number n of the orbit of the satellite. (b) Show from Bohrs angular momentum result and Newtons law of gravitation that the radius of an earth-satellite orbit is directly proportional to the square of the quantum number, where k is the constant of proportionality. (c) Using the result from part (b), find the distance between the orbit of the satellite in this problem and its next allowed orbit. (Calculate a numerical value.) (d) Comment on the possibility of observing the separation of the two adjacent orbits. (e) Do quantized and classical orbits correspond for this satellite? Which is the correct method for calculating the orbits?

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).

CP Light from an ideal spherical blackbody 15.0 cm in diameter is analyzed using a diffraction grating having When you shine this light through the grating, you observe that the peak-intensity wavelength forms a first-order bright fringe at 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?

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 state to the third excited state?

An ideal spherical blackbody 24.0 cm in diameter is maintained at by an internal electrical heater and is immersed in a very large open-faced tank of water that is kept boiling by the energy radiated by the sphere. You can neglect any heat transferred by conduction and convection. Consult Table 17.4 as needed. (a) At what rate, in is water evaporating from the tank? (b) If a physics-wise thermophile organism living in the hot water is observing this process, what will it measure for the peakintensity (i) wavelength and (ii) frequency of the electromagnetic waves emitted by the sphere?

When a photon is emitted by an atom, the atom must recoil to conserve momentum. This means that the photon and the recoiling atom share the transition energy. (a) For an atom with mass m, calculate the correction due to recoil to the wavelength of an emitted photon. Let be the wavelength of the photon if recoil is not taken into consideration. (Hint: The correction is very small, as Problem 39.56 suggests, so Use this fact to obtain an approximate but very accurate expression for ) (b) Evaluate the correction for a hydrogen atom in which an electron in the level returns to the ground level. How does the answer depend on

An Ideal Blackbody. A large cavity with a very small hole and 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 has a hole with area How long does it take for the cavity to radiate 100 J of energy through the hole?

CALC (a) Write the Planck distribution law in terms of the frequency , rather than the wavelength to obtain (b) Show that where is the Planck distribution formula of Eq. (39.24). (Hint: Change the integration variable from to You will need to use the following tabulated integral: (c) The result of part (b) is I and has the form of the Stefan Boltzmann law, (Eq. 39.19). Evaluate the constants in part (b) to show that 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 \(0.10\ \mu\mathrm{m}\) (b) Through approximately what potential difference must electrons be accelerated so that they will exhibit wave nature in passing through a pinhole \(0.10\ \mu\mathrm{m}\) 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?

CP Electrons go through a single slit 150 nm wide and strike a screen 24.0 cm away. You find that at angles of 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 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?

A beam of protons and a beam of alpha particles (of mass \(6.64 \times 10^{-27} \mathrm{~kg}\) and charge +2e ) are accelerated from rest through the same potential difference and pass through identical circular holes in a very thin, opaque film. When viewed far from the hole, the diffracted proton beam forms its first dark ring at \(15^{\circ}\) with respect to its original direction. When viewed similarly, at what angle will the alpha particle form its first dark ring?

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?

Coherent light is passed through two narrow slits whose separation is \(40.0\ \mu\mathrm{m}\). The second-order bright fringe in the interference pattern is located at an angle of 0.0300 rad. If electrons are used instead of light, what must the kinetic energy (in electron volts) of the electrons be if they are to produce an interference pattern for which the second-order maximum is also at 0.0300 rad?

What is the de Broglie wavelength of a red blood cell, with mass that is moving with a speed of Do we need to be concerned with the wave nature of the blood cells when we describe the flow of blood in the body?

Calculate the energy in electron volts of (a) an electron that has de Broglie wavelength 400 nm and (b) a photon that has wavelength 400 nm

High-speed electrons are used to probe the interior structure of the atomic nucleus. For such electrons the expression still holds, but we must use the relativistic expression for momentum, (a) Show that the speed of an electron that has de Broglie wavelength is (b) The quantity equals (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 is small compared to the denominator in the expression found in part (a) is close to unity and the speed is very close to c. In this case it is convenient to write and express the speed of the electron in terms of rather than Find an expression for valid when [Hint: Use the binomial expansion valid for the case ] (c) How fast must an electron move for its de Broglie wavelength to be comparable to the size of a proton? Express your answer in the form and state the value of

Suppose that the uncertainty of position of an electron is equal to the radius of the 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 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: \(E^{2}=(p c)^{2}+\left(m c^{2}\right)^{2}\) and \(K=E-m c^{2}\).) (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 (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 \(5.0\times10^{-15}\mathrm{\ m}\). (a) Estimate the minimum uncertainty in the momentum of an electron if it is confined within a nucleus. (b) Take this uncertainty in momentum to be an estimate of the magnitude of the momentum. Use the relativistic relationship between energy and momentum, Eq. (37.39), to obtain an estimate of the kinetic energy of an electron confined within a nucleus. (c) Compare the energy calculated in part (b) to the mag-nitude of the Coulomb potential energy of a proton and an electron separated by \(5.0\times10^{-15}\mathrm{\ m}\). 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 39.80.)

In a TV picture tube the accelerating voltage is 15.0 kV, and the electron beam passes through an aperture in diameter to a screen away. (a) Calculate the uncertainty in the component of the electrons velocity perpendicular to the line between aperture and screen. (b) What is the uncertainty in position of the point where the electrons strike the screen? (c) Does this uncertainty affect the clarity of the picture significantly? (Use nonrelativistic expressions for the motion of the electrons. This is fairly accurate and is certainly adequate for obtaining an estimate of uncertainty effects.)

The neutral pion \(\left(\pi^{0}\right)\) is an unstable particle produced in high-energy particle collisions. Its mass is about 264 times that of the electron, and it exists for an average lifetime of \(8.4\times10^{-17}\mathrm{\ s}\) before decaying into two gamma-ray photons. Using the relationship \(E=m c^{2}\) between rest mass and energy, find the uncertainty in the mass of the particle and express it as a fraction of the mass.

Quantum Effects in Daily Life? A 1.25-mg insect flies through a 4.00-mm-diameter hole in an ordinary window screen. The thickness of the screen is 0.500 mm. (a) What should be the approximate wavelength and speed of the insect for her to show wave behavior as she goes through the hole? (b) At the speed found in part (a), how long would it take the insect to pass through the 0.500-mm thickness of the hole in the screen? Compare this time to the age of the universe (about 14 billion years). Would you expect to see “insect diffraction” in daily life?

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 \(1.64 \times 10^{-7}\) 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 \(|\Delta\ E/E|=|\Delta\lambda/\lambda|\text{ if }|\Delta\lambda/\lambda|\ \ll\ 1\).Use this to calculate the magnitude of the smallest possible uncertainty in the wavelength of the photon. Give your answer in nanometers.

You intend to use an electron microscope to study the structure of some crystals. For accurate resolution, you want the electron wavelength to be 1.00 nm. (a) Are these electrons relativistic? How do you know? (b) What accelerating potential is needed? (c) What is the kinetic energy of the electrons you are using? To see if it is great enough to damage the crystals you are studying, compare it to the potential energy of a typical NaCl molecule, which is about 6.0 eV. (d) If you decided to use electromagnetic waves as your probe, what energy should their photons have to provide the same resolution as the electrons? Would this energy damage the crystal?

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

CP Electron diffraction can also take place when there is interference between electron waves that scatter from atoms on the surface of a crystal and waves that scatter from atoms in the next plane below the surface, a distance d from the surface (see Fig. 36.23c). (a) Find an equation for the angles at which there is an intensity maximum for electron waves of wavelength (b) The spacing between crystal planes in a certain metal is 0.091 nm. If 71.0-eV electrons are used, find the angle at which there is an intensity maximum due to interference between scattered waves from adjacent crystal planes. The angle is measured as shown in Fig. 36.23c. (c) The actual angle of the intensity maximum is slightly different from your result in part (b). The reason is the work function of the metal (see Section 38.1), which changes the electron potential energy by when it moves from vacuum into the metal. If the effect of the work function is taken into account, is the angle of the intensity maximum larger or smaller than the value found in part (b)? Explain.

A certain atom has an energy level 3.50 eV above the ground state. When excited to this state, it remains \(4.0\ \mu\mathrm{s}\), on the average, before emitting a photon and returning to the ground state. (a) What is the energy of the photon? What is its wavelength? (b) What is the smallest possible uncertainty in energy of the photon?

BIO Structure of a Virus. To investigate the structure of extremely small objects, such as viruses, the wavelength of the probing wave should be about one-tenth the size of the object for sharp images. But as the wavelength gets shorter, the energy of a photon of light gets greater and could damage or destroy the object being studied. One alternative is to use electron matter waves instead of light. Viruses vary considerably in size, but 50 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?

CALC Zero-Point Energy. Consider a particle with mass m moving in a potential as in a massspring system. The total energy of the particle is Assume that p and x are approximately related by the Heisenberg uncertainty principle, so (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?

CALC A particle with mass m moves in a potential 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 as 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 39.92, 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 (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?

(a) Show that in the Bohr model, the frequency of revolution of an electron in its circular orbit around a stationary hydrogen nucleus is (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 to (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 signifi- cant, 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 )

CP CALC 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 in the x-coordinate of the marble when it reaches the ground comes in part from the uncertainty in the x-coordinate initially and in part from the initial uncertainty in The latter gives rise to an uncertainty in the horizontal motion of the marble as it falls. The values of and are related by the uncertainty principle. A small gives rise to a large and vice versa. Find the value of that gives the smallest total uncertainty in x at the ground. Ignore any effects of air resistance.)

Problem 39E 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 \({ }^{2} P_{3 / 2}\) and \({ }^{2} P_{1 / 2}\). Find the ratio of the number of atoms in \({ }^{2} P_{3 / 2}\) a state to the number in a \({ }^{2} P_{1 / 2}\) state for a sodium gas in thermal equilibrium at 500 K. In which state are more atoms found? Equation Transcription: Text Transcription: ^2P_3/2 ^2P_1/2 ^2P_3/2 ^2P_1/2

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

High-speed electrons are used to probe the interior structure of the atomic nucleus. For such electrons the expression \(\lambda=h / p\) still holds, but we must use the relativistic expression for momentum, \(p=m v / \sqrt{1-v^{2} / c^{2}}\). Show that the speed of an electron that has de Broglie wavelength \(\lambda\) is \(v=\frac{c}{\sqrt{1+(m c \lambda / h)^{2}}}\) (b) The quantity \(h / m c\) equals \(2.426 \times 10^{-12}\) m. (As we saw in Section 38.3, this same quantity appears in Eq. (38.7), the expression for Compton scattering of photons by electrons.) If \(\lambda\) is small compared to \(h / m c\), the denominator in the expression found in part (a) is close to unity and the speed \(v\) is very close to In this case it is convenient to write \(v=(1-\Delta) c\) and express the speed of the electron in terms of \(\Delta\) rather than \(v\).Find an expression for \(\Delta\) valid when \(\lambda=<<h / m c\). [Hint: Use the binomial expansion \((1+z)^{n}=1+n z+n(n-1) z^{2} / 2+\cdots\), valid for the case \(|z|<1\).] (c) How fast must an electron move for its de Broglie wavelength to be \(1.00 \times 10^{-15}\) m, comparable to the size of a proton? Express your answer in the form \(v=(1-\Delta) c\), and state the value of \(\Delta\). Equation Transcription: Text Transcription: lambda=h/p p=mv/sqrt 1-v^2/c^2 lambda v=c over sqrt 1+(mc lambda/h)^2 h/mc 2.426x10^-12 lambda h/mc v v=(1-Delta)c Delta v Delta lambda=<<h/mc (1+z)^n=1+nz+n(n-1)z^2/2... |z|<1 1.00x10^-15 v=(1-Delta)c Delta

Problem 28E (a) Show that, as n gets very large, the energy levels of the hydrogen atom get closer and closer together in energy. (b) Do the radii of these energy levels also get closer together?

Problem 29E (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?

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

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 \(m = 6.64 \times 10^{-27} kg\)

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

Problem 4DQ 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?

Problem 4E Wavelength of an Alpha Particle. An alpha particle (m = 6.64 × 10-27 kg) emitted in the radioactive decay of uranium-238 has an energy of 4.20 MeV. What is its de Broglie wavelength?

Problem 5DQ Galaxies tend to be strong emitters of Lyman-? photons (from the n = 2 to n = 1 transition in atomic hydrogen). But the intergalactic medium—the very thin gas between the galaxies— tends to absorb Lyman-? photons. What can you infer from these observations about the temperature in these two environments? Explain.

Problem 5E In the Bohr model of the hydrogen atom, what is the de Broglie wavelength for the electron when it is in (a) the n = 1 level and (b) the n = 4 level? In each ease, compare the de Broglie wavelength to the circumference 2?rn of the orbit.

Problem 6DQ A doubly ionized lithium atom (Li++) 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.

Problem 6E (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?

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? Equation transcription: Text transcription: K r

Problem 7E Why Don’t We Diffract? (a) Calculate the de Broglie wavelength of a typical person walking through a doorway. Make reasonable app oximations for the necessary quantities. (b) Will the person in part(a) exhibit wavelike behavior when walking through the “single slit” of a doorway? Why?

Problem 8DQ How might the energy levels of an atom be measured directly—that is, without recourse to analysis of spectra?

Problem 8E 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.)

Problem 9DQ Elements in the gaseous state emit line spectra with well-defined wavelengths. But hot solid bodies always emit a continuous spectrum—that is, a continuous smear of wavelengths. Can you account for this difference?

Problem 9E (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?

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?

Problem 10E How fast would an electron have to move so that its de Broglie wavelength is 1.00 mm?

The peak-intensity wavelength of red dwarf stars, which have surface temperatures around 3000 K, is about 1000 nm, which is beyond the visible spectrum. So why are we able to see these stars, and why do they appear red?

Problem 11E 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?

Problem 12E Find the wavelengths of a photon and an electron that have the same energy of 25 eV. (Note: The energy of the electron is its kinetic energy.)

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

Problem 13E (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)?

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.

Problem 14E Through what potential difference must electrons be accelerated so they will have (a) the same wavelength as an x ray of wavelength 0.150 nm and (b) the same energy as the x ray in part (a)?

Problem 15DQ 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.

Problem 15E (a) Approximately how fast should an electron move so it has a wavelength that makes it useful to measure the distance between adjacent atoms in typical crystals (about 0.10 nm)? (b) What is the kinetic energy of the electron in part (a)? (c) What would be the energy of a photon of the same wavelength as the electron in part (b)? (d) Which would make a more effective probe of small- scale structures: electrons or photons? Why?

Problem 16DQ 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.

Problem 16E A beam of electrons is accelerated from rest through a potential difference of 0.100 kV and then passes through a thin slit. The diffracted beam shows its first diffraction minima at ±11.5° from the original direction of the beam when viewed far from the slit. (a) Do we need to use relativity formulas? How do you know? (b) How wide is the slit?

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.

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 \(\theta\) in Fig. 39.2 is \(28.6^{\circ} \text {. }\) What is the kinetic energy (in electron volts) of each neutron in the beam?

Problem 18DQ Laser light results from transitions from long-lived metastable states. Why is it more monochromatic than ordinary light?

A beam of 188 -eV electrons is directed at normal incidence onto a crystal surface as shown in Fig. 39.3b. The intensity maximum occurs at an angle \(\Theta=60.6^{\circ}\). (a) What is the spacing between adjacent atoms on the surface? (b) At what other angle or angles is there an intensity maximum? (c) For what electron energy (in electron volts) would the intensity maximum occur at \(\Theta=60.6^{\circ}\)? For this energy, is there an intensity maximum? Explain. Equation transcription: Text transcription: =Theta=60.6^{\circ}

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.

Problem 19E A CD-ROM is used instead of a crystal in an electron-diffraction experiment. The surface of the CD-ROM has tracks of tiny pits with a uniform spacing of 1.60 ?m. (a) If the speed of the electrons is 1.26 × 104 m/s, at which values of ? 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?

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.

Problem 20E (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.)

Problem 21DQ Why can an electron microscope have greater magnification than an ordinary microscope?

Problem 22DQ You want to study a biological specimen by means of a wavelength of 10.0 nm, and you have a choice of using electro-magnetic waves or an electron microscope. (a) Calculate the ratio of the energy of a 10.0-nm-wavelength photon to the kinetic energy of a 10.0-nm-wavelenglh electron. (b) In view of your answer to part (a), which would be less damaging to the specimen you are studying: photons or electrons?

Problem 22E CP 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?

Problem 23E 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 × 10-14 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 × 10-27 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 silicon–silicon 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?

Problem 25E 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?

In a set of experiments on a hypothetical one-electron atom, you measure the wavelengths of the photons emitted from transitions ending in the ground state , as shown in the energy-level diagram in Fig. E39.31. You also observe that it takes to ionize this atom. (a) What is the energy of the atom in each of the levels \((n=1, n=2, \text { etc })\) 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? Equation transcription: Text transcription: (n=1, n=2, { etc }) n=4 n=2

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?

Problem 34E Use Balmer’s formula to calculate (a) the wavelength, (b) the frequency, and (c) the photon energy for the H? line of the Balmer series for hydrogen.

Problem 33E (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?

Problem 35E BIO Laser Surgery. Using a mixture of CO2, N2, and sometimes He, CO2 lasers emit a wavelength of 10.6 ?m. 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?

Problem 36E BIO 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?

Problem 37E How many photons per second are emitted by a 7.50-mW CO2 laser that has a wavelength of 10.6 ?m?

Problem 38E BIO PRK Surgery. Photorefractive keratectomy (PRK) is a laser-based surgical procedure that corrects near- and farsightedness by removing part of the lens of the eye to change its curvature and hence focal length. This procedure can remove layers 0.25 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?

Problem 42E Determine ?m, the wavelength at the peak of the Planck distribution, and the corresponding frequency ƒ, at these temperatures: (a) 3.00 K; (b) 300 K; (c) 3000 K.

Problem 43E 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?

Problem 44E The shortest visible wavelength is about 400 nm. What is the temperature of an ideal radiator whose spectral emittance peaks at this wavelength?

Problem 45E 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?

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 \(1.0\times 10^{25} \mathrm{\ W}\). 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 sun’s 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.

Problem 47E Blue Supergiants. A typical blue supergiant star (the type that explodes and leaves behind a black hole) has a surface temperature of 30,000 K and a visual luminosity 100,000 times that of our sun. Our sun radiates at the rate of 3.86 × 1026 W. (Visual luminosity is the total power radiated at visible wavelengths.) (a) Assuming that this star behaves like an ideal blackbody, what is the principal wavelength it radiates? Is this light visible? Use your answer to explain why these stars are blue. (b) If we assume that the power radiated by the star is also 100,000 times that of our sun, what is the radius of this star? Compare its size to that of our sun, which has a radius of 6.96 × 105 km. (c) Is it really correct to say that the visual luminosity is proportional to the total power radiated? Explain.

Problem 48E 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?

Problem 49E By extremely careful measurement, you determine the x-coordinate of a car’s center of mass with an uncertainty of only 1.00 ?m. The car has a mass of 1200 kg. (a) What is the minimum uncertainty in the x-component of the velocity of the car’s center of mass as prescribed by the Heisenberg uncertainty principle? (b) Does the uncertainty principle impose a practical limit on our ability to make simultaneous measurements of the positions and velocities of ordinary objects like cars, books, and people? Explain.

Problem 50E 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?)

Problem 51E 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.

Problem 52E (a) The x-coordinate of an electron is measured with an uncertainty of 0.30 mm. What is the x-component of the electron’s 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?

Problem 54E (a) The uncertainty in the y-component of a porton’s position is 2.0 × 10?12 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?

Problem 55P The negative muon has a charge equal to that of an electron but a mass that is 207 times as great. Consider a hydrogen-like 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?

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

Problem 57P (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 H? 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.

A large number of hydrogen atoms are in thermal equilibrium. Let \(n_{2} / n_{1}\) be the ratio of the number of atoms in an excited state to the number of atoms in an ground state. At what temperature is \(n_{2} / n_{1}\) 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 star's 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 ) but strong in stars with higher atmospheric temperatures. Equation transcription: Text transcription: n{2} / n{1} 10^{-12} 10^{-8} 10^{-4}

Problem 59P 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?

Problem 60P Bohr Orbits of a Satellite. A 20.0-kg satellite circles the earth once every 2.00 h in an orbit having a radius of 8060 km. (a) Assuming that Bohr’s angular-momentum result (L = nh/2?) applies to satellites just as it does to an electron in the hydrogen atom, find the quantum number n of the orbit of the satellite. (b) Show from Bohr’s angular momentum result and Newton’s law of gravitation that the radius of an earth-satellite orbit is directly proportional to the square of the quantum number, r = kn2, where k is the constant of proportionality. (c) Using the result from part (b), find the distance between the orbit of the satellite in this problem and its next “allowed” orbit. (Calculate a numerical value.) (d) Comment on the possibility of observing the separation of the two adjacent orbits. (e) Do quantized and classical orbits correspond for this satellite? Which is the “correct” method for calculating the orbits?

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).

Problem 62P Light from an ideal spherical blackbody 15.0 cm in diameter is analyzed using a diffraction grating having 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 ±11.6° 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?

Problem 63P What must be the temperature of an ideal blackbody so that photons of its radiated light having the peak-intensity wave-length can excite the electron in the Bohr-model hydrogen atom from the ground state to the third excited state?

An ideal spherical blackbody 24.0 cm in diameter is maintained at \(225^{\circ} \mathrm{C}\) by an internal electrical heater and is immersed in a very large open-faced tank of water that is kept boiling by the energy radiated by the sphere. You can neglect any heat transferred by conduction and convection. Consult Table 17.4 as needed. (a) At what rate, in g/s is water evaporating from the tank? (b) If a physics-wise thermophile organism living in the hot water is observing this process, what will it measure for the peak intensity (i) wavelength and (ii) frequency of the electromagnetic waves emitted by the sphere?

Problem 65P When a photon is emitted by an atom, the atom must recoil to conserve momentum. This means that the photon and the recoiling atom share the transition energy. (a) For an atom with mass m, calculate the correction ?? due to recoil to the wave-length of an emitted photon. Let ? be the wavelength of the photon if recoil is not taken into consideration. (Hint: The correction is very small, as Problem, suggests, so |??|/?? 1. Use this fact to obtain an approximate but very accurate expression for ??.) (b) Evaluate the correction for a hydrogen atom in which an electron in the nth level returns to the ground level. How does the answer depend on n? Problem: An atom with mass m emits a photon of wavelength ?. (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?

An Ideal Blackbody. A large cavity with a very small hole and 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 \(200^{\circ} \mathrm{C}\) 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 \(\lambda\), to obtain \(I(f)\) (b) Show that \(\int_{0}^{\infty} I(\lambda) d \lambda=\frac{2 \pi^{5} k^{4}}{15 c^{2} h^{3}} T^{4}\) where \(I(\lambda)\) is the Planck distribution formula of Eq. (39.24). (Hint: Change the integration variable from \(\lambda\) to \(f\). You will need to use the following tabulated integral: \(\int_{0}^{\infty} \frac{x^{3}}{e^{\alpha x}-1} d x=\frac{1}{240}\left(\frac{2 \pi}{\alpha}\right)^{4}\) (c) The result of part (b) is \(I\) and has the form of the StefanBoltzmann law, \(I=\sigma T^{4}\) (Eq. 39.19). Evaluate the constants in part (b) to show that has the value given in Section . Equation Transcription: Text Transcription: f lamda I(f) Integral_0^Infinity I(lamda)d lamda=2pi^5 k^4/15c^2 h^3 T^4 I(lamda) Integral_0^Infinity x^3/e^alpha x -1 dx=1/240(2pi/alpha)^4 I=sigmaT^4

CP A beam of 40-eV electrons traveling in the x direction passes through a slit that is parallel to the y-axis and \(5.0 \mu m\) 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. Text transcription: Text transcription: 5.0 mu m

Problem 69P (a) What is the energy of a photon that has wavelength 0.10 ?m? (b) Through approximately what potential difference must electrons be accelerated so that they will exhibit wave nature in passing through a pinhole 0.10 ?m 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?

Problem 70P Electrons go through a single slit 150 nm wide and strike a screen 24.0 cm away. You find that 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 last were these electrons moving when they went through the slit? (b) What will be the next larger angles at which no electrons hit the screen?

Problem 71P CP 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 ±18.0o 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?

Problem 72P A beam of protons and a beam of alpha particles (of mass 6.64 × 10?27 kg and charge +2e) are accelerated from rest through the same potential difference and pass through identical circular holes in a very thin, opaque film. When viewed far from the hole, the diffracted proton beam forms its first dark ring at 15° with respect to its original direction. When viewed similarly, at what angle will the alpha particle form its first dark ring?

Problem 73P CP An electron beam and a photon beam pass through identical slits. On a distant screen, the first dark fringe occurs at the same angle for both of the beams. The electron speeds are much slower than that of light. (a) Express the energy of a photon in terms of the kinetic energy K of one of the electrons. (b) Which is greater, the energy of a photon or the kinetic energy of an electron?

Problem 74P Coherent light is passed through two narrow slits whose separation is 40.0 µm. 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?

Problem 75P BIO 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?

Calculate the energy in electron volts of (a) an electron that has de Broglie wavelength 400 nm and (b) a photon that has wavelength 400 nm.

Problem 78P 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: \(E^{2}=(p c)^{2}+\left(m c^{2}\right)^{2} \text { and } K= \left.E-m c^{2} .\right)\) (b) Determine the numerical value of the kinetic energy (in MeV) and the wavelength (in meters) if the particle in part (a) is (i) an electron and (ii) a proton.

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

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

Problem 82P In a TV picture tube the accelerating voltage is 15.0 kV, and the electron beam passes through an aperture 0.50 mm in diameter to a screen 0.300 m away. (a) Calculate the uncertainty in the component of the electron’s velocity perpendicular to the line between aperture and screen. (b) What is the uncertainty in position of the point where the electrons strike the screen? (c) Does this uncertainty affect the clarity of the picture significantly? (Use nonrelativistic expressions for the motion of the electrons. This is fairly accurate and is certainly adequate for obtaining an estimate of uncertainty effects.)

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

Problem 84P Quantum Effects in Daily Life? A 1.25-mg insect flies through a 4.00-mm-diameter hole in an ordinary window screen. The thickness of the screen is 0.500 mm. (a) What should be the approximate wavelength and speed of the insect for her to show wave behaviour as she goes through the hole? (b) At the speed found in part (a), how long would it take the insect to pass through the 0.500-mm thickness of the hole in the screen? Compare this time to the age of the universe (about 14 billion years). Would you expect to see “insect diffraction” in daily life?

Problem 85P 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 |?E/E| = |??/?| if |??/?| << 1. Use this to calculate the magnitude of the smallest possible uncertainty in the wavelength of the photon. Give your answer in nanometers.

Problem 87P You intend to use an electron microscope to study the structure of some crystals. For accurate resolution, you want the electron wavelength to be 1.00 nm. (a) Are these electrons relativistic? How do you know? (b) What accelerating potential is needed? (c) What is the kinetic energy of the electrons you are using? To see if it is great enough to damage the crystals you are studying, compare it to the potential energy of a typical NaCl molecule, which is about 6.0 eV. (d) If you decided to use electromagnetic waves as your probe, what energy should their photons have to provide the same resolution as the electrons? Would this energy damage the crystal?

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

Electron diffraction can also take place when there is interference between electron waves that scatter from atoms on the surface of a crystal and waves that scatter from atoms in the next plane below the surface, a distance d from the surface (see Fig. 36.23c). (a) Find an equation for the angles \(\theta\) at which there is an intensity maximum for electron waves of wavelength \(\lambda\). (b) The spacing between crystal planes in a certain metal is 0.091 nm. If 71.0-eV electrons are used, find the angle at which there is an intensity maximum due to interference between scattered waves from adjacent crystal planes. The angle is measured as shown in Fig. 36.23c. (c) The actual angle of the intensity maximum is slightly different from your result in part (b). The reason is the work function \(\phi\) of the metal (see Section 38.1), which changes the electron potential energy by \(-e \phi\) when it moves from vacuum into the metal. If the effect of the work function is taken into account, is the angle of the intensity maximum larger or smaller than the value found in part (b)? Explain. Text Transcription: theta lambda phi -e phi

Problem 90P A certain atom has an energy level 3.50 eV above the ground state. When excited to this state, it remains 4.0 µs. on the average, before emitting a photon and returning to the ground state. (a) What is the energy of the photon? What is its wave-length? (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 \(U=\frac{1}{2} k x^{2}\), as in a mass–spring system. The total energy of the particle is \(E=p^{2} / 2 m+\frac{1}{2} k x^{2}\). Assume that p and x are approximately related by the Heisenberg uncertainty principle, so \(p x \approx h\). ) 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?

CALC A particle with mass moves in a potential \(U(x)=A|x|\), where 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 represents the separation between a pair of quarks. Because \(U(x) \rightarrow \infty\) as \(x \rightarrow \infty\), it's not possible to separate quarks from each other (a phenomenon called quark confinement). (a) Classically, what is the force acting on this particle as a function of (b) Using the uncertainty principle as in Problem 39.92, determine approximately the zero-point energy of the particle. Equation transcription: Text transcription: U(x)=A|x| U(x) rightarrow infty x rightarrow infty

Problem 94P Imagine another universe in which the value of Planck’s constant is 0.0663 J ? 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 cm3 at the time she throws it? (b) By what horizontal distance could the ball miss the second student?

lution of an electron in its circular orbit around a stationary hydrogen nucleus is \(f=m e^{4} / 4 \varepsilon^{2} n^{3} h^{3}\) (b) In classical physics, the frequency of revolution of the electron is equal to the frequency of the radiation that it emits. Show that when is very large, the frequency of revolution does indeed equal the radiated frequency calculated from Eq. (39.5) for a transition from \(n_{1}=n+1\) to \(n_{2}=n\) (This illustrates Bohr's correspondence principle, which is often used as a check on quantum calculations. When is small, quantum physics gives results that are very different from those of classical physics. When 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 such that it would correspond to classical results for large Equation transcription: Text transcription: f=m e^{4} / 4 \varepsilon^{2} n^{3} h^{3} n{1}=n+1 n{2}=n

CP CALC 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.)