Solved: An electron confined in a one-dimensional box

What is quantization?

What are photons?

What are the consequences of photons?

What are matter waves?

What are the consequences of matter waves?

What is the Bohr model of the atom?

A photoelectric-effect experiment is performed with an aluminum cathode. An electron inside the cathode has a speed of 1.5 * 106 m/s. If the potential difference between the anode and cathode is -2.00 V, what is the highest possible speed with which this electron could reach the anode?

The work function of metal A is 3.0 eV. Metals B and C have work functions of 4.0 eV and 5.0 eV, respectively. Ultraviolet light shines on all three metals, creating photoelectrons. Rank in order, from largest to smallest, the stopping potentials for A, B, and C.

The retina of your eye has three types of color photoreceptors, called cones, with maximum sensitivities at 437 nm, 533 nm, and 575 nm. For each, what is the energy of one quantum of light having that wavelength?

The intensity of a beam of light is increased but the lights frequency is unchanged. Which one (or perhaps more than one) of the following is true? a. The photons travel faster. b. Each photon has more energy. c. The photons are larger. d. There are more photons per second.

What are the threshold frequencies and wavelengths for photoemission from sodium and from aluminum?

What is the quantum number of this particle confined in a box?

What is the maximum photoelectron speed if sodium is illuminated with light of 300 nm?

A photon with a wavelength of 414 nm has energy Ephoton = 3.00 eV. Do you expect to see a spectral line with l = 414 nm in the emission spectrum of the atom represented by this energy-level diagram? If so, what transition or transitions will emit it? Do you expect to see a spectral line with l = 414 nm in the absorption spectrum? If so, what transition or transitions will absorb it?

The 2.0 mW light beam of a red laser pointer (l = 650 nm) shines on a screen. How many photons strike the screen each second?

What is the quantum number of this hydrogen atom?

An x-ray scattering experiment uses 22.0 keV photons. By what percent is the wavelength increased for x rays scattered at 90?

A 30-nm-diameter virus is about the smallest imaginable macroscopic particle. What is the fundamental quantum of energy for this virus if confined in a one-dimensional cell of length 1.0 mm? The density of a virus is very close to that of water.

As a very simple model of a hydrogen atom, consider an electron confined in a one-dimensional box of length 0.10 nm, about the size of an atom. What are the first three allowed energy levels?

An atom has stationary states with energies Ej = 4.00 eV and Ek = 6.00 eV. What is the wavelength of a photon emitted in a quantum jump from state k to state j?

An atom has stationary states E1 = 0.00 eV, E2 = 3.00 eV, and E3 = 5.00 eV. What wavelengths are observed in the absorption spectrum and in the emission spectrum of this atom?

Can an electron in a hydrogen atom have a speed of 3.60 *105 m/s? If so, what are its energy and the radius of its orbit? What about a speed of 3.65 * 105 m/s?

Whenever astronomers look at distant galaxies, they find that the light has been strongly absorbed at the wavelength of the 1 S 2 transition in the Lyman series of hydrogen. This absorption tells us that interstellar space is filled with vast clouds of hydrogen left over from the Big Bang. What is the wavelength of the 1 S 2 absorption in hydrogen?

Fluorescence is the absorption of light at one wavelength followed by emission at a longer wavelength. Suppose a hydrogen atom in its ground state absorbs an ultraviolet photon with a wavelength of 95.10 nm. Immediately after the absorption, the atom undergoes a quantum jump with n = 3. What is the wavelength of the photon emitted in this quantum jump?

a. A negatively charged electroscope can be discharged by shining an ultraviolet light on it. How does this happen? b. You might think that an ultraviolet light shining on an initially uncharged electroscope would cause the electroscope to become positively charged as photoelectrons are emitted. In fact, ultraviolet light has no noticeable effect on an uncharged electroscope. Why not?

a. Explain why the graphs of Figure 38.3 are mostly horizontal for V 7 0. b. Explain why photoelectrons are ejected from the cathode with a range of kinetic energies, rather than all electrons having the same kinetic energy. c. Explain the reasoning by which we claim that the stopping potential Vstop indicates the maximum kinetic energy of the electrons

How would the graph of Figure 38.2 look if classical physics provided the correct description of the photoelectric effect? Draw the graph and explain your reasoning. Assume that the light intensity remains constant as its frequency and wavelength are varied

How would the graphs of Figure 38.3 look if classical physics provided the correct description of the photoelectric effect? Draw the graph and explain your reasoning. Include curves for both weak light and intense light.

FIGURE Q38.5 is the current-versus-potential-difference graph for a photoelectric-effect experiment with an unknown metal. If classical physics provided the correct description of the photoelectric effect, how would the graph look if: a. The light was replaced by an equally intense light with a shorter wavelength? Draw it. b. The metal was replaced by a different metal with a smaller work function? Draw it.

Metal 1 has a larger work function than metal 2. Both are illuminated with the same short-wavelength ultraviolet light. Do photoelectrons from metal 1 have a higher speed, a lower speed, or the same speed as photoelectrons from metal 2? Explain.

Electron 1 is accelerated from rest through a potential difference of 100 V. Electron 2 is accelerated from rest through a potential difference of 200 V. Afterward, which electron has the larger de Broglie wavelength? Explain.

An electron and a proton are each accelerated from rest through a potential difference of 100 V. Afterward, which particle has the larger de Broglie wavelength? Explain

FIGURE Q38.9 is a simulation of the electrons detected behind two closely spaced slits. Each bright dot represents one electron. How will this pattern change if a. The electron-beam intensity is increased? b. The electron speed is reduced? c. The electrons are replaced by neutrons? d. The left slit is closed? Your answers should consider the number of dots on the screen and the spacing, width, and positions of the fringes.

Imagine that the horizontal box of Figure 38.14 is instead oriented vertically. Also imagine the box to be on a neutron star where the gravitational field is so strong that the particle in the box slows significantly, nearly stopping, before it hits the top of the box. Make a qualitative sketch of the n = 3 de Broglie standing wave of a particle in this box. Hint: The nodes are not uniformly spaced.

If an electron is in a stationary state of an atom, is the electron at rest? If not, what does the term mean?

FIGURE Q38.12 shows the energy-level diagram of Element X. a. What is the ionization energy of Element X? b. An atom in the ground state absorbs a photon, then emits a photon with a wavelength of 1240 nm. What was the energy of the photon that was absorbed? c. An atom in the ground state has a collision with an electron, then emits a photon with a wavelength of 1240 nm. What conclusion can you draw about the initial kinetic energy of the electron?

Which metals in Table 38.1 exhibit the photoelectric effect for (a) light with l = 400 nm and (b) light with l = 250 nm

How many photoelectrons are ejected per second in the experiment represented by the graph of FIGURE EX38.2?

Electrons in a photoelectric-effect experiment emerge from an aluminum surface with a maximum kinetic energy of 1.30 eV. What is the wavelength of the light?

Photoelectrons are observed when a metal is illuminated by light with a wavelength less than 388 nm. What is the metals work function?

You need to design a photodetector that can respond to the entire range of visible light. What is the maximum possible work function of the cathode?

A photoelectric-effect experiment finds a stopping potential of 1.56 V when light of 200 nm is used to illuminate the cathode. a. From what metal is the cathode made? b. What is the stopping potential if the intensity of the light is doubled?

What is the wavelength, in nm, of a photon with energy (a) 0.30 eV, (b) 3.0 eV, and (c) 30 eV? For each, is this wavelength visible, ultraviolet, or infrared light?

a. Determine the energy, in eV, of a photon with a 550 nm wavelength. b. Determine the wavelength of a 7.5 keV x-ray photon

What is the energy, in eV, of (a) a 450 MHz radio-frequency photon, (b) a visible-light photon with a wavelength of 450 nm, and (c) an x-ray photon with a wavelength of 0.045 nm?

An FM radio station broadcasts with a power of 10 kW at a frequency of 101 MHz. a. How many photons does the antenna emit each second? b. Should the broadcast be treated as an electromagnetic wave or discrete photons? Explain

A red laser with a wavelength of 650 nm and a blue laser with a wavelength of 450 nm emit laser beams with the same light power. How do their rates of photon emission compare? Answer this by computing Rred/Rblue.

For what wavelength of light does a 100 mW laser deliver 2.50 * 1017 photons per second?

A 100 W incandescent lightbulb emits about 5 W of visible light. (The other 95 W are emitted as infrared radiation or lost as heat to the surroundings.) The average wavelength of the visible light is about 600 nm, so make the simplifying assumption that all the light has this wavelength. How many visible-light photons does the bulb emit per second?

What is the energy, in keV, of 75 keV x-ray photons that are backscattered (i.e., scattered directly back toward the source) by the electrons in a target?

55 keV x-ray photons are incident on a target. At what scattering angle do the scattered photons have an energy of 50 keV?

At what speed is an electrons de Broglie wavelength (a) 1.0 pm, (b) 1.0 nm, (c) 1.0 mm, and (d) 1.0 mm?

What is the de Broglie wavelength of a neutron that has fallen 1.0 m in a vacuum chamber, starting from rest?

Through what potential difference must an electron be accelerated from rest to have a de Broglie wavelength of 500 nm?

a. What is the de Broglie wavelength of a 200 g baseball with a speed of 30 m/s? b. What is the speed of a 200 g baseball with a de Broglie wavelength of 0.20 nm?

What is the quantum number of an electron confined in a 3.0-nm-long one-dimensional box if the electrons de Broglie wavelength is 1.0 nm?

The diameter of the nucleus is about 10 fm. What is the kinetic energy, in MeV, of a proton with a de Broglie wavelength of 10 fm?

The diameter of the nucleus is about 10 fm. A simple model of the nucleus is that protons and neutrons are confined within a one-dimensional box of length 10 fm. What are the first three energy levels, in MeV, for a proton in such a box?

What is the length of a one-dimensional box in which an electron in the n = 1 state has the same energy as a photon with a wavelength of 600 nm?

FIGURE EX38.24 is an energy-level diagram for a simple atom. What wavelengths, in nm, appear in the atoms (a) emission spectrum and (b) absorption spectrum?

An electron with 2.00 eV of kinetic energy collides with the atom shown in FIGURE EX38.24. a. Is the electron able to excite the atom? Why or why not? b. If your answer to part a was yes, what is the electrons kinetic energy after the collision?

The allowed energies of a simple atom are 0.00 eV, 4.00 eV, and 6.00 eV. An electron traveling with a speed of 1.30 * 106 m/s collides with the atom. Can the electron excite the atom to the n = 2 stationary state? The n = 3 stationary state? Explain.

The allowed energies of a simple atom are 0.00 eV, 4.00 eV, and 6.00 eV. a. Draw the atoms energy-level diagram. Label each level with the energy and the quantum number. b. What wavelengths appear in the atoms emission spectrum? c. What wavelengths appear in the atoms absorption spectrum?

What is the radius of a hydrogen atom whose electron is bound by 0.378 eV?

What is the radius of a hydrogen atom whose electron moves at 7.3 * 105 m/s?

a. Calculate the de Broglie wavelength of the electron in the n = 1, 2, and 3 states of the hydrogen atom. Use the information in Table 38.2. b. Show numerically that the circumference of the orbit for each of these stationary states is exactly equal to n de Broglie wavelengths. c. Sketch the de Broglie standing wave for the n = 3 orbit

a. What quantum number of the hydrogen atom comes closest to giving a 100-nm-diameter electron orbit? b. What are the electrons speed and energy in this state?

How much energy does it take to ionize a hydrogen atom that is in its first excited state?

Determine the wavelengths of all the possible photons that can be emitted from the n = 4 state of a hydrogen atom.

Find the radius of the electrons orbit, the electrons speed, and the energy of the atom for the first three stationary states of He+.

What is the third-longest wavelength in the absorption spectrum of hydrogen?

In a photoelectric-effect experiment, the wavelength of light shining on an aluminum cathode is decreased from 250 nm to 200 nm. What is the change in the stopping potential?

A ruby laser emits an intense pulse of light that lasts a mere 10 ns. The light has a wavelength of 690 nm, and each pulse has an energy of 500 mJ. a. How many photons are emitted in each pulse? b. What is the rate of photon emission, in photons per second, during the 10 ns that the laser is on?

The wavelengths of light emitted by a firefly span the visible spectrum but have maximum intensity near 550 nm. A typical flash lasts for 100 ms and has a power output of 1.2 mW. How many photons does a firefly emit in one flash if we assume that all light is emitted at the peak intensity wavelength of 550 nm?

Dinoflagellates are single-cell organisms that float in the worlds oceans. Many types are bioluminescent. When disturbed, a typical bioluminescent dinoflagellate emits 108 photons in a 0.10-s-long flash of wavelength 460 nm. What is the power of the flash?

The maximum kinetic energy of photoelectrons is 2.8 eV. When the wavelength of the light is increased by 50%, the maximum energy decreases to 1.1 eV. What are (a) the work function of the cathode and (b) the initial wavelength of the light?

Potassium and gold cathodes are used in a photoelectriceffect experiment. For each cathode, find: a. The threshold frequency. b. The threshold wavelength. c. The maximum photoelectron ejection speed if the light has a wavelength of 220 nm. d. The stopping potential if the wavelength is 220 nm

The graph in FIGURE P38.42 was measured in a photoelectriceffect experiment. a. What is the work function (in eV) of the cathode? b. What experimental value of Plancks constant is obtained from these data?

In a photoelectric-effect experiment, the stopping potential was measured for several different wavelengths of incident light. The data are as follows: Wavelength (nm) Stopping potential (V) 500 0.19 450 0.48 400 0.83 350 1.28 300 1.89 250 2.74 Use an appropriate graph of the data to determine (a) the metal used for the cathode and (b) an experimental value for Plancks constant

A 75 kW radio transmitter emits 550 kHz radio waves uniformly in all directions. At what rate do photons strike a 1.5-m-tall, 3.0-mm-diameter antenna that is 15 km away?

The cosmic microwave background radiation is light left over from the Big Bang that has been Doppler-shifted to microwave frequencies by the expansion of the universe. It now fills the universe with 450 photons/cm3 at an average frequency of 160 GHz. How much energy from the cosmic microwave background, in MeV, fills a small apartment that has 95 m2 of floor space and 2.5-m-high ceilings?

Compton scattering is relevant not only to x-ray photons but, even more so, to higher energy gamma-ray photons. Suppose a 350 keV gamma-ray photon backscatters (i.e., is scattered back toward the source) from a free electron. Afterward, what is the electrons velocity in m/s? Hint: This problem uses relativity

The electron interference pattern of Figure 38.12 was made by shooting electrons with 50 keV of kinetic energy through two slits spaced 1.0 mm apart. The fringes were recorded on a detector 1.0 m behind the slits. a. What was the speed of the electrons? (The speed is large enough to justify using relativity, but for simplicity do this as a nonrelativistic calculation.) b. Figure 38.12 is greatly magnified. What was the actual spacing on the detector between adjacent bright fringes?

Electrons, all with the same speed, pass through a tiny 15-nm-wide slit and create a diffraction pattern on a detector 50 mm behind the slit. What is the electrons kinetic energy, in eV, if the central maximum has a width of 3.3 mm?

An experiment was performed in which neutrons were shot through two slits spaced 0.10 mm apart and detected 3.5 m behind the slits. FIGURE P38.49 shows the detector output. Notice the 100 mm scale on the figure. To one significant figure, what was the speed of the neutrons?

An electron confined in a one-dimensional box is observed, at different times, to have energies of 12 eV, 27 eV, and 48 eV. What is the length of the box?

A muona subatomic particle with charge -e and a mass 207 times that of an electronis confined in a 15-pm-long, one-dimensional box. (1 pm = 1 picometer = 10-12 m.) What is the wavelength, in nm, of the photon emitted in a quantum jump from n = 2 to n = 1?

A proton confined in a one-dimensional box emits a 2.0 MeV gamma-ray photon in a quantum jump from n = 2 to n = 1. What is the length of the box?

An electron confined in a one-dimensional box emits a 200 nm photon in a quantum jump from n = 2 to n = 1. What is the length of the box?

Consider a small virus having a diameter of 10 nm. The atoms of the intracellular fluid are confined within the virus. Suppose we model the virus as a 10-nm-long box. What is the groundstate energy (in eV) of a sodium ion confined in this box?

The absorption spectrum of an atom consists of the wavelengths 200 nm, 300 nm, and 500 nm. a. Draw the atoms energy-level diagram. b. What wavelengths are seen in the atoms emission spectrum?

The first three energy levels of the fictitious element X are shown in FIGURE P38.56. a. What is the ionization energy of element X? b. What wavelengths are observed in the absorption spectrum of element X? Express your answers in nm. c. State whether each of your wavelengths in part b corresponds to ultraviolet, visible, or infrared ligh

The first three energy levels of the fictitious element X were shown in FIGURE P38.56. An electron with a speed of 1.4 * 106 m/s collides with an atom of element X. Shortly afterward, the atom emits a photon with a wavelength of 1240 nm. What was the electrons speed after the collision? Assume that, because the atom is much more massive than the electron, the recoil of the atom is negligible. Hint: The energy of the photon is not the energy transferred to the atom in the collision.

Starting from Equation 38.32, derive Equation 38.33.

Calculate all the wavelengths of visible light in the emission spectrum of the hydrogen atom. Hint: There are infinitely many wavelengths in the spectrum, so youll need to develop a strategy for this problem rather than using trial and error

An electron with a speed of 2.1 * 106 m/s collides with a hydrogen atom, exciting the atom to the highest possible energy level. The atom then undergoes a quantum jump with n = 1. What is the wavelength of the photon emitted in the quantum jump?

a. What wavelength photon does a hydrogen atom emit in a 200 S 199 transition? b. What is the difference in the wavelengths emitted in a 199 S 2 transition and a 200 S 2 transition?

Consider a hydrogen atom in stationary state n. a. Show that the orbital period of an electron in quantum state n is T = n3 T1, and find a numerical value for T1. b. On average, an atom stays in the n = 2 state for 1.6 ns before undergoing a quantum jump to the n = 1 state. On average, how many revolutions does the electron make before the quantum jump?

Draw an energy-level diagram, similar to Figure 38.21, for the He+ ion. On your diagram: a. Show the first five energy levels. Label each with the values of n and En. b. Show the ionization limit. c. Show all possible emission transitions from the n = 4 energy level. d. Calculate the wavelengths (in nm) for each of the transitions in part c and show them alongside the appropriate arrow

Very large, hot starsmuch hotter than our suncan be identified by the way in which He+ ions in their atmosphere absorb light. What are the three longest wavelengths, in nm, in the Balmer series of He+?

What are the wavelengths of the transitions 3 S 2, 4 S 2, and 5 S 2 in the hydrogen-like ion O+7 ? In what spectral range do these lie?

The muon is a subatomic particle with the same charge as an electron but with a mass that is 207 times greater: mm = 207me. Physicists think of muons as heavy electrons. However, the muon is not a stable particle; it decays with a half-life of 1.5 ms into an electron plus two neutrinos. Muons from cosmic rays are sometimes captured by the nuclei of the atoms in a solid. A captured muon orbits this nucleus, like an electron, until it decays. Because the muon is often captured into an excited orbit 1n 7 12, its presence can be detected by observing the photons emitted in transitions such as 2 S 1 and 3 S 1. Consider a muon captured by a carbon nucleus 1Z = 62. Because of its large mass, the muon orbits well inside the electron cloud and is not affected by the electrons. Thus the muon sees the full nuclear charge Ze and acts like the electron in a hydrogenlike ion. a. What are the orbital radius and speed of a muon in the n = 1 ground state? Note that the mass of a muon differs from the mass of an electron. b. What is the wavelength of the 2 S 1 muon transition? c. Is the photon emitted in the 2 S 1 transition infrared, visible, ultraviolet, or x ray? d. How many orbits will the muon complete during 1.5 ms? Is this a sufficiently large number that the Bohr model makes sense, even though the muon is not stable?

Two hydrogen atoms collide head-on. The collision brings both atoms to a halt. Immediately after the collision, both atoms emit a 121.6 nm photon. What was the speed of each atom just before the collision?

A beam of electrons is incident upon a gas of hydrogen atoms. a. What minimum speed must the electrons have to cause the emission of 656 nm light from the 3 S 2 transition of hydrogen? b. Through what potential difference must the electrons be accelerated to have this speed?

The electrons in a cathode-ray tube are accelerated through a 250 V potential difference and then shot through a 33-nm-diameter circular aperture. What is the diameter of the bright spot on an electron detector 1.5 m behind the aperture?

Ultraviolet light with a wavelength of 70.0 nm shines on a gas of hydrogen atoms in their ground states. Some of the atoms are ionized by the light. What is the kinetic energy of the electrons that are freed in this process?

In the atom interferometer experiment of Figure 38.13, lasercooling techniques were used to cool a dilute vapor of sodium atoms to a temperature of 0.0010 K = 1.0 mK. The ultracold atoms passed through a series of collimating apertures to form the atomic beam you see entering the figure from the left. The standing light waves were created from a laser beam with a wavelength of 590 nm. a. What is the rms speed vrms of a sodium atom 1A = 232 in a gas at temperature 1.0 mK? b. By treating the laser beam as if it were a diffraction grating, calculate the first-order diffraction angle of a sodium atom traveling with the rms speed of part a. c. How far apart are points B and C if the second standing wave is 10 cm from the first? d. Because interference is observed between the two paths, each individual atom is apparently present at both point B and point C. Describe, in your own words, what this experiment tells you about the nature of matter.

Consider an electron undergoing cyclotron motion in a magnetic field. According to Bohr, the electrons angular momentum must be quantized in units of U. a. Show that allowed radii for the electrons orbit are given by rn = 1nU/eB21/2 , where n = 1, 2, 3,. b. Compute the first four allowed radii in a 1.0 T magnetic field. c. Find an expression for the allowed energy levels En in terms of U and the cyclotron frequency fcyc.