| 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.

1. How many photons are emitted in each pulse?

2. What is the rate of photon emission, in photons per second, during the 10 ns that the laser is “on”?

|| 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?

| BIO 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?

|| BIO Dinoflagellates are single-cell organisms that float in the world’s oceans. Many types are bioluminescent. When disturbed, a typical bioluminescent dinoflagellate emits ${10}^8$ 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 photoelectric-effect experiment. For each cathode, find:

1. The threshold frequency.

2. The threshold wavelength.

3. The maximum photoelectron ejection speed if the light has a wavelength of 220 nm.

4. The stopping potential if the wavelength is 220 nm.

|| The graph in FIGURE P38.42 was measured in a photoelectric-effect experiment.

FIGURE P38.42

1. What is the work function (in eV) of the cathode?

2. What experimental value of Planck’s 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:

Use an appropriate graph of the data to determine (a) the metal used for the cathode and (b) an experimental value for Planck’s 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 $450{\text{photons/cm}}^3$ at an average frequency of 160 GHz. How much energy from the cosmic microwave background, in MeV, fills a small apartment that has $95{\text{m}}^2$ of floor space and 2.5-m-high ceilings?

||| INT 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 electron’s velocity in m/s?

Hint: This problem uses relativity.

|| INT The electron interference pattern of Figure 38.12 was made by shooting electrons with 50 keV of kinetic energy through two slits spaced $1.0\mu \text{m}$ apart. The fringes were recorded on a detector 1.0 m behind the slits.

1. 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.)

2. Figure 38.12 is greatly magnified. What was the actual spacing on the detector between adjacent bright fringes?

||| INT 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 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 muon—a subatomic particle with charge $-e$ and a mass 207 times that of an electron—is confined in a 15-pm-long, one-dimensional box. ($1\text{pm}=1\text{picometer}={10}^{-12}\text{ m}$.) 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?

|| BIO 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 ground-state energy (in eV) of a sodium ion confined in this box?

| The first three energy levels of the fictitious element X are shown in FIGURE P38.54.

FIGURE P38.54

1. What is the ionization energy of element X?

2. What wavelengths are observed in the absorption spectrum of element X? Express your answers in nm.

3. State whether each of your wavelengths in part b corresponds to ultraviolet, visible, or infrared light.

|| The absorption spectrum of an atom consists of the wavelengths 200 nm, 300 nm, and 500 nm.

1. Draw the atom’s energy-level diagram.

2. What wavelengths are seen in the atom’s emission spectrum?

|| The first three energy levels of the fictitious element X were shown in Figure P38.54. An electron with a speed of $1.4\times {10}^6\text{m}\text{/}\text{s}$ collides with an atom of element X. Shortly afterward, the atom emits a photon with a wavelength of 1240 nm. What was the electron’s 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 you’ll need to develop a strategy for this problem rather than using trial and error.

||| An electron with a speed of $2.1\times {10}^6\text{m}\text{/}\text{s}$ collides with a hydrogen atom, exciting the atom to the highest possible energy level. The atom then undergoes a transition with $\text{Δ}n=1$. What is the wavelength of the photon emitted in the transition?

|| 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?

||

1. What wavelength photon does a hydrogen atom emit in a $200\to 199$ transition?

2. What is the difference in the wavelengths emitted in a $199\to 2$ transition and a $200\to 2$ transition?

|| Consider a hydrogen atom in stationary state $n$.

1. Show that the orbital period of an electron in quantum state $n$ is $T=n^3{T}_1$, and find a numerical value for $T_1$.

2. On average, an atom stays in the $n=2$ state for 1.6 ns before undergoing a transition to the $n=1$ state. On average, how many revolutions does the electron make before the transition?

|| Draw an energy-level diagram, similar to Figure 38.21, for the ${\text{He}}^{+}$ ion. On your diagram:

1. Show the first five energy levels. Label each with the values of $n$ and $E_n$.

2. Show the ionization limit.

3. Show all possible emission transitions from the $n=4$ energy level.

4. Calculate the wavelengths (in nm) for each of the transitions in part c and show them alongside the appropriate arrow.

| Very large, hot stars—much hotter than our sun—can be identified by the way in which ${\text{He}}^{+}$ ions in their atmosphere absorb light. What are the three longest wavelengths, in nm, in the Balmer series of ${\text{He}}^{+}?$

| What are the wavelengths of the transitions $3\to 2\text{,}4\to 2$, and $5\to 2$ in the hydrogen-like ion $\text{O}{}^{+7}?$ 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: $m_{\mu }=207m_{\text{e}}$. Physicists think of muons as “heavy electrons.” However, the muon is not a stable particle; it decays with a half-life of $1.5\mu \text{s}$ 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 $(n>1)$, its presence can be detected by observing the photons emitted in transitions such as $2\to 1$ and $3\to 1$.

Consider a muon captured by a carbon nucleus $(Z=6)$. 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 hydrogen-like ion.

1. 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.

2. What is the wavelength of the $2\to 1$ muon transition?

3. Is the photon emitted in the $2\to 1$ transition infrared, visible, ultraviolet, or x ray?

4. How many orbits will the muon complete during $1.5\mu \text{s}?$ Is this a sufficiently large number that the Bohr model “makes sense,” even though the muon is not stable?

|| INT 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?

|| INT A beam of electrons is incident upon a gas of hydrogen atoms.

1. What minimum speed must the electrons have to cause the emission of 656 nm light from the $3\to 2$ transition of hydrogen?

2. 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?

||| 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 CP38.70 shows the detector output. Notice the $100\mu \text{m}$ scale on the figure. To one significant figure, what was the speed of the neutrons?

FIGURE CP38.70

||| In the atom interferometer experiment of Figure 38.13, laser-cooling techniques were used to cool a dilute vapor of sodium atoms to a temperature of $0.0010\text{K}=1.0\text{mK}$. 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.

1. What is the rms speed $v_{\text{rms}}$ of a sodium atom $(A=23)$ in a gas at temperature 1.0 mK?

2. 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.

3. How far apart are points B and C if the second standing wave is 10 cm from the first?

4. 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 electron’s angular momentum must be quantized in units of $\hslash$.

1. Show that allowed radii for the electron’s orbit are given by $r_n={\left(n\hslash \text{/}eB\right)}^{1\text{/}2}$, where $n=1\text{,}2\text{,}3,\dots$.

2. Compute the first four allowed radii in a 1.0 T magnetic field.

3. Find an expression for the allowed energy levels $E_n$ in terms of $\hslash$ and the cyclotron frequency $f_{\text{cyc}}$.

||| Suppose the atoms in a gas are moving so slowly that their de Broglie wavelengths are larger than the spacing between atoms. If this happens, the gas enters a new phase of matter, called a Bose-Einstein condensate, in which the entire gas is in a single quantum state and atoms can no longer be identified individually. This can be achieved in the laboratory by using an atom trap. First, laser beams confine approximately $5\times {10}^8{}^{87}\text{Rb}$ atoms in a 1-mm-diameter spherical volume. Then additional laser beams implement a laser cooling technique to lower the temperature of this dilute gas to almost absolute zero. Estimate the critical temperature $T_{\text{c}}$ at which the gas becomes a Bose-Einstein condensate. Give your answer in nK to one significant figure. The rms speed of atoms in an ideal gas is a reasonable estimate of a typical atomic speed. You can estimate the spacing between atoms, a quantity that you’ll need, by imagining that they sit at the corners of a cubic lattice. Your estimate of $T_{\text{c}}$ is very much in the correct range but, because this model is a bit too simple, about a factor of 3 too low.