1. The Lyman series in the line spectra of atomic hydrogen is the
name for the light emitted from transitions from excited states to the n
= 1 hydrogen ground state. The Balmer series refers to the light
emitted from transitions from excited states with n ≥ 3 to the n = 2
energy state.a. Find the wavelengths of the three shortest-wavelength
photons in the Lyman series. In what range of the electromagnetic
spectrum are the spectral lines of the Lyman series?b. Find the
wavelengths of the three shortest-wavelength photons in the Balmer
series. In what range of the electromagnetic spectrum are the spectral
lines of the Balmer series? Get solution
2. a. Will a photon of energy 5 eV likely be absorbed by a hydrogen atom originally in its ground state?b. What is the range of photon wavelengths that could ionize a hydrogen atom that is originally in its ground state?c. What is the range of photon wavelengths that could ionize a hydrogen atom that is originally in its n = 2 energy state? Get solution
3. The allowed rotational energies ... of a diatomic molecule are given by ...In this expression l is the rotational quantum number and can take the values l = 0, 1, 2 …; I is the rotational inertia of the molecule about an axis through its center of mass; and ℏ= h/2π. The equilibrium separation of the two atoms in a diatomic hydrogen molecule H2 is about 0.074 nm. The mass of each hydrogen atom is about 1.67 × 10−27 kg.a. Show that the rotational inertia of the hydrogen molecule about an axis through its center of mass is about I =4.6× 10−48 kg · m2.b. Find the difference in energy between the first excited rotational energy stale and the ground rotational state. That is, find .... Express the answer in both J and eV.c. Find the relative likelihood Pl = 1/Pl = 0 that a hydrogen molecule will be in its first excited rotational state in thermal equilibrium at room temperature, T = 293 K. (Ignore possible state degeneracies.) Get solution
4. The allowed energies ... associated with the vibration of a diatomic molecule are given by ...Here, k is the vibrational quantum number and can take the values k = 0, 1, 2 … and f is the resonant frequency of the vibration. In a simple model of diatomic hydrogen H2, the resonant vibration frequency can be taken as f = 1.3 × 1014 Hz.a. Find the difference in energy between the first excited vibrational energy state and the ground vibrational state of diatomic hydrogen. That is, find .... Express the answer in both J and eV.b. Find the relative likelihood Pk =1/Pk= 0 that a hydrogen molecule will be in its first excited vibrational state in thermal equilibrium at room temperature, T = 293 K. Get solution
5. Referring to problems 1 and 2 and Eq. (6-2), construct an energy level diagram for the H2 molecule that shows the first vibrational and rotational states associated with the ground electronic state of the molecule. (Hint: The molecule can be vibrating and rotating at the same time.)Problems 1–21. The allowed rotational energies ... of a diatomic molecule are given by ...In this expression l is the rotational quantum number and can take the values l = 0, 1, 2 …; I is the rotational inertia of the molecule about an axis through its center of mass; and ℏ= h/2π. The equilibrium separation of the two atoms in a diatomic hydrogen molecule H2 is about 0.074 nm. The mass of each hydrogen atom is about 1.67 × 10−27 kg.a. Show that the rotational inertia of the hydrogen molecule about an axis through its center of mass is about I =4.6× 10−48 kg · m2.b. Find the difference in energy between the first excited rotational energy stale and the ground rotational state. That is, find .... Express the answer in both J and eV.c. Find the relative likelihood Pl = 1/Pl = 0 that a hydrogen molecule will be in its first excited rotational state in thermal equilibrium at room temperature, T = 293 K. (Ignore possible state degeneracies.)2. The allowed energies ... associated with the vibration of a diatomic molecule are given by ...Here, k is the vibrational quantum number and can take the values k = 0, 1, 2 … and f is the resonant frequency of the vibration. In a simple model of diatomic hydrogen H2, the resonant vibration frequency can be taken as f = 1.3 × 1014 Hz.a. Find the difference in energy between the first excited vibrational energy state and the ground vibrational state of diatomic hydrogen. That is, find .... Express the answer in both J and eV.b. Find the relative likelihood Pk =1/Pk= 0 that a hydrogen molecule will be in its first excited vibrational state in thermal equilibrium at room temperature, T = 293 K. Get solution
6. Consider an assembly of atoms that have two energy levels separated by an energy corresponding to a wavelength of 0.6328 μm, as in the He-Ne laser. What is the ratio of the population densities of these two energy levels if the assembly of atoms is in thermal equilibrium as a temperature of T = 300 K? Get solution
7. The rate of decay of an assembly of atoms with population density N2 at excited energy level E2 when spontaneous emission is the only important process is ...Show that an initial population density N20 decreases to a value N20/e in a time τ equal to 1/A21. That is, show that A21 is the inverse of the lifetime of the atomic level. Get solution
8. Derive the Wien displacement law from the Planck black-body spectral radiance formula. Get solution
9. Derive the Stefan-Boltzmann law from the Planck blackbody spectral radiance formula. (Hint: Use a substitution of x = hc/λkBT to facilitate the integration.) Get solution
10. a. At what wavelength does a blackbody at 6000 K radiate the most per unit wavelength?b. If the blackbody is a 1-mm diameter hole in a cavity radiator at this temperature, find the power radiated through the hole in the narrow wavelength region 5500–5510 Ǻ. Get solution
11. At a given temperature, λmax = 550 nm for a blackbody cavity. The cavity temperature is then increased until its total radiant exitance is doubled. What is the new temperature and the new λmax? Get solution
12. What must be the temperature of a graybody with emissivity of 0.45 if it is to have the same total radiant exitance as a blackbody at 5000 K? Get solution
13. Plot the spectral exitance Mλ for a graybody of emissivity 0.4 in thermal equilibrium at 451°F, the temperature at which paper burns. Get solution
14. Why should one expect lasing at ultraviolet wavelengths to be more difficult to attain than lasing at infrared wavelengths? Develop your answer based on the ratio A21/B21 and the meaning of the A21 and B21 coefficients. Get solution
15. The gain bandwidth of the lasing transition (that is, the width of the atomic lineshape g(ν) associated with the transition) in a Nd:YAG gain medium is about Δν = 1.2 × 1011 Hz. Express this bandwidth as a wavelength range Δλ. Use Table 6-1 to find the center wavelength of the Nd:YAG lasing transition. Get solution
16. The output of an argon ion laser can consist of a number of modes of frequency that match the cavity resonance condition and are within the gain bandwidth of the lasing transition. The gain bandwidth of the lasing transition is roughly equal to the width of the atomic lineshape function g(ν) associated with the lasing transition. Take the gain bandwidth of an argon ion laser to be 2 GHz and the linewidth of an individual mode from the argon ion laser to be 100 kHz. The coherence time of a light beam is roughly equal to the reciprocal of the spread of frequencies present in light. Find the coherence lime and the coherence length of the argon ion laser ifa. The laser output consists of a single mode.b. The laser output consists of a number of modes with frequencies spread across the gain bandwidth of the lasing transition. Get solution
17. Find the number of standing wave cavity modes within the gain bandwidth of the argon ion laser of problem 1 if the laser system uses a resonator with flat mirrors separated by a distance d = 0.5 m.The output of an argon ion laser can consist of a number of modes of frequency that match the cavity resonance condition and are within the gain bandwidth of the lasing transition. The gain bandwidth of the lasing transition is roughly equal to the width of the atomic lineshape function g(ν) associated with the lasing transition. Take the gain bandwidth of an argon ion laser to be 2 GHz and the linewidth of an individual mode from the argon ion laser to be 100 kHz. The coherence time of a light beam is roughly equal to the reciprocal of the spread of frequencies present in light. Find the coherence lime and the coherence length of the argon ion laser ifa. The laser output consists of a single mode.b. The laser output consists of a number of modes with frequencies spread across the gain bandwidth of the lasing transition.Problem 1The output of an argon ion laser can consist of a number of modes of frequency that match the cavity resonance condition and are within the gain bandwidth of the lasing transition. The gain bandwidth of the lasing transition is roughly equal to the width of the atomic lineshape function g(ν) associated with the lasing transition. Take the gain bandwidth of an argon ion laser to be 2 GHz and the linewidth of an individual mode from the argon ion laser to be 100 kHz. The coherence time of a light beam is roughly equal to the reciprocal of the spread of frequencies present in light. Find the coherence lime and the coherence length of the argon ion laser ifa. The laser output consists of a single mode.b. The laser output consists of a number of modes with frequencies spread across the gain bandwidth of the lasing transition. Get solution
18. A He-Ne laser has a beam waist (diameter) equal to about 1 mm.a. What is its beam-spread angle in the far field?b. Estimate the diameter of this beam after it has propagated over a distance of 1 km. Get solution
19. To what diameter spot should a He-Ne laser of power 10 mW be focused if the irradiance in the spot is to be the same as the sun’s irradiance at the surface of the earth? (The irradiance of the sun at the earth’s surface is about 1000 W/m2.) Get solution
20. For a Nd:YAG laser, there are four pump levels located at 1.53 eV, 1.653 eV, 2.119 eV, and 2.361 eV above the ground state energy level.a. What is the wavelength associated with the photon energy required to populate each of the pump levels?b. Knowing that a Nd:YAG laser emits photons of wavelength 1.064 μm, determine the quantum efficiency associated with each of the four pump levels. (The quantum efficiency is the ratio of the energy of a single pump event to that of an output photon.) Get solution
21. To operate a Nd:YAG laser, 2500 W of “wall-plug” power are required for a power supply that drives the arc lamps. The arc lamps provide pump energy to create the population inversion. The overall laser system, from power in (to the power supply) to power out (laser output beam), is characterized by the following component efficiencies:80%—power supply operation30%—arc lamps for pump light energy70%—optical reflectors for concentrating pump light on laser rod15%—for spectral match of pump light to Nd:YAG pump levels50%—due to internal cavity/rod lossesa. Taking the efficiencies into account sequentially as they “occur,” how much of the initial 2500 W is available for power in the output beam?b. What is the overall operational efficiency (wall-plug efficiency) for this laser? Get solution
22. Table 6-1 indicates that diode lasers have a large divergence angle. Why is this reasonable? Get solution
24. Why is a Nd:YAG laser system that uses a diode laser as a pump more efficient that a Nd:YAG laser system that uses an arc lamp as a pump? See Table 6.1. Get solution
2. a. Will a photon of energy 5 eV likely be absorbed by a hydrogen atom originally in its ground state?b. What is the range of photon wavelengths that could ionize a hydrogen atom that is originally in its ground state?c. What is the range of photon wavelengths that could ionize a hydrogen atom that is originally in its n = 2 energy state? Get solution
3. The allowed rotational energies ... of a diatomic molecule are given by ...In this expression l is the rotational quantum number and can take the values l = 0, 1, 2 …; I is the rotational inertia of the molecule about an axis through its center of mass; and ℏ= h/2π. The equilibrium separation of the two atoms in a diatomic hydrogen molecule H2 is about 0.074 nm. The mass of each hydrogen atom is about 1.67 × 10−27 kg.a. Show that the rotational inertia of the hydrogen molecule about an axis through its center of mass is about I =4.6× 10−48 kg · m2.b. Find the difference in energy between the first excited rotational energy stale and the ground rotational state. That is, find .... Express the answer in both J and eV.c. Find the relative likelihood Pl = 1/Pl = 0 that a hydrogen molecule will be in its first excited rotational state in thermal equilibrium at room temperature, T = 293 K. (Ignore possible state degeneracies.) Get solution
4. The allowed energies ... associated with the vibration of a diatomic molecule are given by ...Here, k is the vibrational quantum number and can take the values k = 0, 1, 2 … and f is the resonant frequency of the vibration. In a simple model of diatomic hydrogen H2, the resonant vibration frequency can be taken as f = 1.3 × 1014 Hz.a. Find the difference in energy between the first excited vibrational energy state and the ground vibrational state of diatomic hydrogen. That is, find .... Express the answer in both J and eV.b. Find the relative likelihood Pk =1/Pk= 0 that a hydrogen molecule will be in its first excited vibrational state in thermal equilibrium at room temperature, T = 293 K. Get solution
5. Referring to problems 1 and 2 and Eq. (6-2), construct an energy level diagram for the H2 molecule that shows the first vibrational and rotational states associated with the ground electronic state of the molecule. (Hint: The molecule can be vibrating and rotating at the same time.)Problems 1–21. The allowed rotational energies ... of a diatomic molecule are given by ...In this expression l is the rotational quantum number and can take the values l = 0, 1, 2 …; I is the rotational inertia of the molecule about an axis through its center of mass; and ℏ= h/2π. The equilibrium separation of the two atoms in a diatomic hydrogen molecule H2 is about 0.074 nm. The mass of each hydrogen atom is about 1.67 × 10−27 kg.a. Show that the rotational inertia of the hydrogen molecule about an axis through its center of mass is about I =4.6× 10−48 kg · m2.b. Find the difference in energy between the first excited rotational energy stale and the ground rotational state. That is, find .... Express the answer in both J and eV.c. Find the relative likelihood Pl = 1/Pl = 0 that a hydrogen molecule will be in its first excited rotational state in thermal equilibrium at room temperature, T = 293 K. (Ignore possible state degeneracies.)2. The allowed energies ... associated with the vibration of a diatomic molecule are given by ...Here, k is the vibrational quantum number and can take the values k = 0, 1, 2 … and f is the resonant frequency of the vibration. In a simple model of diatomic hydrogen H2, the resonant vibration frequency can be taken as f = 1.3 × 1014 Hz.a. Find the difference in energy between the first excited vibrational energy state and the ground vibrational state of diatomic hydrogen. That is, find .... Express the answer in both J and eV.b. Find the relative likelihood Pk =1/Pk= 0 that a hydrogen molecule will be in its first excited vibrational state in thermal equilibrium at room temperature, T = 293 K. Get solution
6. Consider an assembly of atoms that have two energy levels separated by an energy corresponding to a wavelength of 0.6328 μm, as in the He-Ne laser. What is the ratio of the population densities of these two energy levels if the assembly of atoms is in thermal equilibrium as a temperature of T = 300 K? Get solution
7. The rate of decay of an assembly of atoms with population density N2 at excited energy level E2 when spontaneous emission is the only important process is ...Show that an initial population density N20 decreases to a value N20/e in a time τ equal to 1/A21. That is, show that A21 is the inverse of the lifetime of the atomic level. Get solution
8. Derive the Wien displacement law from the Planck black-body spectral radiance formula. Get solution
9. Derive the Stefan-Boltzmann law from the Planck blackbody spectral radiance formula. (Hint: Use a substitution of x = hc/λkBT to facilitate the integration.) Get solution
10. a. At what wavelength does a blackbody at 6000 K radiate the most per unit wavelength?b. If the blackbody is a 1-mm diameter hole in a cavity radiator at this temperature, find the power radiated through the hole in the narrow wavelength region 5500–5510 Ǻ. Get solution
11. At a given temperature, λmax = 550 nm for a blackbody cavity. The cavity temperature is then increased until its total radiant exitance is doubled. What is the new temperature and the new λmax? Get solution
12. What must be the temperature of a graybody with emissivity of 0.45 if it is to have the same total radiant exitance as a blackbody at 5000 K? Get solution
13. Plot the spectral exitance Mλ for a graybody of emissivity 0.4 in thermal equilibrium at 451°F, the temperature at which paper burns. Get solution
14. Why should one expect lasing at ultraviolet wavelengths to be more difficult to attain than lasing at infrared wavelengths? Develop your answer based on the ratio A21/B21 and the meaning of the A21 and B21 coefficients. Get solution
15. The gain bandwidth of the lasing transition (that is, the width of the atomic lineshape g(ν) associated with the transition) in a Nd:YAG gain medium is about Δν = 1.2 × 1011 Hz. Express this bandwidth as a wavelength range Δλ. Use Table 6-1 to find the center wavelength of the Nd:YAG lasing transition. Get solution
16. The output of an argon ion laser can consist of a number of modes of frequency that match the cavity resonance condition and are within the gain bandwidth of the lasing transition. The gain bandwidth of the lasing transition is roughly equal to the width of the atomic lineshape function g(ν) associated with the lasing transition. Take the gain bandwidth of an argon ion laser to be 2 GHz and the linewidth of an individual mode from the argon ion laser to be 100 kHz. The coherence time of a light beam is roughly equal to the reciprocal of the spread of frequencies present in light. Find the coherence lime and the coherence length of the argon ion laser ifa. The laser output consists of a single mode.b. The laser output consists of a number of modes with frequencies spread across the gain bandwidth of the lasing transition. Get solution
17. Find the number of standing wave cavity modes within the gain bandwidth of the argon ion laser of problem 1 if the laser system uses a resonator with flat mirrors separated by a distance d = 0.5 m.The output of an argon ion laser can consist of a number of modes of frequency that match the cavity resonance condition and are within the gain bandwidth of the lasing transition. The gain bandwidth of the lasing transition is roughly equal to the width of the atomic lineshape function g(ν) associated with the lasing transition. Take the gain bandwidth of an argon ion laser to be 2 GHz and the linewidth of an individual mode from the argon ion laser to be 100 kHz. The coherence time of a light beam is roughly equal to the reciprocal of the spread of frequencies present in light. Find the coherence lime and the coherence length of the argon ion laser ifa. The laser output consists of a single mode.b. The laser output consists of a number of modes with frequencies spread across the gain bandwidth of the lasing transition.Problem 1The output of an argon ion laser can consist of a number of modes of frequency that match the cavity resonance condition and are within the gain bandwidth of the lasing transition. The gain bandwidth of the lasing transition is roughly equal to the width of the atomic lineshape function g(ν) associated with the lasing transition. Take the gain bandwidth of an argon ion laser to be 2 GHz and the linewidth of an individual mode from the argon ion laser to be 100 kHz. The coherence time of a light beam is roughly equal to the reciprocal of the spread of frequencies present in light. Find the coherence lime and the coherence length of the argon ion laser ifa. The laser output consists of a single mode.b. The laser output consists of a number of modes with frequencies spread across the gain bandwidth of the lasing transition. Get solution
18. A He-Ne laser has a beam waist (diameter) equal to about 1 mm.a. What is its beam-spread angle in the far field?b. Estimate the diameter of this beam after it has propagated over a distance of 1 km. Get solution
19. To what diameter spot should a He-Ne laser of power 10 mW be focused if the irradiance in the spot is to be the same as the sun’s irradiance at the surface of the earth? (The irradiance of the sun at the earth’s surface is about 1000 W/m2.) Get solution
20. For a Nd:YAG laser, there are four pump levels located at 1.53 eV, 1.653 eV, 2.119 eV, and 2.361 eV above the ground state energy level.a. What is the wavelength associated with the photon energy required to populate each of the pump levels?b. Knowing that a Nd:YAG laser emits photons of wavelength 1.064 μm, determine the quantum efficiency associated with each of the four pump levels. (The quantum efficiency is the ratio of the energy of a single pump event to that of an output photon.) Get solution
21. To operate a Nd:YAG laser, 2500 W of “wall-plug” power are required for a power supply that drives the arc lamps. The arc lamps provide pump energy to create the population inversion. The overall laser system, from power in (to the power supply) to power out (laser output beam), is characterized by the following component efficiencies:80%—power supply operation30%—arc lamps for pump light energy70%—optical reflectors for concentrating pump light on laser rod15%—for spectral match of pump light to Nd:YAG pump levels50%—due to internal cavity/rod lossesa. Taking the efficiencies into account sequentially as they “occur,” how much of the initial 2500 W is available for power in the output beam?b. What is the overall operational efficiency (wall-plug efficiency) for this laser? Get solution
22. Table 6-1 indicates that diode lasers have a large divergence angle. Why is this reasonable? Get solution
24. Why is a Nd:YAG laser system that uses a diode laser as a pump more efficient that a Nd:YAG laser system that uses an arc lamp as a pump? See Table 6.1. Get solution
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