A population inversion for two energy levels is often

An electron in a hydrogen atom is in a state with . What is the minimum possible value of the semiclassical angle between and Lz?

How many electron states are there in a shell defined by the quantum number n ! 5?

(a) What is the magnitude of the orbital angular momentum in a state with ? (b) What is the magnitude of its largest projection on an imposed z axis?

How many electron states are there in the following shells: (a) n ! 4, (b) n ! 1, (c) n ! 3, (d) n ! 2?

(a) How many values are associated with n 3? (b) How many values are associated with ?

How many electron states are in these subshells: (a) n ! 4, ; (b) n ! 3, ; (c) n ! 4, ; (d) n ! 2, ?

An electron in a multielectron atom has . For this electron, what are (a) the value of , (b) the smallest possible value of n, and (c) the number of possible values of ms?

In the subshell , (a) what is the greatest (most positive) value, (b) how many states are available with the greatest value, and (c) what is the total number of states available in the subshell?

An electron is in a state with . (a) What ! ! 3 m! m! ! ! 3 ! m! ! '4 ! ! 3 ! ! 1 ! ! 1 ! ! 0 m! ! ! 1 ! ! ! ! 3 L : ! ! 5 Tutoring problem available (at instructors discretion) in WileyPLUS and WebAssign SSM Worked-out solution available in Student Solutions Manual Number of dots indicates level of problem difficulty Additional information available in The Flying Circus of Physics and at flyingcircusofphysics.com WWW Worked-out solution is at ILW Interactive solution is at http://www.wiley.com/college/halliday Problems multiple of gives the magnitude of ? (b) What multiple of mB gives the magnitude of ? (c) What is the largest possible value of , (d) what multiple of gives the corresponding value of Lz, and (e) what multiple of mB gives the corresponding value of morb,z? (f) What is the value of the semiclassical angle u between the directions of Lz and ? What is the value of angle u for (g) the second largest possible value of and (h) the smallest (that is, most negative) possible value of ?

An electron is in a state with n ! 3.What are (a) the number of possible values of , (b) the number of possible values of , (c) the number of possible values of ms, (d) the number of states in the n ! 3 shell, and (e) the number of subshells in the n ! 3 shell?

If orbital angular momentum is measured along, say, a z axis to obtain a value for Lz, show that is the most that can be said about the other two components of the orbital angular momentum.

A magnetic field is applied to a freely floating uniform iron sphere with radius The sphere initially had no net magnetic moment, but the field aligns 12% of the magnetic moments of the atoms (that is, 12% of the magnetic moments of the loosely bound electrons in the sphere, with one such electron per atom). The magnetic moment of those aligned electrons is the spheres intrinsic magnetic moment . What is the spheres resulting angular speed ?

What is the acceleration of a silver atom as it passes through the deflecting magnet in the SternGerlach experiment of Fig. 40-8 if the magnetic field gradient is 1.4 T/mm?

Suppose that a hydrogen atom in its ground state moves 80 cm through and perpendicular to a vertical magnetic field that has a magnetic field gradient dB/dz ! 1.6 " 102 T/m. (a) What is the magnitude of force exerted by the field gradient on the atom due to the magnetic moment of the atoms electron, which we take to be 1 Bohr magneton? (b) What is the vertical displacement of the atom in the 80 cm of travel if its speed is 1.2 " 105 m/s?

Calculate the (a) smaller and (b) larger value of the semiclassical angle between the electron spin angular momentum vector and the magnetic field in a SternGerlach experiment. Bear in mind that the orbital angular momentum of the valence electron in the silver atom is zero

Assume that in the SternGerlach experiment as described for neutral silver atoms, the magnetic field has a magnitude of 0.50 T. (a) What is the energy difference between the magnetic moment orientations of the silver atoms in the two subbeams? (b) What is the frequency of the radiation that would induce a transition between these two states? (c) What is the wavelength of this radiation, and (d) to what part of the electromagnetic spectrum does it belong?

In an NMR experiment, the RF source oscillates at 34 MHz and magnetic resonance of the hydrogen atoms in the sample being investigated occurs when the external field has magnitude 0.78 T. Assume that and are in the same direction and take the proton magnetic moment component mz to be 1.41 10#26 J/T. What is the magnitude of ?

A hydrogen atom in its ground state actually has two possible, closely spaced energy levels because the electron is in the magnetic field of the proton (the nucleus). Accordingly, an energy is associated with the orientation of the electrons magnetic moment relative to , and the electron is said to be either spin up (higher energy) or spin down (lower energy) in that field. If the electron is excited to the higher-energy level, it can de-excite by spin-flipping and emitting a photon. The wavelength associated with that photon is 21 cm. (Such a process occurs extensively in the Milky Way galaxy, and reception of the 21 cm radiation by radio telescopes reveals where hydrogen gas lies between stars.) What is the effective magnitude of as experienced by the electron in the ground-state hydrogen atom?

What is the wavelength associated with a photon that will induce a transition of an electron spin from parallel to antiparallel orientation in a magnetic field of magnitude 0.200T? Assume that .

A rectangular corral of widths Lx ! L and Ly ! 2L contains seven electrons. What multiple of h2 /8mL2 gives the energy of the ground state of this system? Assume that the electrons do not interact with one another, and do not neglect spin.

Seven electrons are trapped in a one-dimensional infinite potential well of width L. What multiple of h2 /8mL2 gives the energy of the ground state of this system? Assume that the electrons do not interact with one another,and do not neglect spin.

Figure 40-23 is an energy-level diagram for a fictitious infinite potential well that contains one electron. The number of degenerate states of the levels are indicated: non means nondegenerate (which includes the ground state of the electron), double means 2 states, and triple means 3 states. We put a total of 11 electrons in the well. If the electrostatic forces between the electrons can be neglected, what multiple of gives the energy of the first excited state of the 11-electron system?

A cubical box of widths Lx ! Ly ! Lz ! L contains eight electrons. What multiple of h2 /8mL2 gives the energy of the ground state of this system? Assume that the electrons do not interact with one another, and do not neglect spin.

For Problem 20, what multiple of h2 /8mL2 gives the energy of (a) the first excited state, (b) the second excited state, and (c) the third excited state of the system of seven electrons? (d) Construct an energy-level diagram for the lowest four energy levels.

For the situation of Problem 21, what multiple of h2 /8mL2 gives the energy of (a) the first excited state, (b) the second excited state, and (c) the third excited state of the system of seven electrons? (d) Construct an energy-level diagram for the lowest four energy levels of the system.

For the situation of Problem 23, what multiple of h2 /8mL2 gives the energy of (a) the first excited state, (b) the second excited state, and (c) the third excited state of the system of eight electrons? (d) Construct an energy-level diagram for the lowest four energy levels of the system.

Two of the three electrons in a lithium atom SSM h2 /8mL2 Figure 40-23 Problem 22. Non E (h2/8mL2) 12 11 Triple 7 Double 6 Triple 4 Ground have quantum numbers (n, , ms) of and . What quantum numbers are possible for the third electron if the atom is (a) in the ground state and (b) in the first excited state?

Show that the number of states with the same quantum number n is 2n2 .

A recently named element is darmstadtium (Ds), which has 110 electrons. Assume that you can put the 110 electrons into the atomic shells one by one and can neglect any electron electron interaction. With the atom in ground state, what is the spectroscopic notation for the quantum number for the last electron?

For a helium atom in its ground state, what are quantum numbers ( ) for the (a) spin-up electron and (b) spindown electron?

Consider the elements selenium (Z ! 34), bromine (Z ! 35), and krypton (Z ! 36). In their part of the periodic table, the sub shells of the electronic states are filled in the sequence 1s 2s 2p 3s 3p 3d 4s 4p .... What are (a) the highest occupied subshell for selenium and (b) the number of electrons in it, (c) the highest occupied subshell for bromine and (d) the number of electrons in it, and (e) the highest occupied subshell for krypton and (f) the number of electrons in it?

Suppose two electrons in an atom have quantum numbers n ! 2 and . (a) How many states are possible for those two electrons? (Keep in mind that the electrons are indistinguishable.) (b) If the Pauli exclusion principle did not apply to the electrons, how many states would be possible?

Through what minimum potential difference must an electron in an x-ray tube be accelerated so that it can produce x rays with a wavelength of 0.100 nm?

The wavelength of the Ka line from iron is 193 pm. What is the energy difference between the two states of the iron atom that give rise to this transition?

In Fig. 40-13, the x rays shown are produced when 35.0 keV electrons strike a molybdenum (Z 42) target. If the accelerating potential is maintained at this value but a silver (Z ! 47) target is used instead, what values of (a) lmin, (b) the wavelength of the Ka line, and (c) the wavelength of the Kb line result? The K, L, and M atomic x-ray levels for silver (compare Fig. 40-15) are 25.51, 3.56, and 0.53 keV

When electrons bombard a molybdenum target, they produce both continuous and characteristic x rays as shown in Fig. 40-13. In that figure the kinetic energy of the incident electrons is 35.0 keV. If the accelerating potential is increased to 50.0 keV, (a) what is the value of lmin, and (b) do the wavelengths of the Ka and Kb lines increase, decrease, or remain the same?

Show that a moving electron cannot spontaneously change into an x-ray photon in free space. A third body (atom or nucleus) must be present. Why is it needed? (Hint: Examine the conservation of energy and momentum.)

Here are the Ka wavelengths of a few elements: Element l (pm) Element l (pm) Ti 275 Co 179 V 250 Ni 166 Cr 229 Cu 154 Mn 210 Zn 143 Fe 193 Ga 134 Make a Moseley plot (like that in Fig. 40-16) from these data and verify that its slope agrees with the value given for C in Module 40-6.

Calculate the ratio of the wavelength of the Ka line for niobium (Nb) to that for gallium (Ga). Take needed data from the periodic table of Appendix G.

(a) From Eq. 40-26, what is the ratio of the photon energies due to Ka transitions in two atoms whose atomic numbers are Z and Z4? (b) What is this ratio for uranium and aluminum? (c) For uranium and lithium?

The binding energies of K-shell and L-shell electrons in copper are 8.979 and 0.951 keV, respectively. If a Ka x ray from copper is incident on a sodium chloride crystal and gives a first-order Bragg reflection at an angle of 74.1 measured relative to parallel planes of sodium atoms, what is the spacing between these parallel planes?

From Fig. 40-13, calculate approximately the energy difference EL # EM for molybdenum. Compare it with the value that may be obtained from Fig. 40-15.

A tungsten (Z ! 74) target is bombarded by electrons in an x-ray tube. The K, L, and M energy levels for tungsten (compare Fig. 40-15) have the energies 69.5, 11.3, and 2.30 keV, respectively. (a) What is the minimum value of the accelerating potential that will permit the production of the characteristic Ka and Kb lines of tungsten? (b) For this same accelerating potential, what is lmin? What are the (c) Ka and (d) Kb wavelengths?

A 20 keV electron is brought to rest by colliding twice with target nuclei as in Fig. 40-14. (Assume the nuclei remain stationary.) The wavelength associated with the photon emitted in the second collision is 130 pm greater than that associated with the photon emitted in the first collision. (a) What is the kinetic energy of the electron after the first collision? What are (b) the wavelength l1 and (c) the energy E1 associated with the first photon? What are (d) l2 and (e) E2 associated with the second photon?

X rays are produced in an x-ray tube by electrons accelerated through an electric potential difference of 50.0 kV. Let K0 be the kinetic energy of an electron at the end of the acceleration. The electron collides with a target nucleus (assume the nucleus remains stationary) and then has kinetic energy K1 ! 0.500K0. (a) What wavelength is associated with the photon that is emitted? The electron collides with another target nucleus (assume it, too, remains stationary) and then has kinetic energy K2 ! 0.500K1. (b) What wavelength is associated with the photon that is emitted?

Determine the constant C in Eq. 40-27 to five significant figures by finding C in terms of the fundamental constants in Eq. 40-24 and then using data from Appendix B to evaluate those constants. Using this value of C in Eq. 40-27, determine the theoretical energy Etheory of the Ka photon for the low-mass elements listed in the following table. The table includes the value (eV) of the measured energy Eexp of the Ka photon for each listed element. The percentage deviation between Etheory and Eexp can be calculated as What is the percentage deviation for (a) Li, (b) Be, (c) B, (d) C, (e) N, (f) O, (g) F, (h) Ne, (i) Na, and (j) Mg? Li 54.3 O 524.9 Be 108.5 F 676.8 B 183.3 Ne 848.6 C 277 Na 1041 N 392.4 Mg 1254 (There is actually more than one Ka ray because of the splitting of the L energy level, but that effect is negligible for the elements listed here.)

The active volume of a laser constructed of the semiconductor GaAlAs is only 200 mm3 (smaller than a grain of sand), and yet the laser can continuously deliver 5.0 mW of power at a wavelength of 0.80 mm.At what rate does it generate photons?

A high-powered laser beam (l ! 600 nm) with a beam diameter of 12 cm is aimed at the Moon, 3.8 " 105 km distant.The beam spreads only because of diffraction. The angular location of the edge of the central diffraction disk (see Eq. 36-12) is given by where d is the diameter of the beam aperture. What is the diameter of the central diffraction disk on the Moons surface?

Assume that lasers are available whose wavelengths can be precisely tuned to anywhere in the visible rangethat is, in the range 450 nm / l / 650 nm. If every television channel occupies a bandwidth of 10 MHz, how many channels can be accommodated within this wavelength range?

A hypothetical atom has only two atomic energy levels, separated by 3.2 eV. Suppose that at a certain altitude in the atmosphere of a star there are 6.1 " 1013/cm3 of these atoms in the higher-energy state and 2.5 " 1015/cm3 in the lower-energy state. What is the temperature of the stars atmosphere at that altitude?

A hypothetical atom has only two atomic energy levels, separated by 3.2 eV. Suppose that at a certain altitude in the atmosphere of a star there are 6.1 " 1013/cm3 of these atoms in the higher-energy state and 2.5 " 1015/cm3 in the lower-energy state. What is the temperature of the stars atmosphere at that altitude?

A laser emits at 424 nm in a single pulse that lasts 0.500 ms. The power of the pulse is 2.80 MW. If we assume that the atoms contributing to the pulse underwent stimulated emission only once during the 0.500 ms,how many atoms contributed?

A heliumneon laser emits laser light at a wavelength of 632.8 nm and a power of 2.3 mW.At what rate are photons emitted by this device?

A certain gas laser can emit light at wavelength 550 nm, which involves population inversion between ground state and an excited state. At room temperature, how many moles of neon are needed to put 10 atoms in that excited state by thermal agitation?

A pulsed laser emits light at a wavelength of 694.4 nm. The pulse duration is 12 ps, and the energy per pulse is 0.150 J. (a) What is the length of the pulse? (b) How many photons are emitted in each pulse?

A population inversion for two energy levels is often described by assigning a negative Kelvin temperature to the system. What negative temperature would describe a system in which the population of the upper energy level exceeds that of the lower level by 10% and the energy difference between the two levels is 2.26 eV?

A hypothetical atom has two energy levels, with a transition wavelength between them of 580 nm. In a particular sample at 300 K, 4.0 " 1020 such atoms are in the state of lower energy. (a) How many atoms are in the upper state, assuming conditions of thermal equilibrium? (b) Suppose, instead, that 3.0 " 1020 of these atoms are pumped into the upper state by an external process, with 1.0 " 1020 atoms remaining in the lower state. What is the maximum energy that could be released by the atoms in a single laser pulse if each atom jumps once between those two states (either via absorption or via stimulated emission)?

The mirrors in the laser of Fig. 40-20, which are separated by 8.0 cm, form an optical cavity in which standing waves of laser light can be set up. Each standing wave has an integral number n of half wavelengths in the 8.0 cm length, where n is large and the waves differ slightly in wavelength. Near l ! 533 nm, how far apart in wavelength are the standing waves?

Figure 40-24 shows the energy levels of two types of atoms. Atoms A are in one tube, and atoms B are in another tube. The energies (relative to a ground-state energy of zero) are indicated; the average lifetime of atoms in each level is also indicated. All the atoms are initially pumped to levels higher than the levels shown in the figure. The atoms then drop down through the levels, and many become stuck on certain levels, leading to population inversion and lasing. The light emitted by A illuminates B and can cause stimulated emission of B. What is the energy per photon of that stimulated emission of B?

The beam from an argon laser (of wavelength 515 nm) has a diameter d of 3.00 mm and a continuous energy output rate of 5.00 W. The beam is focused onto a diffuse surface by a lens whose focal length f is 3.50 cm. A diffraction pattern such as that of Fig. 36-10 is formed, the radius of the central disk being given by

The active medium in a particular laser that generates laser light at a wavelength of 694 nm is 6.00 cm long and 1.00 cm in diameter. (a) Treat the medium as an optical resonance cavity analogous to a closed organ pipe. How many standing-wave nodes are there along the laser axis? (b) By what amount )f would the beam frequency have to shift to increase this number by one? (c) Show that )f is just the inverse of the travel time of laser light for one round trip back and forth along the laser axis. (d) What is the corresponding fractional frequency shift )f/f? The appropriate index of refraction of the lasing medium (a ruby crystal) is 1.75.

Ruby lases at a wavelength of 694 nm. A certain ruby crystal has Cr ions (which are the atoms that lase).The lasing transition is between the first excited state and the ground state, and the output is a light pulse lasting 2.00 ms.As the pulse begins, 60.0% of the Cr ions are in the first excited state and the rest are in the ground state. What is the average power emitted during the pulse? (Hint: Dont just ignore the ground-state ions.)

Figure 40-25 is an energy-level diagram for a fictitious three-dimensional infinite potential well that contains one electron. The number of degenerate states of the levels are indicated: non means nondegenerate (which includes the ground state) and triple means 3 states. If we put a total of 22 electrons in the well, what multiple of gives the energy of the ground state of the 22-electron system? Assume that the electrostatic forces between the electrons are negligible

ight shines on the atmosphere of Mars, carbon dioxide molecules at an altitude of about 75 km undergo natural laser action. The energy levels involved in the action are shown in Fig. 40-26; population inversion occurs between energy levels E2 and E1. (a) What wavelength of sunlight excites the molecules in the lasing action? (b) At what wavelength does lasing occur? (c) In what region of the electromagnetic spectrum do the excitation and lasing wavelengths lie?

Excited sodium atoms emit two closely spaced spectrum lines called the sodium doublet (Fig. 40-27) with wavelengths 588.995 nm and 589.592 nm. (a) What is the difference in energy between the two upper energy levels (n ! 3, )? (b) This energy difference occurs because the electrons spin magnetic moment can be oriented either parallel or antiparallel to the internal magnetic field associated with the electrons orbital motion. Use your result in (a) to find the magnitude of this internal magnetic field

Comet stimulated emission. When a comet approaches the Sun, the increased warmth evaporates water from the ice on the surface of the comet nucleus, producing a thin atmosphere of water vapor around the nucleus. Sunlight can then dissociate H2O molecules in the vapor to H atoms and OH molecules. The sunlight can also excite the OH molecules to higher energy levels. When the comet is still relatively far from the Sun, the sunlight causes equal excitation to the E2 and E1 levels (Fig. 40- 28a). Hence, there is no population inversion between the two levels. However, as the comet approaches the Sun, the excitation to the E1 level decreases and population inversion occurs. The reason has to do with one of the many wavelengthssaid to be Fraunhofer linesthat are missing in sunlight because, as the light travels outward through the Suns atmosphere, those particular wavelengths are absorbed by the atmosphere. As a comet approaches the Sun, the Doppler effect due to the comets speed relative to the Sun shifts the Fraunhofer lines in wavelength, apparently overlapping one of them with the wavelength required for excitation to the E1 level in OH molecules. Population inversion then occurs in those molecules, and they radiate stimulated emission (Fig. 40- 28b). For example, as comet Kouhoutek approached the Sun in December 1973 and January 1974, it radiated stimulated emission at about 1666 MHz during mid-January. (a) What was the energy difference E2 # E1 for that emission? (b) In what region of the electromagnetic spectrum was the emission?

Show that the cutoff wavelength (in picometers) in the continuous x-ray spectrum from any target is given by lmin ! 1240/V, where V is the potential difference (in kilovolts) through which the electrons are accelerated before they strike the target.

By measuring the go-and-return time for a laser pulse to travel from an Earth-bound observatory to a reflector on the Moon, it is possible to measure the separation between these bodies. (a) What is the predicted value of this time? (b) The separation can be measured to a precision of about 15 cm. To what uncertainty in travel time does this correspond? (c) If the laser beam forms a spot on the Moon 3 km in diameter, what is the angular divergence of the beam?

Can an incoming intercontinental ballistic missile be destroyed by an intense laser beam? A beam of intensity 108 W/m2 would probably burn into and destroy a nonspinning missile in 1 s. (a) If the laser had 5.0 MW power, 3.0 mm wavelength, and a 4.0 m beam diameter (a very powerful laser indeed), would it destroy a missile at a distance of 3000 km? (b) If the wavelength could be changed, what maximum value would work? Use the equation for the central diffraction maximum as given by Eq. 36- 12 (sin u ! 1.22l/d).

A molybdenum (Z ! 42) target is bombarded with 35.0 keV electrons and the x-ray spectrum of Fig. 40-13 results. The Kb and Ka wavelengths are 63.0 and 71.0 pm, respectively. What photon energy corresponds to the (a) Kb and (b) Ka radiation? The two radiations are to be filtered through one of the substances in the following table such that the substance absorbs the Kb line more strongly than the Ka line. A substance will absorb radiation x1 more strongly than it absorbs radiation x2 if a photon of x1 has enough energy to eject a K electron from an atom of the substance but a photon of x2 does not.The table gives the ionization energy of the K electron in molybdenum and four other substances. Which substance in the table will serve (c) best and (d) second best as the filter? Zr Nb Mo Tc Ru Z 40 40 42 43 44 EK (keV) 18.00 18.99 20.00 21.04 22.12

An electron in a multielectron atom is known to have the quantum number . What are its possible , and ms quantum numbers?

Show that if the 63 electrons in an atom of europium were assigned to shells according to the logical sequence of quantum numbers, this element would be chemically similar to sodium.

Lasers can be used to generate pulses of light whose durations are as short as 10 fs. (a) How many wavelengths of light (l ! 500 nm) are contained in such a pulse? (b) In what is the missing quantity X (in years)?

Show that + ! 1.06 " 10 #34 J$s ! 6.59 " 10 #16 eV$s

Suppose that the electron had no spin and that the Pauli exclusion principle still held. Which, if any, of the present noble gases would remain in that category?

(A correspondence principle problem.) Estimate (a) the quantum number for the orbital motion of Earth around the Sun and (b) the number of allowed orientations of the plane of Earths orbit. (c) Find umin, the half-angle of the smallest cone that can be swept out by a perpendicular to Earths orbit as Earth revolves around the Sun.

Knowing that the minimum x-ray wavelength produced by 40.0 keV electrons striking a target is 31.1 pm, determine the Planck constant h

Consider an atom with two closely spaced excited states A and B. If the atom jumps to ground state from A or from B, it emits a wavelength of 500 nm or 510 nm, respectively. What is the energy difference between states A and B?

In 1911, Ernest Rutherford modeled an atom as being a point of positive charge Ze surrounded by a negative charge #Ze uniformly distributed in a sphere of radius R centered at the point. At distance r within the sphere, the electric potential is (a) From this formula, determine the magnitude of electric field for 0 0 r 0 R.What are the (b) electric field and (c) potential for ?