Chapter 40

An electron in a hydrogen atom is in a state with \(\ell=5\). What is the minimum possible value of the semiclassical angle between \(\vec{L}\) and \(L_{z} ?\)

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 \(\ell=3 ?\) (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) \(n=1,(\mathrm{c}) n=3,(\) d \() n=2 ?\)

(a) How many \(\ell\) values are associated with \(n=3\) ? (b) How many \(m_{\ell}\) values are associated with \(\ell=1 ?\)

How many electron states are in these subshells: (a) \(n=4\), \(\ell=3 ;\) (b) \(n=3, \ell=1 ;\) (c) \(n=4, \ell=1 ;\) (d) \(n=2, \ell=0\) ?

In the subshell \(\ell=3\), (a) what is the greatest (most positive) \(m_{e}\) value, (b) how many states are available with the greatest \(m_{\ell}\) value, and (c) what is the total number of states available in the subshell?

An electron is in a state with \(\ell=3\). (a) What multiple of \(\hbar\) gives the magnitude of \(\vec{L} ?\) (b) What multiple of \(\mu_{\mathrm{B}}\) gives the magnitude of \(\vec{\mu} ?(\mathrm{c})\) What is the largest possible value of \(m_{e}\), (d) what multiple of \(\hbar\) gives the corresponding value of \(L_{z}\), and (e) what multiple of \(\mu_{\mathrm{B}}\) gives the corresponding value of \(\mu_{\text {orb } z}\) ? (f) What is the value of the semiclassical angle \(\theta\) between the directions of \(L\), and \(\vec{L}\) ? What is the value of angle \(\theta\) for \((\mathrm{g})\) the second largest possible value of \(m_{\ell}\) and (h) the smallest (that is, most negative) possible value of \(m_{i}\) ?

An electron is in a state with \(n=3\). What are (a) the number of possible values of \(\ell,(\mathrm{b})\) the number of possible values of \(m_{e},(\mathrm{c})\) the number of possible values of \(m_{s},(\mathrm{~d})\) the number of states in the \(n=3\) shell, and (e) the number of subshells in the \(n=3\) shell?

If orbital angular momentum \(\vec{L}\) is measured along, say, \(a z\) axis to obtain a value for \(L_{z}\), show that $$ \left(L_{x}^{2}+L_{y}^{2}\right)^{1 / 2}=\left[\ell(\ell+1)-m_{\ell}^{2}\right]^{1 / 2} \hbar $$ is the most that can be said about the other two components of the orbital angular momentum.

Suppose that a hydrogen atom in its ground state moves 80 \(\mathrm{cm}\) through and perpendicular to a vertical magnetic field that has a magnetic field gradient \(d B / d z=1.6 \times 10^{2} \mathrm{~T} / \mathrm{m} .\) (a) What is the magnitude of force exerted by the field gradient on the atom due to the magnetic moment of the atom's electron, which we take to be 1 Bohr magneton? (b) What is the vertical displacement of the atom in the \(80 \mathrm{~cm}\) of travel if its speed is \(1.2 \times 10^{5} \mathrm{~m} / \mathrm{s}\) ?

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 Stern-Gerlach experiment. Bear in mind that the orbital angular momentum of the valence electron in the silver atom is zero.

In an NMR experiment, the RF source oscillates at \(34 \mathrm{MHz}\) and magnetic resonance of the hydrogen atoms in the sample being investigated occurs when the external field \(\vec{B}_{\text {ext }}\) has magnitude \(0.78 \mathrm{~T}\). Assume that \(\vec{B}_{\text {int }}\) and \(\vec{B}_{\text {ext }}\) are in the same direction and take the proton magnetic moment component \(\mu_{z}\) to be \(1.41 \times 10^{-26} \mathrm{~J} / \mathrm{T}\). What is the magnitude of \(\vec{B}_{\text {int }} ?\)

A rectangular corral of widths \(L_{x}=L\) and \(L_{y}=2 L\) contains seven electrons. What multiple of \(h^{2} / 8 m L^{2}\) 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 \(h^{2} / 8 m L^{2}\) 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.

A cubical box of widths \(L_{x}=L_{y}=L_{z}=L\) contains eight electrons. What multiple of \(h^{2} / 8 m L^{2}\) 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.

Two of the three electrons in a lithium atom have quantum numbers \(\left(n, \ell, m_{\varepsilon}, m_{s}\right)\) of \(\left(1,0,0,+\frac{1}{2}\right)\) and \(\left(1,0,0,-\frac{1}{2}\right)\). 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 \(2 n^{2}\).

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 electronelectron interaction. With the atom in ground state, what is the spectroscopic notation for the quantum number \(\ell\) for the last electron?

For a helium atom in its ground state, what are quantum numbers \(\left(n, \ell, m_{\ell}\right.\), and \(\left.m_{s}\right)\) 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 subshells of the electronic states are filled in the sequence $$ 1 s 2 s 2 p 3 s 3 p 3 d 4 s 4 p \ldots $$ 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 \(\ell=1 .\) (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 \(0.100 \mathrm{~nm}\) ?

The wavelength of the \(K_{\alpha}\) line from iron is \(193 \mathrm{pm}\). What is the energy difference between the two states of the iron atom that give rise to this transition?

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.)

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

\( X\) rays are produced in an \(x\) -ray tube by electrons accelerated through an electric potential difference of \(50.0 \mathrm{kV}\). Let \(K_{0}\) 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 \(K_{1}=\) \(0.500 K_{0}\). (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 \(K_{2}=0.500 K_{1} .\) (b) What wavelength is associated with the photon that is emitted?

The active volume of a laser constructed of the semiconductor GaAlAs is only \(200 \mu \mathrm{m}^{3}\) (smaller than a grain of sand), and yet the laser can continuously deliver \(5.0 \mathrm{~mW}\) of power at \(a\) wavelength of \(0.80 \mu \mathrm{m}\). At what rate does it gencrate photons?

A high-powered laser beam \((\lambda=600 \mathrm{~nm})\) with a beam diameter of \(12 \mathrm{~cm}\) is aimed at the Moon, \(3.8 \times 10^{5} \mathrm{~km}\) 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 $$ \sin \theta=\frac{1.22 \lambda}{d} $$ where \(d\) is the diameter of the beam aperture. What is the diameter of the central diffraction disk on the Moon's surface?

Assume that lasers are available whose wavelengths can be precisely "tuned" to anywhere in the visible range \(-\) that is, in the range \(450 \mathrm{~nm}<\lambda<650 \mathrm{~nm}\). If every television channel occupies a bandwidth of \(10 \mathrm{MHz}\), how many channels can be accommodated within this wavelength range?

A hypothetical atom has only two atomic energy levels, separated by \(3.2 \mathrm{eV}\). Suppose that at a certain altitude in the atmosphere of a star there are \(6.1 \times 10^{13} / \mathrm{cm}^{3}\) of these atoms in the higher- energy state and \(2.5 \times 10^{15} / \mathrm{cm}^{3}\) in the lower-energy state. What is the temperature of the star's atmosphere at that altitude?

A hypothetical atom has energy levels uniformly separated by \(1.2 \mathrm{eV}\). At a temperature of \(2000 \mathrm{~K}\), what is the ratio of the number of atoms in the 13 th excited state to the number in the 11th excited state?

A laser emits at \(424 \mathrm{~nm}\) in a single pulse that lasts \(0.500 \mu \mathrm{s}\). The power of the pulse is \(2.80 \mathrm{MW}\). If we assume that the atoms contributing to the pulse underwent stimulated emission only once during the \(0.500 \mu \mathrm{s}\), how many atoms contributed?

A helium-neon laser emits laser light at a wavelength of \(632.8 \mathrm{~nm}\) and a power of \(2.3 \mathrm{~mW}\). At what rate are photons emitted by this device?

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

A hypothetical atom has two energy levels, with a transition wavelength between them of \(580 \mathrm{~nm}\). In a particular sample at 300 K, \(4.0 \times 10^{20}\) 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 \(3.0 \times 10^{20}\) of these atoms are "pumped" into the upper state by an external process, with \(1.0 \times 10^{20}\) 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 beam from an argon laser (of wavelength \(515 \mathrm{~nm}\) ) has a diameter \(d\) of \(3.00 \mathrm{~mm}\) and a continuous energy output rate of \(5.00 \mathrm{~W}\). The beam is focused onto a diffuse surface by a lens whose focal length \(f\) is \(3.50 \mathrm{~cm}\). A diffraction pattern such as that of Fig. \(36-10\) is formed. the radius of the central disk being given by $$ R=\frac{1.22 f \lambda}{d} $$ (see Eq. 36-12 and Fig. 36-14). The central disk can be shown to con\(\operatorname{tain} 84 \%\) of the incident power. (a) What is the radius of the central disk? (b) What is the average intensity (power per unit area) in the incident beam? (c) What is the average intensity in the central disk?

The active medium in a particular laser that generates laser light at a wavelength of \(694 \mathrm{~nm}\) is \(6.00 \mathrm{~cm}\) long and \(1.00 \mathrm{~cm}\) 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 \(\Delta f\) would the beam frequency have to shift to increase this number by one? (c) Show that \(\Delta f\) 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 \(\Delta f l f ?\) The appropriate index of refraction of the lasing medium (a ruby crystal) is \(1.75\).

Ruby lases at a wavelength of \(694 \mathrm{~nm}\). A certain ruby crystal has \(4.00 \times 10^{19} \mathrm{Cr}\) 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 \(2.00 \mu\) s. As the pulse begins, \(60.0 \%\) of the \(\mathrm{Cr}\) 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: Don't just ignore the ground-state ions.)

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 \(\mathrm{H}_{2} \mathrm{O}\) molecules in the vapor to \(\mathrm{H}\) atoms and \(\mathrm{OH}\) molecules. The sunlight can also excite the \(\mathrm{OH}\) molecules to higher energy levels. When the comet is still relatively far from the Sun, the sunlight causes equal excitation to the \(E_{2}\) and \(E_{1}\) levels (Fig. \(40-28 a\) ). Hence, there is no population inversion between the two levels. However, as the comet approaches the Sun, the excitation to the \(E_{1}\) level decreases and population inversion occurs. The reason has to do with one of the many wavelengths - said to be Fraunhofer lines-that are missing in sunlight because, as the light travels outward through the Sun's atmosphere, those particular wavelengths are absorbed by the atmosphere. As a comet approaches the Sun, the Doppler effect due to the comet's speed relative to the Sun shifts the Fraunhofer lines in wavelength, apparently overlapping one of them with the wavelength required for excitation to the \(E_{1}\) level in OH molecules. Population inversion then occurs in those molecules, and they radiate stimulated emission (Fig. \(40-28 b\) ). 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 \(E_{2}-E_{1}\) for that emission? (b) In what region of the electromagnetic spectrum was the emission?

Show that the cutoff wavelength (in picometers) in the continuous \(\mathrm{x}\) -ray spectrum from any target is given by \(\lambda_{\min }=1240 / \mathrm{V}\), 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 bodjes. (a) What is the predicted value of this time? (b) The separation can be measured to a precision of about \(15 \mathrm{~cm}\). To what uncertainty in travel time does this correspond? (c) If the laser beam forms a spot on the Moon \(3 \mathrm{~km}\) in diameter, what is the angular divergence of the beam? x

A molybdenum \((Z=42)\) target is bombarded with \(35.0 \mathrm{keV}\) electrons and the \(x\) -ray spectrum of Fig. \(40-13\) results. The \(K_{\beta}\) and \(K_{a}\) wavelengths are \(63.0\) and \(71.0 \mathrm{pm}\), respectively. What photon energy corresponds to the (a) \(K_{\beta}\) and (b) \(K_{\alpha}\) radiation? The two radiations are to be filtered through one of the substances in the following table such that the substance absorbs the \(K_{\beta}\) line more strongly than the \(K_{\alpha}\) line. A substance will absorb radiation \(x_{1}\) more strongly than it absorbs radiation \(x_{2}\) if a photon of \(x_{1}\) has enough energy to eject a \(K\) electron from an atom of the substance but a photon of \(x_{2}\) 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? $$ \begin{array}{llllll} \hline & \mathrm{Zr} & \mathrm{Nb} & \mathrm{Mo} & \mathrm{Tc} & \mathrm{Ru} \\\ \hline Z & 40 & 40 & 42 & 43 & 44 \\ E_{K}(\mathrm{keV}) & 18.00 & 18.99 & 20.00 & 21.04 & 22.12 \end{array} $$

An electron in a multielectron atom is known to have the quantum number \(\ell=3 .\) What are its possible \(n, m_{\ell}\), and \(m_{s}\) 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 \(10 \mathrm{fs}\). (a) How many wavelengths of light \((\lambda=500 \mathrm{~nm})\) are contained in such a pulse? (b) In $$ \frac{10 \mathrm{fs}}{1 \mathrm{~s}}=\frac{1 \mathrm{~s}}{X} $$ what is the missing quantity \(X\) (in years)?

Show that \(\hbar=1.06 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}=6.59 \times 10^{-16} \mathrm{eV} \cdot \mathrm{s}\)

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 \(\ell\) for the orbital motion of Earth around the Sun and (b) the number of allowed orientations of the plane of Earth's orbit. (c) Find \(\theta_{\min }\), the half-angle of the smallest cone that can be swept out by a perpendicular to Earth's orbit as Earth revolves around the Sun.

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