University Physics with Modern Physics In SI Units

Hugh D Young; Roger A Freedman

Chapter 41

Quantum Mechanics II: Atomic Structure - all with Video Answers

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Chapter Questions

Problem 1

For a particle in a three-dimensional cubical box, what is the degeneracy (number of different quantum states with the same energy) of the energy levels
(a) $3 \pi^{2} \hbar^{2} / 2 m L^{2}$ and
(b) $9 \pi^{2} \hbar^{2} / 2 m L^{2} ?$

Khaled Yasein

Khaled Yasein

Numerade Educator

Problem 2

Model a hydrogen atom as an electron in a cubical box with side length $L$. Set the value of $L$ so that the volume of the box equals the volume of a sphere of radius $a=5.29 \times 10^{-11} \mathrm{~m}$, the Bohr radius. Calculate the energy separation between the ground and first excited levels, and compare the result to this energy separation calculated from the Bohr model.

Ommair Ishaque

Numerade Educator

Problem 3

A photon is emitted when an electron in a three-dimensional cubical box of side length $9.00 \times 10^{-11} \mathrm{~m}$ makes a transition from the $n_{X}=2, n_{Y}=2, n_{Z}=1$ state to the $n_{X}=1, n_{Y}=1, n_{Z}=1$ state.
What is the wavelength of this photon?

Ommair Ishaque

Numerade Educator

Problem 4

For each of the following states of a particle in a three-dimensional cubical box, at what points is the probability distribution function a maxi-
mum:
(a) $n_{X}=1, n_{Y}=1, n_{7}=1$ and
(b) $n_{X}=2, n_{Y}=2, n_{7}=1 ?$

Khaled Yasein

Numerade Educator

Problem 5

A photon with wavelength $8.00 \mathrm{nm}$ is absorbed when an electron in a three-dimensional cubical box makes a transition from the ground state to the second excited state. What is the side length $L$ of the box?

Nathan Silvano

Numerade Educator

Problem 6

What is the energy difference between the two lowest energy levels for a proton in a cubical box with side length $1.21 \times$ $10^{-14} \mathrm{~m}$, the approximate diameter of a nucleus?

Narayan Hari

Numerade Educator

Problem 7

A particle is in a three-dimensional cubical box that has side length $L$. For the state $n_{X}=3, n_{Y}=2,$ and $n_{Z}=1,$ for what planes (in addition to the walls of the box) is the probability distribution function zero?

Nathan Silvano

Numerade Educator

Problem 8

A hydrogen atom is in a state with quantum number $n=3 .$ In the quantum-mechanical description of the atom, what is the largest possible magnitude of the orbital angular momentum? What is the magnitude of the orbital angular momentum of the electron in the Bohr-model description of the atom? What is the percentage difference between these two values? (b) Answer the same questions as in part (a) for $n=30$.

Nathan Silvano

Numerade Educator

Problem 9

Consider an electron in the $N$ shell. (a) What is the smallest orbital angular momentum it could have? (b) What is the largest orbital angular momentum it could have? Express your answers in terms of $\hbar$ and in SI units. (c) What is the largest orbital angular momentum this electron could have in any chosen direction? Express your answers in terms of $\hbar$ and in SI units. (d) What is the largest spin angular momentum this electron could have in any chosen direction? Express your answers in terms of $\hbar$ and in SI units. (e) For the electron in part (c), what is the ratio of its spin angular momentum in the $z$ -direction to its orbital angular momentum in the $z$ -direction?

Ommair Ishaque

Numerade Educator

Problem 10

An electron is in the hydrogen atom with $n=5$. (a) Find the possible values of $L$ and $L_{z}$ for this electron, in units of $\hbar$. (b) For each value of $L,$ find all the possible angles between $\vec{L}$ and the $z$ -axis.
(c) What are the maximum and minimum values of the magnitude of the angle between $\vec{L}$ and the $z$ -axis?

Nathan Nowack

Numerade Educator

Problem 11

The orbital angular momentum of an electron has a magnitude of $4.716 \times 10^{-34} \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}$. What is the angular momentum quantum number $I$ for this electron?

Nathan Nowack

Numerade Educator

Problem 12

Consider states with angular momentum quantum number $I=2$. (a) In units of $\hbar$, what is the largest possible value of $L_{z}$ ? (b) In units of $\hbar,$ what is the value of $L$ ? Which is larger: $L$ or the maximum possible $L_{z}$ ? (c) For each allowed value of $L_{z},$ what angle does the vector $\vec{L}$ make with the $+z$ -axis? How does the minimum angle for $l=2$ compare to the minimum angle for $l=3$ calculated in Example 41.3 ?

Khaled Yasein

Numerade Educator

Problem 13

In a particular state of the hydrogen atom, the angle between the angular momentum vector $\vec{L}$ and the $z$ -axis is $\theta=26.6^{\circ} .$ If this is the smallest angle for this particular value of the orbital quantum number $l$ what is $\ ?$

Nathan Silvano

Numerade Educator

Problem 14

A hydrogen atom is in a state that has $L_{z}=2 \hbar$. In the semiclassical vector model, the angular momentum vector $\vec{L}$ for this state makes an angle $\theta_{L}=63.4^{\circ}$ with the $+z$ -axis. (a) What is the $l$ quantum number for this state? (b) What is the smallest possible $n$ quantum num-

Khaled Yasein

Numerade Educator

Problem 15

Consider the seventh excited level of the hydrogen atom.
(a) What is the energy of this level? (b) What is the largest magnitude of the orbital angular momentum? (c) What is the largest angle between the orbital angular momentum and the $z$ -axis?

Nathan Silvano

Numerade Educator

Problem 16

(a) Make a chart showing all possible sets of quantum numbers $l$ and $m_{l}$ for the states of the electron in the hydrogen atom when $n=4$. How many combinations are there? (b) What are the energies of these states?

Zhaojie Xu

Numerade Educator

Problem 17

(a) How many different $6 d$ states does hydrogen have?
(b) Which of the states in part (a) has the largest angle between $\vec{L}$ and the $z$ -axis, and what is that angle? (c) Which of the states in part (a) has the smallest angle between $\vec{L}$ and the $z$ -axis, and what is that angle?

Khaled Yasein

Numerade Educator

Problem 18

(a) What is the probability that an electron in the $1 s$ state of a hydrogen atom will be found at a distance less than $a / 2$ from the nucleus? (b) Use the results of part (a) and of Example 41.4 to calculate the probability that the electron will be found at distances between $a / 2$ and $a$ from the nucleus.

Ommair Ishaque

Numerade Educator

Problem 19

Show that $\Phi(\phi)=e^{i m \phi}=\Phi(\phi+2 \pi)$ (that is, show that $\Phi(\phi)$ is periodic with period $2 \pi$ ) if and only if $m_{l}$ is restricted to the values $0,\pm 1,\pm 2, \ldots$ (Hint: Euler's formula states that $\left.e^{j \phi}=\cos \phi+i \sin \phi .\right)$

Nathan Silvano

Numerade Educator

Problem 20

(a) The radial probability distribution function for a hydrogen atom state has one peak, at $r=0.476 \mathrm{nm} .$ What is the $n l$ spectroscopic notation of this state? (b) What is $r$ for the one peak of a $4 f$ state?

Nathan Silvano

Numerade Educator

Problem 21

A hydrogen atom in a $3 p$ state is placed in a uniform external magnetic field $\overrightarrow{\boldsymbol{B}}$. Consider the interaction of the magnetic field with the atom's orbital magnetic dipole moment.
(a) What field magnitude $B$ is required to split the $3 p$ state into multiple levels with an energy difference of $2.93 \times 10^{-5} \mathrm{eV}$ between adjacent levels? (b) How many levels will there be?

Eduard Sanchez

Numerade Educator

Problem 22

A hydrogen atom is in a $d$ state. In the absence of an external magnetic field, the states with different $m_{l}$ values have (approximately) the same energy. Consider the interaction of the magnetic field with the atom's orbital magnetic dipole moment.
(a) Calculate the splitting (in electron volts) of the $m_{l}$ levels when the atom is put in a $0.400 \mathrm{~T}$ magnetic field that is in the $+z$ -direction. (b) Which $m_{l}$ level will have the lowest energy? (c) Draw an energy-level diagram that shows the $d$ levels with and without the external magnetic field.

Zachary Warner

Numerade Educator

Problem 23

A hydrogen atom in the $5 g$ state is placed in a magnetic field of $0.250 \mathrm{~T}$ that is in the $z$ -direction. (a) Into how many levels is this state split by the interaction of the atom's orbital magnetic dipole moment with the magnetic field? (b) What is the energy separation between adjacent levels?
(c) What is the energy separation between the level of lowest energy and the level of highest energy?

Khaled Yasein

Numerade Educator

Problem 24

A hydrogen atom undergoes a transition from a $2 p$ state to the $1 s$ ground state. In the absence of a magnetic field, the wavelength of the photon emitted is $122 \mathrm{nm}$. The atom is then placed in a strong magnetic field in the $z$ -direction. Ignore spin effects; consider only the interaction of the magnetic field with the atom's orbital magnetic moment.
(a) How many different photon wavelengths are observed for the $2 p \rightarrow 1 s$ transition? What are the $m_{l}$ values for the initial and final states for the transition that leads to each photon wavelength? (b) One observed wavelength is exactly the same with the magnetic field as without. What are the initial and final $m_{l}$ values for the transition that produces a photon of this wavelength?
(c) One observed wavelength with the field is longer than the wavelength without the field. What are the initial and final $m_{l}$ values for the transition that produces a photon of this wavelength? (d) Repeat part (c) for the wavelength that is shorter than the wavelength in the absence of the field.

Lainey Roebuck

Numerade Educator

Problem 25

Classical Electron Spin. (a) If you treat an electron as a classical spherical object with a radius of $1.8 \times 10^{-17} \mathrm{~m}$, what angular speed is necessary to produce a spin angular momentum of magnitude $\sqrt{3} \hbar ?(b)$ Use $v=r \omega$ and the result of part (a) to calculate the speed $v$ of a point at the electron's equator. What does your result suggest about the validity of this model?

Lainey Roebuck

Numerade Educator

Problem 26

The hyperfine interaction in a hydrogen atom between the magnetic dipole moment of the proton and the spin magnetic dipole moment of the electron splits the ground level into two levels separated by $5.9 \times 10^{-6} \mathrm{eV}$. (a) Calculate the wavelength and frequency of the photon emitted when the atom makes a transition between these states, and compare your answer to the value given at the end of Section $41.5 .$ In what part of the electromagnetic spectrum does this lie? Such photons are emitted by cold hydrogen clouds in interstellar space; by detecting these photons, astronomers can learn about the number and density of such clouds. (b) Calculate the effective magnetic field experienced by the electron in these states (see Fig. 41.18 ). Compare your result to the effective magnetic field due to the spin-orbit coupling calculated in Example 41.7 .

Kathleen Tatem

Numerade Educator

Problem 27

Calculate the energy difference between the $m_{s}=\frac{1}{2}$ ("spin up") and $m_{s}=-\frac{1}{2}$ ("spin down") levels of a hydrogen atom in the Is state when it is placed in a $1.50 \mathrm{~T}$ magnetic field in the negative $z$ -direction. Which level, $m_{s}=\frac{1}{2}$ or $m_{s}=-\frac{1}{2},$ has the lower energy?

Khaled Yasein

Numerade Educator

Problem 28

A hydrogen atom in the $n=1, m_{s}=-\frac{1}{2}$ state is placed in a magnetic field with a magnitude of $0.54 \mathrm{~T}$ in the $+z$ -direction.
(a) Find the magnetic interaction energy (in electron volts) of the electron with the field. (b) Is there any orbital magnetic dipole moment interaction for this state? Explain. Can there be an orbital magnetic dipole moment interaction for $n \neq 1 ?$

Eduard Sanchez

Numerade Educator

Problem 29

Make a list of the four quantum numbers $n, l, m_{l},$ and $m_{s}$ for each of the 10 electrons in the ground state of the neon atom. Do not refer to Table 41.2 or 41.3 .

Zachary Warner

Numerade Educator

Problem 30

For germanium $(\mathrm{Ge}, Z=32),$ make a list of the number of electrons in each subshell $(1 s, 2 s, 2 p, \ldots)$. Use the allowed values of the quantum numbers along with the exclusion principle; do not refer to Table 41.3 .

Zachary Warner

Numerade Educator

Problem 31

(a) Write out the ground-state electron configuration $\left(1 s^{2}, 2 s^{2}, \ldots\right)$ for the beryllium atom. (b) What element of next-larger $Z$ has chemical properties similar to those of beryllium? Give the groundstate electron configuration of this element. (c) Use the procedure of part (b) to predict what element of next-larger $Z$ than in (b) will have chemical properties similar to those of the element you found in part
(b), and give its ground-state electron configuration.

Nathan Silvano

Numerade Educator

Problem 32

(a) Write out the ground-state electron configuration $\left(1 s^{2}, 2 s^{2}, \ldots\right)$ for the carbon atom. (b) What element of next-larger $Z$ has chemical properties similar to those of carbon? Give the groundstate electron configuration for this element.

Lien Le

Numerade Educator

Problem 33

The $5 s$ electron in rubidium (Rb) sees an effective charge of $2.771 e .$ Calculate the ionization energy of this electron.

Zachary Warner

Numerade Educator

Problem 34

The energies of the $4 s, 4 p,$ and $4 d$ states of potassium are given in Example $41.10 .$ Calculate $Z_{\text {eff }}$ for each state. What trend do your results show? How can you explain this trend?

Zachary Warner

Numerade Educator

Problem 35

(a) The doubly charged ion $\mathrm{N}^{2+}$ is formed by removing two electrons from a nitrogen atom. What is the ground-state electron configuration for the $\mathrm{N}^{2+}$ ion? (b) Estimate the energy of the least strongly bound level in the $L$ shell of $\mathrm{N}^{2+} .(\mathrm{c})$ The doubly charged ion $\mathrm{P}^{2+}$ is formed by removing two electrons from a phosphorus atom. What is the ground-state electron configuration for the $\mathrm{P}^{2+}$ ion? (d) Estimate the energy of the least strongly bound level in the $M$ shell of $\mathrm{P}^{2+}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 36

\cdot \mathrm{A} K_{a} \mathrm{x}$ ray emitted from a sample has an energy of $7.46 \mathrm{keV} .$ Of which element is the sample made?

Aparna Shakti

Numerade Educator

Problem 37

Calculate the frequency, energy (in keV), and wavelength of the $K_{\alpha} \mathrm{x}$ ray for the elements
(a) calcium (Ca, $Z=20$ ); (b) cobalt (Co, $Z=27) ;(c)$ cadmium $(\mathrm{Cd}, Z=48)$

Zachary Warner

Numerade Educator

Problem 38

The energies for an electron in the $K, L,$ and $M$ shells of the tungsten atom are $-69,500 \mathrm{eV},-12,000 \mathrm{eV},$ and $-2200 \mathrm{eV},$ respectively. Calculate the wavelengths of the $K_{\alpha}$ and $K_{\beta} \mathrm{x}$ rays of tungsten.

Kyle Godbey

Numerade Educator

Problem 39

In terms of the ground-state energy $E_{1,1,1},$ what is the energy of the highest level occupied by an electron when 10 electrons are placed into a cubical box?

Zachary Warner

Numerade Educator

Problem 40

An electron is in a three-dimensional box with side lengths $L_{X}=0.600 \mathrm{nm}$ and $L_{Y}=L_{Z}=2 L_{X} .$ What are the quantum numbers $n_{X}, n_{Y},$ and $n_{Z}$ and the energies, in $\mathrm{eV},$ for the four lowest energy levels? What is the degeneracy of each (including the degeneracy due to spin)?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 41

A particle is in the three-dimensional cubical box of Section $41.2 .$ (a) Consider the cubical volume defined by $0 \leq x \leq L / 4,$ $0 \leq y \leq L / 4,$ and $0 \leq z \leq L / 4 .$ What fraction of the total volume of the box is this cubical volume? (b) If the particle is in the ground state $\left(n_{X}=1, n_{Y}=1, n_{Z}=1\right),$ calculate the probability that the particle will be found in the cubical volume defined in part (a).
(c) Repeat the calculation of part (b) when the particle is in the state

Khaled Yasein

Numerade Educator

Problem 42

An electron is in a three-dimensional box. The $x$ - and $z$ -sides of the box have the same length, but the $y$ -side has a different length. The two lowest energy levels are $2.24 \mathrm{eV}$ and $3.47 \mathrm{eV},$ and the degeneracy of each of these levels (including the degeneracy due to the electron spin) is two. (a) What are the $n_{X}, n_{Y},$ and $n_{Z}$ quantum numbers for each of these two levels? (b) What are the lengths $L_{X}, L_{Y},$ and $L_{Z}$ for each side of the box? (c) What are the energy, the quantum numbers, and the degeneracy Gincluding the spin degeneracy) for the next higher energy state?

Khaled Yasein

Numerade Educator

Problem 43

A particle in the three-dimensional cubical box of Section 41.2 is in the ground state, where $n_{X}=n_{Y}=n_{Z}=1$.
(a) Calculate the probability that the particle will be found somewhere between $x=0$ and $x=L / 2 .$ (b) Calculate the probability that the particle will be found somewhere between $x=L / 4$ and $x=L / 2 .$ Compare your results to the result of Example 41.1 for the probability of finding the particle in the region $x=0$ to $x=L / 4$.

Nathan Silvano

Numerade Educator

Problem 44

A Three-Dimensional Isotropic Harmonic }\end{array}$ Oscillator. An isotropic harmonic oscillator has the potential-energy function $U(x, y, z)=\frac{1}{2} k^{\prime}\left(x^{2}+y^{2}+z^{2}\right)$. (Isotropic means that the force constant $k^{\prime}$ is the same in all three coordinate directions.)
(a) Show that for this potential, a solution to Eq. (41.5) is given by $\psi=\psi_{n_{x}}(x) \psi_{n_{2}}(y) \psi_{n_{z}}(z) .$ In this expression, $\psi_{n_{s}}(x)$ is a solution
to the one-dimensional harmonic-oscillator Schrödinger equation, Eq. (40.44) , with energy $E_{n_{x}}=\left(n_{x}+\frac{1}{2}\right) \hbar \omega .$ The functions $\psi_{n_{n}}(y)$ and $\psi_{n_{z}}(z)$ are analogous one-dimensional wave functions for oscillations in the $y$ - and $z$ -directions. Find the energy associated with this $\psi .$ (b) From your results in part (a) what are the ground-level and first-excited-level energies of the three-dimensional isotropic oscillator? (c) Show that there is only one state (one set of quantum numbers $n_{x}, n_{y},$ and $n_{2}$ ) for the ground level but three states for the first excited level.

Lainey Roebuck

Numerade Educator

Problem 45

Three-Dimensional Anisotropic Harmonic }\end{array}$ Oscillator. An oscillator has the potential-energy function $U(x, y, z)=\frac{1}{2} k_{1}^{\prime}\left(x^{2}+y^{2}\right)+\frac{1}{2} k_{2}^{\prime} z^{2},$ where $k_{1}^{\prime}>k_{2}^{\prime} .$ This oscillator is
called anisotropic because the force constant is not the same in all three coordinate directions. (a) Find a general expression for the energy levels of the oscillator (see Problem 41.44 ). (b) From your results in part (a), what are the ground-level and first-excited-level energies of this oscillator? (c) How many states (different sets of quantum numbers $n_{x}, n_{y}$, and $n_{z}$ ) are there for the ground level and for the first excited level? Compare to part (c) of Problem 41.44 .

Lainey Roebuck

Numerade Educator

Problem 46

A particle is in a three-dimensional box. The $y$ length of the box is twice the $x$ length, and the $z$ length is one-third of the $y$ length. (a) What is the energy difference between the first excited level and the ground level?
(b) Is the first excited level degenerate? (c) In terms of the $x$ length, where is the probability distribution the greatest in the lowest-energy level?

Keshav Singh

Numerade Educator

Problem 47

(a) Show that the total number of atomic states (including different spin states) in a shell of principal quantum number $n$ is $2 n^{2}$.

Zhaojie Xu

Numerade Educator

Problem 48

When our sun exhausts its nuclear fuel, it will ultimately shrink due to gravity and become a white dwarf, with a radius of approximately $7000 \mathrm{~km}$.
(a) Using the mass of the sun, $M_{\text {san }}=2.0 \times 10^{30} \mathrm{~kg}$, and the mass of a proton, $M_{\text {proton }}=1.7 \times 10^{-27} \mathrm{~kg}$, estimate the number of electrons in the sun. (b) From the radius given, estimate the average volume to be occupied by each electron in the eventual white dwarf. (c) The white dwarf will consist of mostly carbon. Since there are six electrons in each carbon atom, multiply the volume of an electron by 6 to obtain the volume of each carbon atom. (d) Model the atomic arrangement as a cubical lattice and use your earlier estimate to determine the distance $L$ between adjacent carbon nuclei. (e) View each atom as an $L \times L \times L$ box filled with six electrons. Since no
two electrons can be in the same quantum state, what quantum numbers $\left(n_{X}, n_{Y}, n_{Z}, m_{z}\right)$ are associated with these six electrons, where $m_{z}$ is the quantum number for the component of the electron spin along the z-axis? (f) Ignoring contributions from spin couplings and Coulomb interactions between the electrons, what would be the energy of the higher-energy electrons?

Lainey Roebuck

Numerade Educator

Problem 49

Consider a hydrogen atom in the $1 s$ state.
(a) For what value of $r$ is the potential energy $U(r)$ equal to the total energy $E$ ? Express your answer in terms of $a$. This value of $r$ is called the classical turning point, since this is where a Newtonian particle would stop its motion and reverse direction. (b) For $r$ greater than the classical turning point, $U(r)>E$. Classically, the particle cannot be in this region, since the kinetic energy cannot be negative. Calculate the probability of the electron being found in this classically forbidden region.

Lainey Roebuck

Numerade Educator

Problem 50

For a hydrogen atom, the probability $P(r)$ of finding the electron within a spherical shell with inner radius $r$ and outer radius $r+d r$ is given by Eq. $(41.25) .$ For a hydrogen atom in the $1 s$ ground state, at what value of $r$ does $P(r)$ have its maximum value? How does your result compare to the distance between the electron and the nucleus for the $n=1$ state in the Bohr model, Eq. (41.26)$?$

Nathan Silvano

Numerade Educator

Problem 51

The normalized radial wave function for the $2 p$ state of the hydrogen atom is $R_{2 p}=\left(1 / \sqrt{24 a^{5}}\right) r e^{-r / 2 a}$. After we average over the angular variables, the radial probability function becomes $P(r) d r=\left(R_{2 p}\right)^{2} r^{2} d r .$ At what value of $r$ is $P(r)$ for the $2 p$ state a maximum? Compare your results to the radius of the $n=2$ state in the Bohr model.

Zachary Warner

Numerade Educator

Problem 52

Interstellar hydrogen emits characteristic microwave radiation that can be understood as follows: (a) Model the magnetic moment of a proton using $\mu_{\mathrm{p}}=\left(e / 2 m_{\mathrm{p}}\right) S_{z},$ where $m_{\mathrm{p}}=1.7 \times 10^{-27} \mathrm{~kg}$ and
$S_{z}=\pm \hbar / 2 .$ Model the proton as a current loop with radius $R_{\mathrm{p}}=0.85 \mathrm{fm} .$ What current would produce the expected magnetic moment? (b) The magnetic field a distance $x$ along the axis of a current loop is given by Eq. (28.15). Use this equation to estimate the magnetic field $B$ at a distance $x=0.40 a_{0}$, where $a_{0} \gg R_{\mathrm{p}}$ is the Bohr radius. (c) The potential energy associated with the electron spin coupled to this magnetic field is $U_{\text {hyperfine }}=\pm \mu_{z} B$. Use Eq. (41.38) to show that $\mu_{z}=\mu_{B}$ where $\mu_{B}$ is the Bohr magneton. The sign depends on whether the electron spin is aligned or antialigned to that of the proton. Estimate the magnitude of this energy. (d) If an electron's spin were to flip, it would radiate a photon with energy $2\left|U_{\text {hyperfine }}\right|$. What would be the wavelength of such a photon? (e) Note that $21 \mathrm{~cm}$ radiation is observed in hydrogen clouds. Could electron spin flips explain this radiation?

Khaled Yasein

Numerade Educator

Problem 53

The hydrogen spectrum includes four visible lines. Of these, the blue line corresponds to a transition from the $n=5$ shell to the $n=2$ shell and has a wavelength of $434 \mathrm{nm}$. If we look closer, this line is broadened by fine structure due to spin-orbit coupling and relativistic effects. (a) How many different sets of $l$ and $j$ quantum numbers are there for the $n=5$ shell and for the $n=2$ shell? (b) How many different energy levels are there for $n=5$ and for $n=2$ ? For each of these levels, what is their energy difference in eV from $-(13.6 \mathrm{eV}) / n^{2} ?$
(c) In a transition that emits a photon the quantum number $I$ must change by ±1 . Which transition in the fine structure of the hydrogen blue line emits a photon of the shortest wavelength? For this photon what is the shift in wavelength due to the fine structure? (d) Which transition in the fine structure emits a photon of the longest wavelength? For this photon what is the shift in wavelength due to the fine structure? (e) By what total extent, in $\mathrm{nm}$, is the wavelength of the blue line broadened around the $434 \mathrm{nm}$ value?

Lainey Roebuck

Numerade Educator

Problem 54

An atom in a $3 d$ state emits a photon of wavelength $475.082 \mathrm{nm}$ when it decays to a $2 p$ state. (a) What is the energy (in electron volts) of the photon emitted in this transition?
(b) Use the selection rules described in Section 41.4 to find the allowed transitions if the atom is now in an external magnetic field of $3.200 \mathrm{~T}$. Ignore the effects of the electron's spin. (c) For the case in part (b), if the energy of the $3 d$ state was originally $-8.50000 \mathrm{eV}$ with no magnetic field present, what will be the energies of the states into which it splits in the magnetic field? (d) What are the allowed wavelengths of the light emitted during transition in part (b)?

Lainey Roebuck

Numerade Educator

Problem 55

Spectral Analysis. While studying the spectrum of a gas cloud in space, an astronomer magnifies a spectral line that results from a transition from a $p$ state to an $s$ state. She finds that the line at $575.050 \mathrm{nm}$ has actually split into three lines, with adjacent lines $0.0440 \mathrm{nm}$ apart, indicating that the gas is in an external magnetic field. (Ignore effects due to electron spin.) What is the strength of the external magnetic field?

Narayan Hari

Numerade Educator

Problem 56

Stern-Gerlach Experiment. In a Stern-Gerlach experiment, the deflecting force on the atom is $F_{z}=-\mu_{z}\left(d B_{z} / d z\right),$ where $\mu_{z}$ is given by Eq. (41.38) and $d B_{z} / d z$ is the magnetic-field gradient. $\operatorname{In}$ a particular experiment, the magnetic-field region is $50.0 \mathrm{~cm}$ long; assume the magnetic-field gradient is constant in that region. A beam of silver atoms enters the magnetic field with a speed of $375 \mathrm{~m} / \mathrm{s}$. What value of $d B_{z} / d z$ is required to give a separation of $1.0 \mathrm{~mm}$ between the two spin components as they exit the field? (Note: The magnetic dipole moment of silver is the same as that for hydrogen, since its valence electron is in an $I=0$ state.)

Penny Riley

Numerade Educator

Problem 57

The magnetic moment $\vec{\mu}$ of a spinning object with charge $Q$ and mass $M$ is proportional to its angular momentum $\vec{L}$ according to $\overrightarrow{\boldsymbol{\mu}}=g(Q / 2 M) \overrightarrow{\boldsymbol{L}}$. The dimensionless coefficient $g$ is known as the $g$ -factor of the object. Consider a spherical shell with radius $R$ and uniform charge density $\sigma$ spinning with angular velocity $\omega$ around the z-axis. (a) What is the differential area $d a$ of the ring at latitude $\theta$, with width $R d \theta,$ as shown in Fig. $P 41.57 ?$ (b) The current $d I$ carried by the ring is its charge $\sigma$ da divided by the period of rotation. Determine $d I$ in terms of $R, \sigma, \omega,$ and $\theta .(\mathrm{c})$ Determine the magnetic moment $d \overrightarrow{\boldsymbol{\mu}}$ of the ring by multiplying $d I$ by the area enclosed by the ring. (d) Determine the magnetic moment of the spherical shell by integrating over the sphere. Express your result in terms of the total charge $Q=4 \pi R^{2} \sigma$.
(e) Now consider a solid sphere of radius $R$ with volume charge density $\rho=\rho(r),$ where $r$ is the distance from the center, spinning with angular velocity $\omega$ about the z-axis. Determine its magnetic moment by integrating $\overrightarrow{\boldsymbol{\mu}}=\int d \overrightarrow{\boldsymbol{\mu}},$ where $d \overrightarrow{\boldsymbol{\mu}}$ is now the magnetic moment associated with the shell at radius $r$ with differential width $d r$. Express your answer in terms of an integral $\int_{0}^{R} r^{4} \rho(r) d r .$ (f) The angular momentum of the sphere is $\vec{L}=I \vec{\omega},$ where $I=c M R^{2}$ is its moment of inertia, and $c$ is a dimensionless factor determined by the mass distribution. Determine the $g$ -factor in terms of the $\rho$ integral and the value of $c .(g)$ If the charge is uniformly distributed so that $\rho=Q /\left(\frac{4}{3} \pi R^{3}\right)$ and if the mass is uniformly distributed so that $c=\frac{2}{5},$ then what is the value of $g ?$

Lainey Roebuck

Numerade Educator

Problem 58

Effective Magnetic Field. An electron in a hydrogen atom is in the $2 p$ state. In a simple model of the atom, assume that the electron circles the proton in an orbit with radius $r$ equal to the Bohr-model radius for $n=2$. Assume that the speed $v$ of the orbiting electron can be calculated by setting $L=m w r$ and taking $L$ to have the quantum-mechanical value for a $2 p$ state. In the frame of the electron, the proton orbits with radius $r$ and speed $v .$ Model the orbiting proton as a circular current loop, and calculate the magnetic field it produces at the location of the electron.

Lainey Roebuck

Numerade Educator

Problem 59

Weird Universe. In another universe, the electron is a spin-\frac{3} { 2 }
rather than a spin-\frac{1} { 2 } \text { particle, but all other physics are the same as in our } universe. In this universe, (a) what are the atomic numbers of the lightest two inert gases?
(b) What is the ground-state electron configuration of sodium?

Zachary Warner

Numerade Educator

Problem 60

A lithium atom has three electrons, and the ${ }^{2} S_{1 / 2}$ groundstate electron configuration is $1 s^{2} 2 s$. The $1 s^{2} 2 p$ excited state is split into two closely spaced levels, ${ }^{2} P_{3 / 2}$ and ${ }^{2} P_{1 / 2}$, by the spin-orbit interaction (see Example 41.7 in Section 41.5 ). A photon with wavelength $67.09608 \mu \mathrm{m}$ is emitted in the ${ }^{2} P_{3 / 2} \rightarrow{ }^{2} S_{1 / 2}$ transition, and a photon with wavelength $67.09761 \mu \mathrm{m}$ is emitted in the ${ }^{2} P_{1 / 2} \rightarrow{ }^{2} S_{1 / 2}$ transition. Calculate the effective magnetic field seen by the electron in the $1 s^{2} 2 p$ state of the lithium atom. How does your result compare to that for the
$3 p$ level of sodium found in Example $41.7 ?$

Nathan Silvano

Numerade Educator

Problem 61

A hydrogen atom in an $n=2, l=1, m_{l}=-1$ state emits a photon when it decays to an $n=1, l=0, m_{l}=0$ ground state.
(a) In the absence of an external magnetic field, what is the wavelength of this photon? (b) If the atom is in a magnetic field in the $+z$ -direction and with a magnitude of $2.20 \mathrm{~T}$, what is the shift in the wavelength of the photon from the zero-field value? Does the magnetic field increase or decrease the wavelength? Disregard the effect of electron spin. [Hint:
Use the result of Problem $39.74(\mathrm{c}) .]$

Zachary Warner

Numerade Educator

Problem 62

Electron Spin Resonance. Electrons in the lower of two spin states in a magnetic field can absorb a photon of the right frequency and move to the higher state.
(a) Find the magnetic-field magnitude $B$ required for this transition in a hydrogen atom with $n=1$ and $l=0$ to be induced by microwaves with wavelength $\lambda .$ (b) Calculate the value of $B$ for a wavelength of $4.20 \mathrm{~cm} .$

Rashmi Sinha

Numerade Educator

Problem 63

While working in a magnetics lab, you conduct an experiment in which a hydrogen atom in the $n=1$ state is in a magnetic field of magnitude $B$. A photon of wavelength $\lambda$ (in air) is absorbed in a transition from the $m_{s}=-\frac{1}{2}$ to the $m_{s}=+\frac{1}{2}$ state. The wavelengths $\lambda$ as a function of $B$ are given in the table.
\begin{tabular}{l|ccccccc}
$\boldsymbol{B}(\mathbf{T})$ & 0.51 & 0.74 & 1.03 & 1.52 & 2.02 & 2.48 & 2.97 \\
\hline $\boldsymbol{\lambda}(\mathrm{mm})$ & 21.4 & 14.3 & 10.7 & 7.14 & 5.35 & 4.28 & 3.57
\end{tabular}
(a) Graph the data in the table as photon frequency $f$ versus $B,$ where $f=c / \lambda .$ Find the slope of the straight line that gives the best fit to the data. (b) Use your results of part (a) to calculate $\left|\mu_{z}\right|,$ the magnitude of the spin magnetic moment. (c) Let $\gamma=\left|\mu_{z}\right| /\left|S_{z}\right|$ denote the gyromagnetic ratio for electron spin. Use your result of part (b) to calculate $\gamma$. What is the value of $\gamma /(e / 2 m)$ given by your experimental data?

Lainey Roebuck

Numerade Educator

Problem 64

transition to the $n=2, l=0, j=\frac{1}{2}$ state. Find the difference in wavelength between the following two photons: one emitted in a transition that starts in the $n=3, l=1, j=\frac{3}{2}$ state and one that starts instead in the $n=3, I=1, j=\frac{1}{2}$ state. Which photon has the longer wavelength?

Lainey Roebuck

Numerade Educator

Problem 65

In studying electron screening in multielectron atoms, you begin with the alkali metals. You look up experimental data and find the results given in the table.
\begin{tabular}{l|cccccc}
Element & $\mathrm{Li}$ & $\mathrm{Na}$ & $\mathrm{K}$ & $\mathrm{Rb}$ & $\mathrm{Cs}$ & $\mathrm{Fr}$ \\
\hline Ionization energy $(\mathbf{k} \mathbf{J} / \mathbf{m o l})$ & 520.2 & 495.8 & 418.8 & 403.0 & 375.7 & 380 \\
& & & & & &
\end{tabular}
The ionization energy is the minimum energy required to remove the least-bound electron from a ground-state atom. (a) The units $\mathrm{kJ} / \mathrm{mol}$ given in the table are the minimum energy in $\mathrm{kJ}$ required to ionize $1 \mathrm{~mol}$ of atoms. Convert the given values for ionization energy to the energy in $\mathrm{eV}$ required to ionize one atom. (b) What is the value of the nuclear charge $Z$ for each element in the table? What is the $n$ quantum number for the least-bound electron in the ground state? (c) Calculate $Z_{\text {eff }}$ for this electron in each alkali-metal atom. (d) The ionization energies decrease as $Z$ increases. Does $Z_{\text {eff }}$ increase or decrease as $Z$ increases? Why does $Z_{\text {sff }}$ have this behavior?

Lainey Roebuck

Numerade Educator

Problem 66

You are studying the absorption of electromagnetic radiation by electrons in a crystal structure. The situation is well described by an electron in a cubical box of side length $L .$ The electron is initially in the ground state.
(a) You observe that the longest-wavelength photon that is absorbed has a wavelength in air of $\lambda=624 \mathrm{nm} .$ What is $L ?$
(b) You find that $\lambda=234 \mathrm{nm}$ is also absorbed when the initial state is still the ground state. What is the value of $n^{2}$ for the final state in the transition for which this wavelength is absorbed, where $n^{2}=n_{X}^{2}+n_{Y}^{2}+n_{Z}^{2}$ ? What is the degeneracy of this energy level (including the degeneracy due to electron spin)?

Keshav Singh

Numerade Educator

Problem 67

$\mathrm{Spin}-\frac{3}{2}$ particles have four distinct spin states corresponding to $S_{z}=m_{z} \hbar,$ where $m_{2}$ may be $\pm \frac{1}{2}$ or $\pm \frac{3}{2}$. We can write the corresponding normalized wave functions as $\psi_{m z}(\vec{r})$.
(a) What is the magnitude of such a particle's spin $\vec{S} ?$
(b) Suppose we have three identical entangled $\operatorname{spin}-\frac{3}{2}$ particles, each sharing an equal probability for exhibiting $m_{z}=\pm \frac{1}{2}$ or $m_{z}=-\frac{3}{2}$ but a zero probability for being in the state $m_{z}=+\frac{3}{2}$. The three-particle wave function is a sum of products such as $\psi_{-1 / 2}\left(\vec{r}_{1}\right) \psi_{+1 / 2}\left(\vec{r}_{2}\right) \psi_{-3 / 2}\left(\vec{r}_{3}\right),$ which we abbreviate as
$\psi_{-1 / 2} \psi_{+1 / 2} \psi_{-3 / 2}$. (It is understood that the $n$ th factor corresponds to the position $\vec{r}_{n} .$ Write the normalized three-particle wave function for

Laszlo Zalavari

Laszlo Zalavari

Numerade Educator

Problem 68

Each of $2 N$ electrons (mass $m$ ) is free to move along the $x$ -axis. The potential-energy function for each electron is $U(x)=\frac{1}{2} k^{\prime} x^{2}$, where $k^{\prime}$ is a positive constant. The electric and magnetic interactions between electrons can be ignored. Use the exclusion principle to show that the minimum energy of the system of $2 N$ electrons is $\hbar N^{2} \sqrt{k^{+} / m}$.

Nathan Silvano

Numerade Educator

Problem 69

Consider a simple model of the helium atom in which two electrons, each with mass $m,$ move around the nucleus (charge $+2 e)$ in the same circular orbit. Each electron has orbital angular momen$\operatorname{tum} \hbar$ (that is, the orbit is the smallest-radius Bohr orbit), and the two electrons are always on opposite sides of the nucleus. Ignore the effects of spin. (a) Determine the radius of the orbit and the orbital speed of each electron. [Hint: Follow the procedure used in Section 39.3 to derive Eqs. (39.8) and $(39.9) .$ Each electron experiences an attractive force from the nucleus and a repulsive force from the other electron.]
(b) What is the total kinetic energy of the electrons?
(c) What is the potential energy of the system (the nucleus and the two electrons)? (d) In this model, how much energy is required to remove both electrons to infinity? How does this compare to the experimental value of $79.0 \mathrm{eV} ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 70

In the Bohr model, what is the principal quantum number $n$ at which the excited electron is at a radius of $1 \mu \mathrm{m}$ ? (a) $140 ;$ (b) 400 ;
(c) $20 ;$ (d) 81 .

Ankur S

Numerade Educator

Problem 71

Take the size of a Rydberg atom to be the diameter of the orbit of the excited electron. If the researchers want to perform this experiment with the rubidium atoms in a gas, with atoms separated by a distance 10 times their size, the density of atoms per cubic centimeter should
be about (a) $10^{5}$ atoms $/ \mathrm{cm}^{3}$; (b) $10^{8}$ atoms $/ \mathrm{cm}^{3}$;
(c) $10^{11}$ atoms $/ \mathrm{cm}^{3}$;
(d) $10^{21}$ atoms/cm $^{3}$.

Nathan Silvano

Numerade Educator

Problem 72

Assume that the researchers place an atom in a state with $n=100, l=2 .$ What is the magnitude of the orbital angular momentum $\vec{L}$ associated with this state?
(a) $\sqrt{2} \hbar ;$
(b) $\sqrt{6} \hbar$
(c) $\sqrt{200} \hbar$;
(d) $\sqrt{10,100} \hbar$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 73

How many different possible electron states are there in the $n=100, l=2$ subshell? (a) $2 ;(b) 100 ;(c) 10,000 ;$ (d) $10 .$

Nathan Silvano

Numerade Educator