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University Physics with Modern Physics

Hugh D. Young

Chapter 41

Atomic Structure - all with Video Answers

Educators

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Chapter Questions

03:10

Problem 1

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

Zachary Warner
Zachary Warner
Numerade Educator
06:14

Problem 2

CP 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, the box, the 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.

Zachary Warner
Zachary Warner
Numerade Educator
04:08

Problem 3

CP A photon is emitted when an electron in a three-dimensional box of side length $8.00 \times 10^{-11} \mathrm{m}$ 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?

Zachary Warner
Zachary Warner
Numerade Educator
09:46

Problem 4

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

Zachary Warner
Zachary Warner
Numerade Educator
03:42

Problem 5

A particle is in the three-dimensional box of Section 41.1 For the state $n_{X}=2, n_{Y}=2, n_{Z}=1,$ for what planes (in addition to the walls of the box) is the probability distribution function zero? Compare this number of planes to the corresponding number of planes where $|\psi|^{2}$ is zero for the lower-energy state $n_{X}=2, n_{Y}=1, n_{Z}=1$ and for the ground state $n_{X}=1, n_{Y}=1$ $n_{Z}=1 .$

Zhaojie Xu
Zhaojie Xu
Numerade Educator
02:28

Problem 6

What is the energy difference between the two lowest energy levels for a proton in a cubical box with side length $1.00 \times 10^{-14} \mathrm{m},$ the approximate diameter of a nucleus?

Ajay Singhal
Ajay Singhal
Numerade Educator
06:02

Problem 7

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?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
06:33

Problem 8

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
Nathan Nowack
Numerade Educator
00:33

Problem 9

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 $/$ for this electron?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
07:42

Problem 10

Consider states with angular-momentum quantum number $l=2 .$ (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 $\vec{\boldsymbol{L}}$ 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$?$

Zachary Warner
Zachary Warner
Numerade Educator
02:56

Problem 11

Calculate, in units of $\hbar,$ the magnitude of the maximum orbital angular momentum for an electron in a hydrogen atom for states with a principal quantum number of $2,20,$ and $200 .$ Compare each with the value of $n \hbar$ postulated in the Bohr model. What trend do you see?

Kyle Godbey
Kyle Godbey
Numerade Educator
03:33

Problem 12

(a) Make a chart showing all the possible sets of quantum numbers $l$ and $m_{l}$ for the states of the electron in the hydrogen atom when $n=5 .$ How many combinations are there? (b) What are the energies of these states?

Zachary Warner
Zachary Warner
Numerade Educator
04:34

Problem 13

(a) How many different 5$g$ 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?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
04:26

Problem 14

CALC (a) What is the probability that an electron in the 1s 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.

Zachary Warner
Zachary Warner
Numerade Educator
04:13

Problem 15

CALC In Example 41.4 fill in the missing details that show that $P=1-5 e^{-2}$

Zachary Warner
Zachary Warner
Numerade Educator
03:11

Problem 16

Show that $\Phi(\phi)=e^{i m_{r} \phi}=\Phi(\phi+2 \pi)$ (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 $e^{i \phi}=\cos \phi+i \sin \phi . )$

Zachary Warner
Zachary Warner
Numerade Educator
05:28

Problem 17

A hydrogen atom in a 3$p$ state is placed in a uniform external magnetic field $B$ . 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 $2.71 \times 10^{-5}$ eV between adjacent levels? (b) How many levels will there be?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:19

Problem 18

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
Zachary Warner
Numerade Educator
05:34

Problem 19

A hydrogen atom in the 5$g$ state is placed in a magnetic field of 0.600 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?

Shoukat Ali
Shoukat Ali
Other Schools
01:02

Problem 20

CP 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 energy of the photon emitted is 122 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 wave-
lengths 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 wave-length? (d) Repeat part (c) for the wavelength that is shorter than the wavelength in the absence of the field.

Zachary Warner
Zachary Warner
Numerade Educator
02:42

Problem 21

CP Classical Electron Spin. (a) If you treat an electron as a classical spherical object with a radius of $1.0 \times 10^{-17} \mathrm{m}$ , what angular speed is necessary to produce a spin angular momentum of magnitude $\sqrt{\frac{3}{4}} \hbar ?$ (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?

Zachary Warner
Zachary Warner
Numerade Educator
04:09

Problem 22

A hydrogen atom in the $n=1, m_{s}=-\frac{1}{2}$ state is placed in a magnetic field with a magnitude of 0.480 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 $n \neq 1 ?$

Zachary Warner
Zachary Warner
Numerade Educator
03:44

Problem 23

Calculate the energy difference between the $m_{s}=\frac{1}{2}$ $\left($ "spin up $^{\prime \prime}\right)$ and $m_{s}=-\frac{1}{2}($ "spin down") levels of a hydrogen atom in the 1 state when it is placed in a $1.45-$ 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
Khaled Yasein
Numerade Educator
03:52

Problem 24

CP 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 intotwo levels separated by $5.9 \times 10^{-6}$ 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 $41.7 .$

Zachary Warner
Zachary Warner
Numerade Educator
00:38

Problem 25

A hydrogen atom in a particular orbital angular momentum state is found to have $j$ quantum numbers $\frac{7}{2}$ and $\frac{4}{2} .$ What is the letter that labels the value of $l$ for the state?

Zachary Warner
Zachary Warner
Numerade Educator
00:44

Problem 26

For germanium $(\mathrm{Ge}, \mathrm{Z}=32),$ 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
Zachary Warner
Numerade Educator
01:51

Problem 27

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
Zachary Warner
Numerade Educator
00:57

Problem 28

(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
ground-state electron configuration for this element.

Zachary Warner
Zachary Warner
Numerade Educator
00:58

Problem 29

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

Zachary Warner
Zachary Warner
Numerade Educator
02:07

Problem 30

For magnesium, the first ionization potential is 7.6 $\mathrm{eV}$ The second ionization potential (additional energy required to remove a second electron is almost twice this, $15 \mathrm{eV},$ and the third ionization potential is much larger, about 80 $\mathrm{eV} .$ How can these numbers be understood?

Zachary Warner
Zachary Warner
Numerade Educator
00:55

Problem 31

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

Zachary Warner
Zachary Warner
Numerade Educator
03:08

Problem 32

The energies of the $4 s, 4 p,$ and 4$d$ states of potassium are given in Example $41.9 .$ Calculate $Z_{\text { eff for each state. What trend do }}$ your results show? How can you explain this trend?

Zachary Warner
Zachary Warner
Numerade Educator
03:19

Problem 33

(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 $N^{2+}$ ion? (b) Estimate the energy of the least strongly bound level in the $L$ shell of $N^{2+} .(c)$ 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 $\mathrm{P}^{2+}$ .

Zachary Warner
Zachary Warner
Numerade Educator
03:07

Problem 34

(a) The energy of the 2$s$ state of lithium is $-5.391$ eV. Calculate the value of $Z_{\text { eff }}$ for this state. (b) The energy of the 4$s$ state of potassium is $-4.339$ eV. Calculate the value of $Z_{\text { eff for this }}$ state. (c) Compare $Z_{\text { eff }}$ for the 2$s$ state of lithium, the 3$s$ state of sodium (see Example 41.8$)$ , and the 4$s$ state of potassium. What trend do you see? How can you explain this trend?

Zachary Warner
Zachary Warner
Numerade Educator
01:14

Problem 35

Estimate the energy of the highest-l state for (a) the $L$ shell of $\mathrm{Be}^{+}$ and $(\mathrm{b})$ the $N$ shell of $\mathrm{Ca}^{+}$

Zachary Warner
Zachary Warner
Numerade Educator
02:23

Problem 36

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

Zachary Warner
Zachary Warner
Numerade Educator
06:30

Problem 37

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

Zachary Warner
Zachary Warner
Numerade Educator
03:54

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
Kyle Godbey
Numerade Educator
00:54

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
Zachary Warner
Numerade Educator
07:06

Problem 40

CALC A particle in the three-dimensional 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 $x=L / 4$ .

Zachary Warner
Zachary Warner
Numerade Educator
13:08

Problem 41

A particle is in the three-dimensional 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 $\left(n_{Y}=1, n_{Y}=1, n_{Z}=1\right)$ 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 $n_{X}=2, n_{Y}=1, n_{Z}=1.$

Zachary Warner
Zachary Warner
Numerade Educator
03:19

Problem 42

CALC A particle is described by the normalized wave function $\psi(x, y, z)=A x e^{-\alpha x^{2}} e^{-\beta y^{2}} e^{-\gamma y^{2}},$ where $A, \alpha, \beta,$ and $\gamma$ are all real, positive constants. The probability that the particle will be found in the infinitesimal volume $d x d y d z$ centered at the point
$\left(x_{0}, y_{0}, z_{0}\right)$ is $\left|\psi\left(x_{0}, y_{0}, z_{0}\right)\right|^{2} d x d y d z z$ (a) At what value of $x_{0}$ is the particle most likely to be found? (b) Are there values of $x_{0}$ for which
the probability of the particle being found is zero? If so, at what $x_{0} ?$

Zhaojie Xu
Zhaojie Xu
Numerade Educator
13:27

Problem 43

A particle is described by the normalized wave function $\psi(x, y, z)=A e^{-\alpha\left(x^{2}+y^{2}+z^{2}\right)},$ where $A$ and $\alpha$ are real, positive constants. (a) Determine the probability of finding the particle at a distance between $r$ and $r+d r$ from the origin. (Hint: See
Problem $41.42 .$ Consider a spherical shell centered on the origin with inner radius $r$ and thickness $d r$ . $)$ For what value of $r$ does the probability in part (a) have its maximum value? Is this the
same value of $r$ for which $|\psi(x, y, z)|^{2}$ is a maximum? Explain any differences.

Zachary Warner
Zachary Warner
Numerade Educator
02:23

Problem 44

A Three-Dimensional Isotropic HarmonicOscillator. 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 $\psi=\psi_{n_{x}}(x) \psi_{n_{y}}(y) \psi_{n_{z}}(z) .$ In this expression, $\psi_{n_{s}}(x)$ is a solution to the one-dimensional harmonic oscillator Schrodinger equation, Eq. $(40.44),$ with energy $E_{n_{x}}=\left(n_{x}+\frac{1}{2}\right) \hbar \omega$ . The functions $\psi_{n_{y}}(y)$ 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 $n_{z} )$ for the ground level but three states for the first excited level.

Zhaojie Xu
Zhaojie Xu
Numerade Educator
12:47

Problem 45

Three-Dimensional Anisotropic Harmonic 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 $n_{x x}, n_{y},$ and $n_{z} )$ are there for the ground level and for the first excited level? Compare to part (c) of Problem $41.44 .$

Guilherme Barros
Guilherme Barros
Numerade Educator
03:44

Problem 46

An electron in hydrogen is in the 5$f$ state. (a) Find the largest possible value of the $z$ -component of its angular momentum. (b) Show that for the electron in part (a), the corresponding $x$ -and $y$ -components of its angular momentum satisfy the equation $\sqrt{L_{x}^{2}+L_{y}^{2}}=\hbar \sqrt{3}$

Kyle Godbey
Kyle Godbey
Numerade Educator
04:48

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$n^{2} .[$Hint : The sum of the first $N$ integers $1+2+3+\cdots+N$ is
equal to $N(N+1) / 2 .$ (b) Which shell has 50 states?

Zachary Warner
Zachary Warner
Numerade Educator
04:41

Problem 48

(a) What is the lowest possible energy ( in electron volts) of an electron in hydrogen if its orbital angular momentum is $\sqrt{12 \hbar ?}$ (b) What are the largest and smallest values of the $z$ -component of the orbital angular momentum (in terms of $\hbar )$ for the electron in part (a)? (c) What are the largest and smallest values of the spin angular momentum (in terms of $\hbar$ ) for the electron in part
(a)? (d) What are the largest and smallest values of the orbital angular momentum (in terms of $\hbar$ ) for an electron in the $M$ shell of hydrogen?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
03:21

Problem 49

Consider an electron in hydrogen having total energy $-0.5440 \mathrm{eV} .$ (a) What are the possible values of its orbital angular momentum (in terms of $\hbar ) ?$ (b) What wavelength of light would it
take to excite this electron to the next higher shell? Is this photon visible to humans?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
08:20

Problem 50

(a) Show all the distinct states for an electron in the $N$ shell of hydrogen. Include all four quantum numbers. (b) For an $f$ electron in the $N$ shell, what is the largest possible orbital angular momentum and the greatest positive value for the component of this angular momentum along any chosen direction (the z-axis)? What is the magnitude of its spin angular momentum? Express these quantities in units of $\hbar$ . (c) For an electron in the $d$ state of the $N$ shell, what are the maximum and minimum angles between its angular momentum vector and any chosen direction (the $z$ -axis)? (d) What is the largest value of the orbital angular momentum for an $f$ electron in the $M$ shell?

Zachary Warner
Zachary Warner
Numerade Educator
04:55

Problem 51

(a) The energy of an electron in the 4 state of sodium is $-1.947$ eV. What is the effective net charge of the nucleus "seen" by this electron? On the average, how many electrons screen the nucleus? (b) For an outer electron in the 4$p$ state of potassium, on the average 17.2 inner electrons screen the nucleus. (i) What is the effective net charge of the nucleus "seen" by this outer electron? (ii) What is the energy of this outer electron?

Zachary Warner
Zachary Warner
Numerade Educator
04:31

Problem 52

CALC 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, Eg. $(41.26) ?$

Zachary Warner
Zachary Warner
Numerade Educator
07:33

Problem 53

CALC 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.

Zachary Warner
Zachary Warner
Numerade Educator
03:18

Problem 54

CP Rydberg Atoms. Rydberg atoms are atoms whose outermost electron is in an excited state with a very large principal quantum number. Rydberg atoms have been produced in the laboratory and detected in interstellar space. (a) Why do all neutral Rydberg atoms with the same $n$ value have essentially the same ionization energy, independent of the total number of electrons in the atom? (b) What is the ionization energy for a Rydberg atom with a principal quantum number of 350$?$ What is the radius in the Bohr model of the Rydberg electron's orbit? (c) Repeat part (b) for $n=650$ .

Zachary Warner
Zachary Warner
Numerade Educator
08:06

Problem 55

CALC The wave function for a hydrogen atom in the 2$s$ state is $$\psi_{2 s}(r)=\frac{1}{\sqrt{32 \pi a^{3}}}\left(2-\frac{r}{a}\right) e^{-r / 2 a}$$ (a) Verify that this function is normalized. (b) In the Bohr model, the distance between the electron and the nucleus in the $n=2$ state is exactly 4$a$ . Calculate the probability that an electron in the 2$s$ state will be found at a distance less than 4$a$ from the nucleus.

Zachary Warner
Zachary Warner
Numerade Educator
09:37

Problem 56

CALC The normalized wave function for a hydrogen atom in the 2$s$ state is given in Problem 41.55 . (a) For a hydrogen atom in the 2$s$ state, at what value of $r$ is $P(r)$ maximum? How does your result compare to $4 a,$ the distance between the electron and the nucleus in the $n=2$ state of the Bohr model? (b) At what value of $r($ other than $r=0$ or $r=\infty)$ is $P(r)$ equal to zero, so that the probability of finding the electron at that separation from the nucleus is zero? Compare your result to Fig, 41.9 .

Zachary Warner
Zachary Warner
Numerade Educator
04:46

Problem 57

(a) For an excited state of hydrogen, show that the smallest angle that the orbital angular momentum vector $\vec{L}$ can have with the $z$ -axis is $$\left(\theta_{L}\right)_{\min }=\arccos \left(\frac{n-1}{\sqrt{n(n-1)}}\right)$$ (b) What is the corresponding expression for $\left(\theta_{L}\right)_{\max },$ the largest possible angle between $\vec{L}$ and the $z$ -axis?

Zachary Warner
Zachary Warner
Numerade Educator
04:44

Problem 58

(a) If the value of $L_{z}$ is known, we cannot know either $L_{x}$ or $L_{y}$ precisely. But we can know the value of the quantity $\sqrt{L_{x}^{2}+L_{y}^{2}}$ . Write an expression for this quantity in terms of $l$ $m_{l},$ and $\hbar .$ (b) What is the meaning of $\sqrt{L_{\mathrm{r}}^{2}+L_{\mathrm{y}}^{2}} ?(\mathrm{c})$ For a state of nonzero orbital angular momentum, find the maximum and minimum values of $\sqrt{L_{x}^{2}+L_{y}^{2}}$ . Explain your results.

Zachary Warner
Zachary Warner
Numerade Educator
04:04

Problem 59

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
Zachary Warner
Numerade Educator
07:53

Problem 60

CP Stern-Gerlach Experiment. In a Stern-Gerlach experiment, the deffecting force on the atom is $F_{z}=-\mu_{z}\left(d B_{z} / d z\right)$ where $\mu_{z}$ is given by Eq. $(41.40)$ and $d B_{z} / d z$ is the magnetic-field gradient. In a particular experiment the magnetic-field region is 50.0 $\mathrm{cm}$ long; assume the magnetic-field gradient is constant in this region. A beam of silver atoms enters the magnetic field with a speed of 525 $\mathrm{m} / \mathrm{s} .$ 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 $l=0$ state.)

Zachary Warner
Zachary Warner
Numerade Educator
03:42

Problem 61

Consider the transition from a 3$d$ to a 2$p$ state of hydrogen in an external magnetic field. Assume that the effects of electron spin can be ignored (which is not actually the case) so that the magnetic field interacts only with the orbital angular momentum. Identify each allowed transition by the $m_{l}$ values of the initial and final states. For each of these allowed transitions, determine the shift of the transition energy from the zero-field value and show that there are three different transition energies.

Zachary Warner
Zachary Warner
Numerade Educator
00:03

Problem 62

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 3.500 $\mathrm{T}$ . 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 $-8.50000$ 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)?

Arihant Jain
Arihant Jain
Numerade Educator
02:50

Problem 63

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 nm has actually split into three lines, with adjacent lines 0.0462 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?

Zachary Warner
Zachary Warner
Numerade Educator
04:35

Problem 64

A hydrogen atom makes a transition from an $n=3$ state to an $n=2$ state (the Balmer $\mathrm{H}_{\alpha}$ line $)$ while in a magnetic field in the $+z$ -direction and with magnitude 1.40 $\mathrm{T}$ . (a) If the magnetic quantum number is $m_{l}=2$ in the initial $(n=3)$ state and $m_{l}=1$ in the final $(n=2)$ state, by how much is each energy level shifted from the zero-field value? (b) By how much is the wave-length of the $\mathrm{H}_{\alpha}$ line shifted from the zero-field value? Is the wavelength increased or decreased? Disregard the effect of electron spin. [Hint: Use the result of Problem 39.86$(\mathrm{c}) . ]$

Zachary Warner
Zachary Warner
Numerade Educator
04:49

Problem 65

CP A large number of hydrogen atoms in I$s$ states are placed in an external magnetic field that is in the $+z$ -direction.Assume that the atoms are in thermal equilibrium at room temperature, $T=300 \mathrm{K}$ . According to the Maxwell-Boltzmann distribution (see Section $39.4 ),$ what is the ratio of the number of atoms in the $m_{s}=\frac{1}{2}$ state to the number in the $m_{s}=-\frac{1}{2}$ state when the magnetic-field magnitude is (a) $5.00 \times 10^{-5} \mathrm{T}$ (approximately the earth's field); (b) $0.500 \mathrm{T} ;(\mathrm{c}) 5.00 \mathrm{T}$ ?

Zachary Warner
Zachary Warner
Numerade Educator
04:04

Problem 66

Effective Magnetic Field. An electron in a hydrogenatom 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 $L=$ mur 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.

Zachary Warner
Zachary Warner
Numerade Educator
04:48

Problem 67

Weird Universe. In another universe, the electron is a $\operatorname{spin}-\frac{3}{2}$ rather than a spin- $\frac{1}{2}$ 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
Zachary Warner
Numerade Educator
06:17

Problem 68

For an ion with nuclear charge $Z$ and a single electron, the electric potential energy is $-Z e^{2} / 4 \pi \epsilon_{0} r$ and the expression for the energies of the states and for the normalized wave functions are obtained from those for hydrogen by replacing $e^{2}$ by $Z e^{2} .$ Consider the $\mathrm{N}^{6+}$ ion, with seven protons and one electron. (a) What is the ground-state energy in electron volts? (b) What is the ionization energy, the energy required to remove the electron from the $\mathrm{N}^{6+}$ ion if it is initially in the ground state? (c) What is the distance $a$ [given for hydrogen by Eq. (41.26)] for this ion? (d) What is the wavelength of the photon emitted when the $N^{6+}$ ion makes a transition from the $n=2$ state to the $n=1$ ground state?

Zachary Warner
Zachary Warner
Numerade Educator
04:22

Problem 69

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 39.86$(\mathrm{c}) . ]$

Zachary Warner
Zachary Warner
Numerade Educator
05:11

Problem 70

A lithium atom has three electrons, and the $S_{1 / 2 \text { ground- }}$ state 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$?$

Zachary Warner
Zachary Warner
Numerade Educator
04:39

Problem 71

Estimate the minimum and maximum wavelengths of the characteristic $x$ rays emitted by (a) vanadium $(Z=23)$ and (b) rhenium $(Z=45) .$ Discuss any approximations that you make.

Zachary Warner
Zachary Warner
Numerade Educator
01:30

Problem 72

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 3.50 $\mathrm{cm} .$

Zachary Warner
Zachary Warner
Numerade Educator
04:05

Problem 73

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 $\hbar N^{2} \sqrt{k^{\prime} / m} .$ Hint: See Section 40.5 and the hint given in Problem $41.47 . )$

Keshav Singh
Keshav Singh
Numerade Educator
07:40

Problem 74

CP 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 momentum $\hbar$ (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. $J$ (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 $\mathrm{eV}$ ?

Keshav Singh
Keshav Singh
Numerade Educator
02:55

Problem 75

CALC Repeat the calculation of Problem 41.53 for a one-electron ion with nuclear charge $Z$ ( See Problem $41.68 . )$ How does the probability of the electron being found in the classically forbidden region depend on $Z$ ?

Ronald Prasad
Ronald Prasad
Numerade Educator