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University Physics with Modern Physics

Hugh D. Young

Chapter 41

Quantum Mechanics II: Atomic Structure - all with Video Answers

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Chapter Questions

04:25

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$\pi$ $\hslash$$^2$/$2mL$$^2$ and (b) 9$\pi$$^2$$\hslash$$^2$/$2mL$$^2$?

Ommair Ishaque
Ommair Ishaque
Numerade Educator
07:18

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$^{-11}$ 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
Ommair Ishaque
Numerade Educator
04:54

Problem 3

A photon is emitted when an electron in a threedimensional cubical box of side length 8.00 $\times$ 10$^{-11}$ 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
Ommair Ishaque
Numerade Educator
10:40

Problem 4

For each of the following states of a particle in a threedimensional cubical 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?

Ommair Ishaque
Ommair Ishaque
Numerade Educator
09:32

Problem 5

A particle is in the three-dimensional cubical 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.

Ommair Ishaque
Ommair Ishaque
Numerade Educator
03:40

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$^{-14}$ m, the approximate diameter of a nucleus?

Ommair Ishaque
Ommair Ishaque
Numerade Educator
06:27

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 $\hslash$ 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 $\hslash$ 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 $\hslash$ 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
Ommair Ishaque
Numerade Educator
08:42

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 $\hslash$. (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 $L$ S and the z-axis?

Ommair Ishaque
Ommair Ishaque
Numerade Educator
03:27

Problem 9

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

Ommair Ishaque
Ommair Ishaque
Numerade Educator
05:43

Problem 10

Consider states with angular momentum quantum number $l$ = 2. (a) In units of $\hslash$, what is the largest possible value of L$_z$? (b) In units of $\hslash$, 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?

Ommair Ishaque
Ommair Ishaque
Numerade Educator
04:31

Problem 11

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

Ommair Ishaque
Ommair Ishaque
Numerade Educator
03:50

Problem 12

A hydrogen atom is in a state that has $L$$_z$ = 2$\hslash$. 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 number for this state?

Ommair Ishaque
Ommair Ishaque
Numerade Educator
03:11

Problem 13

Calculate, in units of $\hslash$, 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$\hslash$ postulated in the Bohr model. What trend do you see?

Ommair Ishaque
Ommair Ishaque
Numerade Educator
03:10

Problem 14

(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?

Ommair Ishaque
Ommair Ishaque
Numerade Educator
04:10

Problem 15

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?

Ommair Ishaque
Ommair Ishaque
Numerade Educator
04:58

Problem 16

(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.

Ommair Ishaque
Ommair Ishaque
Numerade Educator
03:58

Problem 17

Show that $\Phi$($\phi$) = $e$$^{im_l}$$^\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$2,.... ($Hint$: Euler's formula states that $e$$^i$$^\phi$ = cos $\phi$ + $i$ sin $\phi$.)

Ommair Ishaque
Ommair Ishaque
Numerade Educator
05:03

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 0.800-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.

Kathleen Tatem
Kathleen Tatem
Numerade Educator
04:29

Problem 19

A hydrogen atom in a 3$p$ state is placed in a uniform external magnetic field $\vec B$. 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?

Surendra Kumar
Surendra Kumar
Numerade Educator
10:52

Problem 20

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 wavelengths are observed for the 2p $\rightarrow$ 1s 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.

Kathleen Tatem
Kathleen Tatem
Numerade Educator
04:53

Problem 21

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?

Kathleen Tatem
Kathleen Tatem
Numerade Educator
04:56

Problem 22

A hydrogen atom in the $n = 1$, $ms = - {1\over2}$ state is placed in a magnetic field with a magnitude of 1.60 $T$ 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 $n\neq 1$?

Kathleen Tatem
Kathleen Tatem
Numerade Educator
04:25

Problem 23

$\textbf{Classical Electron Spin}$. (a) If you treat an electron as a classical spherical object with a radius of 1.0 $\times$ 10$^{-17}$ m, what angular speed is necessary to produce a spin angular momentum of magnitude $\sqrt{3\over4}\hslash$ ? (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?

Kathleen Tatem
Kathleen Tatem
Numerade Educator
05:10

Problem 24

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 $5.9 \times 10^{-6} eV$. (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
Kathleen Tatem
Numerade Educator
05:51

Problem 25

Calculate the energy difference between the $m_s = {1\over2}$ ("spin up") and $m_s = - {1\over2}$ ("spin down") levels of a hydrogen atom in the $1s$ state when it is placed in a 1.45-T magnetic field in the $negative$ $z$-direction. Which level, $m_s = {1\over2}$ or $ms = - {1\over2}$ , has the lower energy?

Kathleen Tatem
Kathleen Tatem
Numerade Educator
10:25

Problem 26

A hydrogen atom in a particular orbital angular momentum state is found to have $j$ quantum numbers ${7\over2}$ and ${9\over2}$ . (a) What is the letter that labels the value of $l$ for the state? (b) If $n = 5$, what is the energy difference between the $j = {7\over2}$ and $j = {9\over2}$ levels?

Khoobchandra Agrawal
Khoobchandra Agrawal
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
00:44

Problem 28

For germanium ($Ge, Z = 32$), make a list of the number of electrons in each subshell ($1s, 2s, 2p,\dots$). 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
08:04

Problem 29

(a) Write out the ground-state electron configuration ($1s^2, 2s^2,\dots$) for the beryllium atom. (b) What element of nextlarger $Z$ 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 nextlarger $Z$ 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.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
00:57

Problem 30

(a) Write out the ground-state electron configuration ($1s^2, 2s^2,\dots$) for the carbon atom. (b) What element of nextlarger $Z$ has chemical properties similar to those of carbon? Give the ground-state electron configuration for this element.

Zachary Warner
Zachary Warner
Numerade Educator
00:55

Problem 31

The $5s$ 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 $4s, 4p,$ and $4d$ states of potassium are given in Example 41.10. Calculate $Z_{eff}$ for each state. What trend do your results show? How can you explain this trend?

Zachary Warner
Zachary Warner
Numerade Educator
08:03

Problem 33

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

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
07:39

Problem 34

(a) The energy of the $2s$ state of lithium is -5.391 eV. Calculate the value of $Z_{eff}$ for this state. (b) The energy of the $4s$ state of potassium is -4.339 eV. Calculate the value of $Z_{eff}$ for this state. (c) Compare $Z_{eff}$ for the $2s$ state of lithium, the $3s$ state of sodium (see Example 41.9), and the $4s$ state of potassium. What trend do you see? How can you explain this trend?

DM
Dominique Madrid
Numerade Educator
01:14

Problem 35

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

Zachary Warner
Zachary Warner
Numerade Educator
02:23

Problem 36

A $K_\alpha$ x ray emitted from a sample has an energy of 7.46 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 ($Ca, Z = 20$); (b) cobalt ($Co, Z = 27$); (c) cadmium ($Cd, Z = 48$).

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 eV, -12,000 eV, and -2200 eV, respectively. Calculate the wavelengths of the $K_\alpha$ and $K_\beta$ 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
16:41

Problem 40

An electron is in a three-dimensional box with side lengths $L_X =$ 0.600 nm and $L_Y = L_Z = 2L_X$. What are the quantum numbers $n_X, n_Y,$ and $n_Z$ and the energies, in 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
12:55

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 $(n_X = 1, n_Y = 1, n_Z = 1)$, 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 $n_X = 2, n_Y = 1, n_Z = 1$.

Guilherme Barros
Guilherme Barros
Numerade Educator
14:49

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 eV and 3.47 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 (including the spin degeneracy) for the next higher energy state?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
10:25

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
Nathan Silvano
Numerade Educator
08:57

Problem 44

$\textbf{A Three-Dimensional Isotropic Harmonic Oscillator}$. An isotropic harmonic oscillator has the potentialenergy function $U(x, y, z) = {1\over2}k'(x^2 + y^2 + z^2)$. ($Isotropic$ means that the force constant $k'$ 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, $\psi _{n_x}(x)$ is a solution to the one-dimensional harmonic-oscillator Schrodinger equation, Eq. (40.44), with energy $E_{n_x} = (n_x + {1\over2} )\hslash \omega$. 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.

Nathan Silvano
Nathan Silvano
Numerade Educator
View

Problem 45

$\textbf{Three-Dimensional Anisotropic Harmonic Oscillator}$. An oscillator has the potential-energy function $U(x, y, z) = {1\over2} k{'_1}(x^2 + y^2) + {1\over 2} k{'_2}z^2$, where $k{'_1} > k{'_2}$. 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 $nx, ny,$ and $nz$) are there for the ground level and for the first excited level? Compare to part (c) of Problem 41.44.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
04:43

Problem 46

A particle is described by the normalized wave function $\psi$$(x, y, z)$ = $Axe$${^-}{^a}{^x}^2$$e$${^-}{^\beta}$${^y}^2$$e$${^-}{^y}^z$$^2$, where $A$, $\alpha$,$\beta$, and $\gamma$ are all real, positive constants. The probability that the particle will be found in the infinitesimal volume $dx$ $dy$ $dz$ centered at the point $(x_0$, $y_0$, $z_0$) is $\mid$$\psi$$(x_0$, $y_0$, $z_0$)$\mid$$^2$ $dx$ $dy$ $dz$. (a) At what value of $x_0$ is the particle most likely to be found? (b) Are there values of $x_0$ for which the probability of the particle being found is zero? If so,at what $x_0$$?$

Guilherme Barros
Guilherme Barros
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 $2n^2$. $[Hint$: 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
06:38

Problem 48

(a) What is the lowest possible energy (in electron volts) of an electron in hydrogen if its orbital angular momentum is $\sqrt{20}$ $\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?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
07:26

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.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:44

Problem 50

For a hydrogen atom, the probability $P$$(r)$ of finding the electron within a spherical shell with inner radius $r$ and outer radius $r$ + $dr$ is given by Eq. (41.25). For a hydrogen atom in the 1$s$ ground state, at what value of $r$ does $P$$(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)?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:42

Problem 51

The normalized radial wave function for the $2p$ state of the hydrogen atom is $R_2{_p}$ = $( 1/ \sqrt{24a^5}$)$re$$^-{^r}{^/}{^2}{^a}$. After we average over the angular variables, the radial probability function becomes $P$$(r)$ $dr$ = $(R_2{_p}$$)^2$r$^2$ $dr$. At what value of $r$ is $P$$(r)$ for the $2p$ state a maximum? Compare your results to the radius of the $n$ = 2 state in the Bohr model.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:31

Problem 52

$\textbf{Rydberg Atoms.}$ $Rydberg$ $atoms$ 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 300? By the Bohr model, what is the radius of the Rydberg electron's orbit? (c) Repeat part (b) for $n$ = 600.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
14:18

Problem 53

An atom in a $3$$d$ state emits a photon of wavelength 475.082 nm when it decays to a $2$$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 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)?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:42

Problem 54

An atom in a $3d$ state emits a photon of wavelength 475.082 nm when it decays to a $2p$ 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 T. Ignore the effects of the electron's spin. (c) For the case in part (b),if the energy of the $3d$ 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
07:29

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 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?

Ross Blackburn
Ross Blackburn
Numerade Educator
06:31

Problem 56

$\textbf{Stern-Gerlach Experiment}$. In a Stern-Gerlach experiment, the deflecting force on the atom is $F$$_z$ = -$\mu$$_z$($dB$$_z$/$dz$), where $\mu$$_z$ is given by Eq. (41.38) and $dB$$_z$/$dz$ is the magnetic-field gradient. In a particular experiment, the magnetic-field region is 50.0 cm long; assume the magnetic-field gradient is constant in that region. A beam of silver atoms enters the magnetic field with a speed of 375 m/s. What value of $dB$$_z$/$d$$_z$ is required to give a separation of 1.0 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.)

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
08:16

Problem 57

A large number of hydrogen atoms in $1s$ 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 K. According to the Maxwell-Boltzmann distribution (see Section 39.4), what is the ratio of the number of atoms in the $m_s = {1\over2}$ state to the number in the $m_s = - {1\over2}$ state when the magnetic-field magnitude is (a) $5.00 \times 10^{-5}$ T (approximately the earth's field); (b) 0.500 T; (c) 5.00 T?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:24

Problem 58

$\textbf{Effective Magnetic Field.}$ An electron in a hydrogen atom is in the $2p$ 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 $L$ = $mvr$ and taking $L$ to have the quantum-mechanical value for a $2p$ 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. 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)?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:30

Problem 59

$\textbf{Weird Universe.}$ In another universe, the electron is a 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 groundstate electron configuration of sodium?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:34

Problem 60

A lithium atom has three electrons, and the $^2$S$_1$$_/$$_2$ 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 mm is emitted in the $^2$P$_3$$_/$$_2$ $\rightarrow$ $^2$S$_1$$_/$$_2$ transition, and a photon with wavelength 67.09761 $\mu$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 2p$ state of the lithium atom. How does your result compare to that for the $3p$ level of sodium found in Example 41.7?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:22

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 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 39.76(c).]

Zachary Warner
Zachary Warner
Numerade Educator
03:47

Problem 62

$\textbf{Electron Spin Resonance.}$ 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 magneticfield 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 cm.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:39

Problem 63

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
12:31

Problem 64

A hydrogen atom initially in an $n$ = $3,$ $l$ = 1 state makes a 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 $n$ = $3$, $l$ = $1$, $j$ = \(\frac{1}{2}\) state. Which photon has the longer wavelength?

Nathan Silvano
Nathan Silvano
Numerade Educator
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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.
The ionization energy is the minimum energy required to remove the least-bound electron from a ground-state atom. (a) The units kJ/mol given in the table are the minimum energy in kJ required to ionize 1 mol of atoms. Convert the given values for ionization energy to the energy in 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$$_{eff}$ for this electron in each alkali-metal atom. (d) The ionization energies decrease as $Z$ increases. Does $Z$$_{eff}$ increase or decrease as $Z$ increases? Why does $Z$$_{eff}$ have this behavior?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
05:12

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 nm. What is $L$? (b) You find that $\lambda$ = 234 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
Keshav Singh
Numerade Educator
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Problem 67

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.
(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 $\mid$$\mu$$_z$$\mid$, the magnitude of the spin magnetic moment. (c) Let $\gamma$ = $\mid$$\mu$$_z$$\mid$$/$$\mid$$S$$_z$$\mid$ denote the gyromagnetic ratio for electron spin. Use your result of part (b) to calculate $\gamma$. What is the value of $\gamma$$/$($e$$/$2$m$) given by your experimental data?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 68

Each of $2N$ 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'$$x$$^2$, where $k$' 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 $2N$ electrons is $\hslash$$N^2$$\sqrt{k'/m}$. ($Hint:$ See Section 40.5 and the hint given in Problem 41.47.)

Keshav Singh
Keshav Singh
Numerade Educator
06:27

Problem 69

Consider a simple model of the helium atom in which two electrons, each with mass $m$, move around the nucleus (charge $+2e$) in the same circular orbit. Each electron has orbital angular momentum $\hslash$ (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 eV ?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
00:55

Problem 70

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

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:18

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/cm$^3$; (b) 10$^8$ atoms/cm$^3$; (c) 10$^{11}$ atoms/cm$^3$; (d) 10$^{21}$ atoms/cm$^3$.

Nathan Silvano
Nathan Silvano
Numerade Educator
00:52

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 $L$ associated with this state? (a) $\sqrt{2} \space\hslash $; (b) $\sqrt{6} \space\hslash$; (c) $\sqrt{200}\space \hslash$; (d) $\sqrt{10,100}\space \hslash $.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:36

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.

Sanat Mukherjee
Sanat Mukherjee
Numerade Educator