Modern Physics

Paul A. Tipler, Ralph A. Llewellyn

Chapter 7

Atomic Physics - all with Video Answers

Educators

Chapter Questions

Problem 1

Find the energies $E_{311}, E_{222},$ and $E_{321}$ and construct an energy-level diagram for the three-dimensional cubic well that includes the third, fourth, and fifth excited states. Which of the states on your diagram are degenerate?

Suzanne W.

Suzanne W.

Numerade Educator

Problem 2

A particle is confined to a three-dimensional box that has sides $L_{1}, L_{2}=2 L_{1},$ and $L_{3}=3 L_{1} .$ Give the sets of quantum numbers $n_{1}, n_{2},$ and $n_{3}$ that correspond to the lowest 10 energy levels of this box.

Suzanne W.

Numerade Educator

Problem 3

A particle moves in a potential well given by $V(x, y, z)=0$ for $-L / 2<x<L / 2$, $0<y<L,$ and $0<z<L$ and $V=\infty$ outside these ranges. $(a)$ Write an expression for the ground-state wave function for this particle. ( $b$ ) How do the allowed energies compare with those for a box having $V=0$ for $0<x<L,$ rather than for $-L / 2<x<L / 2 ?$

Suzanne W.

Numerade Educator

Problem 4

Write down the wave functions for the five lowest energy levels of the particle in Problem 7 - 2 .

Suzanne W.

Numerade Educator

Problem 5

( $a$ ) Repeat Problem $7-2$ for the case $L_{2}=2 L_{1}$ and $L_{3}=4 L_{1} .$ (b) What sets of quantum numbers correspond to degenerate energy levels?

Suzanne W.

Numerade Educator

Problem 6

Write down the wave functions for the lowest 10 quantized energy states for the particle in Problem $7-5 .$

Suzanne W.

Numerade Educator

Problem 7

Suppose the particle in Problem $7-1$ is an electron and $L=0.10 \mathrm{nm}$. Compute the energy of the transitions from each of the third, fourth, and fifth excited states to the ground state.

Suzanne W.

Numerade Educator

Problem 8

Consider a particle moving in a two-dimensional space defined by $V=0$ for $0<x<L$ and $0<y<L$ and $V=\infty$ elsewhere. ( $a$ ) Write down the wave functions for the particle in this well. ( $b$ ) Find the expression for the corresponding energies. ( $c$ ) What are the sets of quantum numbers for the lowest-energy degenerate state?

Suzanne W.

Numerade Educator

Problem 9

If $n=3,(a)$ what are the possible values of $\ell ?(b)$ For each value of $\ell$ in $(a)$, list the possible values of $m$. ( $c$ ) Using the fact that there are two quantum states for each combination of values of $\ell$ and $m$ because of electron spin, find the total number of electron states with $n=3$.

Suzanne W.

Numerade Educator

Problem 10

Determine the minimum angle that $\mathbf{L}$ can make with the $z$ axis when the angular momentum quantum number is $(a) \ell=4$ and $(b) \ell=2$.

Narayan Hari

Numerade Educator

Problem 11

The moment of inertia of a compact disc is about $10^{-5} \mathrm{~kg}-\mathrm{m}^{2}$. ( $a$ ) Find the angular momentum $L=I \omega$ when the disc rotates at $\omega / 2 \pi=735$ rev $/$ min and $(b)$ find the approximate value of the quantum number $\ell$.

Narayan Hari

Numerade Educator

Problem 12

Draw an accurately scaled vector model diagram illustrating the possible orientations of the angular momentum vector $\mathbf{L}$ for $(a) \ell=1,(b) \ell=2,(c) \ell=4 .(d)$ Compute the magnitude of $\mathbf{L}$ in each case.

Suzanne W.

Numerade Educator

Problem 13

For $\ell=2,(a)$ what is the minimum value of $L_{x}^{2}+L_{v}^{2} ?$ (b) What is the maximum value of $L_{x}^{2}+L_{y}^{2} ?$ (c) What is $L_{x}^{2}+L_{y}^{2}$ for $\ell=2$ and $m=1$ ? Can either $L_{x}$ or $L_{y}$ be determined from this? ( $d$ ) What is the minimum value of $n$ that this state can have?

Suzanne W.

Numerade Educator

Problem 14

For $\ell=1$, find $(a)$ the magnitude of the angular momentum $L$ and $(b)$ the possible values of $m$. ( $c$ ) Draw to scale a vector diagram showing the possible orientations of $\mathbf{L}$ with the $z$ axis. $(d)$ Repeat the above for $\ell=3$.

Suzanne W.

Numerade Educator

Problem 15

Show that, if $V$ is a function only of $r$, then $d \mathbf{L} / d t=0$, that is, $\mathbf{L}$ is conserved.

Suzanne W.

Numerade Educator

Problem 16

What are the possible values of $m$ and the smallest value of $n$ if $(a) \ell=3,$ and (b) $\ell=4,$ and $(c) \ell=0 ?(d)$ Compute the minimum possible energy for each case.

Narayan Hari

Numerade Educator

Problem 17

A hydrogen atom electron is in the $6 f$ state. ( $a$ ) What are the values of $n$ and $\ell$ ? (b) Compute the energy of the electron. (c) Compute the magnitude of L. ( $d$ ) Compute the possible values of $L_{z}$ in this situation.

Anand Jangid

Numerade Educator

Problem 18

At what values of $r / a_{0}$ is the radial function $R_{30}$ equal to zero? (See Table $7-2 .$ )

Suzanne W.

Numerade Educator

Problem 19

Use dimensional analysis to show that the expression for the energy levels of hydrogenlike atoms given by Equation $7-25$ has the units of energy.

Suzanne W.

Numerade Educator

Problem 20

For the ground state of the hydrogen atom, find the values of $(a) \psi,(b) \psi^{2},$ and (c) the radial probability density $P(r)$ at $r=a_{0}$. Give your answers in terms of $a_{0}$.

Suzanne W.

Numerade Educator

Problem 21

For the ground state of the hydrogen atom, find the probability of finding the electron in the range $\Delta r=0.03 a_{0}$ at $(a) r=a_{0}$ and at $(b) r=2 a_{0}$.

Suzanne W.

Numerade Educator

Problem 22

The radial probability distribution function for the hydrogen ground state can be written $P(r)=C r^{2} e^{-2 Z r / a_{0}}$ where $C$ is a constant. Show that $P(r)$ has its maximum value at $r=a_{0} / Z$.

Suzanne W.

Numerade Educator

Problem 23

Compute the normalization constant $C_{210}$ in Equation $7-34$.

Suzanne W.

Numerade Educator

Problem 24

Find the probability of finding the electron in the range $\Delta r=0.02 a_{0}$ at $(a) r=a_{0}$ and (b) $r=2 a_{0}$ for the state $n=2, \ell=0, m=0$ in hydrogen. (See Problem $7-26$ for the value of $C_{200}$.

Suzanne W.

Numerade Educator

Problem 25

Show that the radial probability density for the $n=2, \ell=1, m=0$ state of a one-electron atom can be written as
$$
P(r)=A \cos ^{2} \theta r^{4} e^{-Z r / a_{0}}
$$
where $A$ is a constant.

Suzanne W.

Numerade Educator

Problem 26

The value of the constant $C_{200}$ in Equation $7-33$ is
$$
C_{200}=\frac{1}{\sqrt{2 \pi}}\left(\frac{Z}{a_{0}}\right)^{3 / 2}
$$
Find the values of $(a) \psi,(b) \psi^{2},$ and $(c)$ the radial probability density $P(r)$ at $r=a_{0}$ for the state $n=2, \ell=0, m=0$ in hydrogen. Give your answers in terms of $a_{0}$.

Suzanne W.

Numerade Educator

Problem 27

Show that an electron in the $n=2, \ell=1$ state of hydrogen is most likely to be found at $r=4 a_{0}$.

Suzanne W.

Numerade Educator

Problem 28

Write down the wave function for the hydrogen atom when the electron's quantum numbers are $n=3, \ell=4,$ and $m_{\ell}=-1 .$ Check to be sure that the wave function is normalized.

Suzanne W.

Numerade Educator

Problem 29

Verify that the wave function found in Problem $7-28$ is a solution of the timeindependent Schrödinger equation, Equation $7-9 .$

Suzanne W.

Numerade Educator

Problem 30

If a classical system does not have a constant charge-to-mass ratio throughout the system, the magnetic moment can be written
$$
\mu=g \frac{Q}{2 M} L
$$
where $Q$ is the total charge, $M$ is the total mass, and $g \neq 1$. ( $a$ ) Show that $g=2$ for a solid cylinder $\left(I=\frac{1}{2} M R^{2}\right)$ that spins about its axis and has a uniform charge on its cylindrical surface. ( $b$ ) Show that $g=2.5$ for a solid sphere $\left(I=2 M R^{2} / 5\right)$ that has a ring of charge on the surface at the equator, as shown in Figure $7-33 .$

Narayan Hari

Numerade Educator

Problem 31

Assuming the electron to be a classical particle, a sphere of radius $10^{-15} \mathrm{~m}$ and a uniform mass density, use the magnitude of the spin angular momentum $|S|=[s(s+1)]^{1 / 2} \hbar=(3 / 4)^{1 / 2} \hbar$ to compute the speed of rotation at the electron's equator. How does your result compare with the speed of light?

Suzanne W.

Numerade Educator

Problem 32

How many lines would be expected on the detector plate of a Stern-Gerlach experiment (see Figure $7-15$ ) if we use a beam of $(a)$ potassium atoms, $(b)$ calcium atoms, (c) oxygen atoms, and ( $d$ ) tin atoms?

Suzanne W.

Numerade Educator

Problem 33

The force on a magnetic moment with $z$ component $\mu_{z}$ moving in an inhomogeneous magnetic field is given by Equation $7-51 .$ If the silver atoms in the SternGerlach experiment traveled horizontally $1 \mathrm{~m}$ through the magnet and $1 \mathrm{~m}$ in a field-free region at a speed of $250 \mathrm{~m} / \mathrm{s},$ what must have been the gradient of $B_{z}, d B_{z} / d z,$ in order that the beams each be deflected a maximum of $0.5 \mathrm{~mm}$ from the central, or no-field, position?

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 34

( $a$ ) The angular momentum of the yttrium atom in the ground state is characterized by the quantum number $j=3 / 2$. How many lines would you expect to see if you could do a Stern-Gerlach experiment with yttrium atoms? (b) How many lines would you expect to see if the beam consisted of atoms with zero spin, but $\ell=1 ?$

Suzanne W.

Numerade Educator

Problem 35

The spin-orbit effect removes a symmetry in the hydrogen atom potential, splitting the energy levels. ( $a$ ) Considering the state with $n=4$, write down in spectroscopic notation the identification of each state and list them in order of increasing energy. $(b)$ If a weak external magnetic field is applied to the atoms, into how many levels will each state in $(a)$ be split?

Suzanne W.

Numerade Educator

Problem 36

Suppose the outer electron in a potassium atom is in a state with $\ell=2$. Compute the magnitude of $\mathbf{L}$. What are the possible values of $j$ and the possible magnitudes of $\mathbf{J}$ ?

Narayan Hari

Numerade Educator

Problem 37

A hydrogen atom is in the $3 d$ state $(n=3, \ell=2)$. ( $a$ ) What are the possible values of $j ?(b)$ What are the possible values of the magnitude of the total angular momentum? (c) What are the possible $z$ components of the total angular momentum?

Narayan Hari

Numerade Educator

Problem 38

Compute the angle between $\mathbf{L}$ and $\mathbf{S}$ in $(a)$ the $d_{5 / 2}$ and $(b)$ the $d_{3 / 2}$ states of atomic hydrogen.

Suzanne W.

Numerade Educator

Problem 39

Write down all possible sets of quantum numbers for an electron in a ( $a$ ) $4 f,$ (b) $3 d$, and $(c) 2 p$ subshell.

Suzanne W.

Numerade Educator

Problem 40

Consider a system of two electrons, each with $\ell=1$ and $s=\frac{1}{2}$. ( $a$ ) What are the possible values of the quantum number for the total orbital angular momentum $\mathbf{L}=\mathbf{L}_{1}+\mathbf{L}_{2} ?(b)$ What are the possible values of the quantum number $S$ for the total spin $\mathbf{S}=\mathbf{S}_{1}+\mathbf{S}_{2} ?(c)$ Using the results of parts $(a)$ and $(b),$ find the possible quantum numbers $j$ for the combination $\mathbf{J}=\mathbf{L}+\mathbf{S}$. ( $d$ ) What are the possible quantum numbers $j_{1}$ and $j_{2}$ for the total angular momentum of each particle? $(e)$ Use the results of part $(d)$ to calculate the possible values of $j$ from the combinations of $j_{1}$ and $j_{2} .$ Are these the same as in part $(c) ?$

Suzanne W.

Numerade Educator

Problem 41

The Lamb shift energy difference between the $2^{2} \mathrm{~S}_{1 / 2}$ and $2^{2} \mathrm{P}_{1 / 2}$ levels in atomic hydrogen is $4.372 \times 10^{-6} \mathrm{eV} .$ ( $a$ ) What is the frequency of the photon emitted in this transition? $(b)$ What is the photon's wavelength? $(c)$ In what part of the electromagnetic spectrum does this transition lie?

Narayan Hari

Numerade Educator

Problem 42

The prominent yellow doublet lines in the spectrum of sodium result from transitions from the $3 P_{3 / 2}$ and $3 P_{1 / 2}$ states to the ground state. The wavelengths of these two lines are $589.0 \mathrm{nm}$ and $589.6 \mathrm{nm} .$ (a) Calculate the energies in $\mathrm{eV}$ of the photons corresponding to these wavelengths. (b) The difference in energy of these photons equals the difference in energy $\Delta E$ of the $3 P_{3 / 2}$ and $3 P_{1 / 2}$ states. This energy difference is due to the spin-orbit effect. Calculate $\Delta E .(c)$ If the $3 p$ electron in sodium sees an internal magnetic field $B$, the spin-orbit energy splitting will be of the order of $\Delta E=2 \mu_{B} B,$ where $\mu_{B}$ is the Bohr magneton. Estimate $B$ from the energy difference $\Delta E$ found in part $(b)$.

Narayan Hari

Numerade Educator

Problem 43

Show that the wave function of Equation $7-57$ satisfies the Schrödinger equation (Equation $7-55)$ with $V=0$ and find the energy of this state.

Suzanne W.

Numerade Educator

Problem 44

Two neutrons are in an infinite square well with $L=2.0 \mathrm{fm}$. What is the minimum total energy that the system can have? (Neutrons, like electrons, have antisymmetric wave functions. Ignore spin.)

Narayan Hari

Numerade Educator

Problem 45

Five identical noninteracting particles are place in an infinite square well with $L=1.0 \mathrm{nm} .$ Compare the lowest total energy for the system if the particles are $(a)$ electrons and $(b)$ pions. Pions have symmetric wave functions and their mass is $264 m_{e}$.

Suzanne W.

Numerade Educator

Problem 46

Write the ground-state electron configuration of $(a)$ carbon, $(b)$ oxygen, and $(c)$ argon.

Narayan Hari

Numerade Educator

Problem 47

Using Figure $7-34,$ determine the ground-state electron configurations of tin $(\mathrm{Sn}, Z=50),$ neodymium $(\mathrm{Nd}, Z=60),$ and ytterbium $(\mathrm{Yb}, Z=70) .$ Check your answers with Appendix C. Are there any disagreements? If so, which one(s)?

Suzanne W.

Numerade Educator

Problem 48

In Figure $7-20$ there are small dips in the ionization potential curve at $Z=31$ (gallium), $Z=49$ (indium), and $Z=81$ (thallium) that are not labeled in the figure. Explain these dips, using the electron configuration of these atoms given in Appendix $\mathrm{C}$.

Suzanne W.

Numerade Educator

Problem 49

Which of the following atoms would you expect to have its ground state split by the spin-orbit interaction: $\mathrm{Li}, \mathrm{B}, \mathrm{Na}, \mathrm{Al}, \mathrm{K}, \mathrm{Ag}, \mathrm{Cu}, \mathrm{Ga} ?$ (Hint: Use Appendix $\mathrm{C}$ to see which elements have $\ell=0$ in their ground state and which do not.)

Suzanne W.

Numerade Educator

Problem 50

If the $3 s$ electron in sodium did not penetrate the inner core, its energy would be $-13.6 \mathrm{eV} / 3^{2}=-1.51 \mathrm{eV} .$ Because it does penetrate, it sees a higher effective $Z$ and its energy is lower. Use the measured ionization potential of $5.14 \mathrm{~V}$ to calculate $Z_{\text {eff }}$ for the $3 s$ electron in sodium.

Suzanne W.

Numerade Educator

Problem 51

What elements have these ground-state electron configurations? (a) $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{2}$ and (b) $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{2} ?$

Suzanne W.

Numerade Educator

Problem 52

Give the possible values of the $z$ component of the orbital angular momentum of (a) a $d$ electron, $(b)$ an $f$ electron, and $(c)$ an $s$ electron.

Suzanne W.

Numerade Educator

Problem 53

Which of the following elements should have an energy-level diagram similar to that of sodium and which should be similar to mercury: $\mathrm{Li}, \mathrm{He}, \mathrm{Ca}, \mathrm{Ti}, \mathrm{Rb}, \mathrm{Ag}, \mathrm{Cd}, \mathrm{Mg},$ $\mathrm{Cs}, \mathrm{Ba}, \mathrm{Fr}, \mathrm{Ra} ?$

Suzanne W.

Numerade Educator

Problem 54

The optical spectra of atoms with two electrons in the same outer shell are similar, but they are quite different from the spectra of atoms with just one outer electron because of the interaction of the two electrons. Separate the following elements into two groups such that those in each group have similar spectra: lithium, beryllium, sodium, magnesium, potassium, calcium, chromium, nickel, cesium, and barium.

Suzanne W.

Numerade Educator

Problem 55

Which of the following elements should have optical spectra similar to that of hydrogen and which should have optical spectra similar to that of helium: $\mathrm{Li}, \mathrm{Ca}, \mathrm{Ti}, \mathrm{Rb},$ $\mathrm{Ag}, \mathrm{Cd}, \mathrm{Ba}, \mathrm{Hg}, \mathrm{Fr}, \mathrm{Ra} ?$

Suzanne W.

Numerade Educator

Problem 56

The quantum numbers $n, \ell,$ and $j$ for the outer electron in potassium have the values $4,0,$ and $\frac{1}{2}$ respectively in the ground state; $4,1,$ and $\frac{1}{2}$ in the first excited state; and $4,1,$ and $3 / 2$ in the second excited state. Make a table giving the $n, 1,$ and $j$ values for the 12 lowest energy states in potassium (see Figure $7-24$ ).

Suzanne W.

Numerade Educator

Problem 57

Which of the following transitions in sodium do not occur as electric dipole transitions? (Give the selection rule that is violated.)
$$
\begin{array}{llll}
4 S_{1 / 2} \rightarrow 3 S_{1 / 2} & 4 S_{1 / 2} \rightarrow 3 P_{3 / 2} & 4 P_{3 / 2} \rightarrow 3 S_{1 / 2} & 4 D_{5 / 2} \rightarrow 3 P_{1 / 2} \\
4 D_{3 / 2} \rightarrow 3 P_{1 / 2} & 4 D_{3 / 2} \rightarrow 3 S_{1 / 2} & 5 D_{3 / 2} \rightarrow 4 S_{1 / 2} & 5 P_{1 / 2} \rightarrow 3 S_{1 / 2}
\end{array}
$$

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 58

Transitions between the inner electron levels of heavier atoms result in the emission of characteristic $x$ rays, as was discussed in Section $4-4 .$ ( $a$ ) Calculate the energy of the electron in the $K$ shell for tungsten using $Z-1$ for the effective nuclear charge. $(b)$ The experimental result for this energy is $69.5 \mathrm{keV}$. Assume that the effective nuclear charge is $(\mathrm{Z}-\sigma),$ where $\sigma$ is called the screening constant, and calculate $\sigma$ from the experimental result for the energy.

Suzanne W.

Numerade Educator

Problem 59

Since the $P$ states and the $D$ states of sodium are all doublets, there are four possible energies for transitions between these states. Indicate which three transitions are allowed and which one is not allowed by the selection rule of Equation $7-65 .$

Suzanne W.

Numerade Educator

Problem 60

The relative penetration of the inner-core electrons by the outer electron in sodium can be described by the calculation of $Z_{\text {eff }}$ from $E=-\left[Z_{\text {eff }}^{2}(13.6) \mathrm{eV}\right] / n^{2}$ and comparing with $E=-13.6 \mathrm{eV} / n^{2}$ for no penetration (see Problem $7-47$ ). ( $a$ ) Find the energies of the outer electron in the $3 s, 3 p,$ and $3 d$ states from Figure $7-22$. (Hint: An accurate method is to use $-5.14 \mathrm{eV}$ for the ground state as given and find the energy of the $3 p$ and $3 d$ states from the photon energies of the indicated transitions.) $(b)$ Find $Z_{\text {eff }}$ for the $3 p$ and $3 d$ states. (c) Is the approximation $-13.6 \mathrm{eV} / \mathrm{n}^{2}$ good for any of these states?

Suzanne W.

Numerade Educator

Problem 61

A hydrogen atom in the ground state is placed in a magnetic field of strength $B_{z}=0.55 \mathrm{~T} .(a)$ Compute the energy splitting of the spin states.
(b) Which state has the higher energy? ( $c$ ) If you wish to excite the atom from the lower- to the higher-energy state with a photon, what frequency must the photon have? In what part of the electromagnetic spectrum does this lie?

Suzanne W.

Numerade Educator

Problem 62

Show that the change in wavelength $\Delta \lambda$ of a transition due to a small change in energy is
$$
\Delta \lambda \approx-\frac{\lambda^{2}}{h c} \Delta E
$$
(Hint: Differentiate $E=h c / \lambda .)$

Suzanne W.

Numerade Educator

Problem 63

( $a$ ) Find the normal Zeeman energy shift $\Delta E=e \hbar B / 2 m_{e}$ for a magnetic field of strength $B=0.05 \mathrm{~T} .(b)$ Use the result of Problem $7-62$ to calculate the wavelength changes for the singlet transition in mercury of wavelength $\lambda=579.07 \mathrm{nm}$.
(c) If the smallest wavelength change that can be measured in a spectrometer is $0.01 \mathrm{nm}$, what is the strength of the magnetic field needed to observe the Zeeman effect in this transition?

Suzanne W.

Numerade Educator

Problem 64

If the outer electron in lithium moves in the $n=2$ Bohr orbit, the effective nuclear charge would be $Z_{\text {eff }} e=1 e,$ and the energy of the electron would be $-13.6 \mathrm{eV} / 2^{2}=-3.4 \mathrm{eV}$. However, the ionization energy of lithium is $5.39 \mathrm{eV},$ not $3.4 \mathrm{eV}$ Use this fact to calculate the effective nuclear charge $Z_{\text {eff }}$ seen by the outer electron in lithium. Assume that $r=4 a_{0}$ for the outer electron.

Suzanne W.

Numerade Educator

Problem 65

Show that the expectation value of $r$ for the electron in the ground state of a oneelectron atom is $\langle r\rangle=(3 / 2) a_{0} / Z$.

Suzanne W.

Numerade Educator

Problem 66

If a rigid body has moment of inertia $I$ and angular velocity $\omega$, its kinetic energy is
$$
E=\frac{1}{2} I \omega^{2}=\frac{(I \omega)^{2}}{2 I}=\frac{L^{2}}{2 I}
$$
where $L$ is the angular momentum. The solution of the Schrödinger equation for this problem leads to quantized energy values given by
$$
E_{\ell}=\frac{\ell(\ell+1) \hbar^{2}}{2 I}
$$
(a) Make an energy-level diagram of these energies, and indicate the transitions that obey the selection rule $\Delta \ell=\pm 1 .(b)$ Show that the allowed transition energies are $E_{1}$, $2 E_{1}, 3 E_{1}, 4 E_{1},$ etc., where $E_{1}=\hbar^{2} / I .(c)$ The moment of inertia of the $\mathrm{H}_{2}$ molecule is $I=\frac{1}{2} m_{p} r^{2},$ where $m_{p}$ is the mass of the proton and $r \approx 0.074 \mathrm{nm}$ is the distance between the protons. Find the energy of the first excited state $\ell=1$ for $\mathrm{H}_{2}$, assuming it is a rigid rotor. $(d)$ What is the wavelength of the radiation emitted in the transition $\ell=1$ to $\ell=0$ for the $\mathrm{H},$ molecule?

Sheh Lit Chang

University of Washington

Problem 67

In a Stern-Gerlach experiment hydrogen atoms in their ground state move with speed $v_{x}=14.5 \mathrm{~km} / \mathrm{s}$. The magnetic field is in the $z$ direction and its maximum gradient is given by $d B_{z} / d z=600 \mathrm{~T} / \mathrm{m} .$ ( $a$ ) Find the maximum acceleration of the hydrogen atoms. $(b)$ If the region of the magnetic field extends over a distance $\Delta x=75 \mathrm{~cm}$ and there is an additional $1.25 \mathrm{~m}$ from the edge of the field to the detector, find the maximum distance between the two lines on the detector.

Suzanne W.

Numerade Educator

Problem 68

Find the minimum value of the angle between the angular momentum $\mathbf{L}$ and the $z$ axis for a general value of $\ell,$ and show that for large values of $\ell, \theta_{\min } \approx 1 / \ell^{1 / 2}$.

Suzanne W.

Numerade Educator

Problem 69

The wavelengths of the photons emitted by potassium corresponding to transitions from the $4 P_{3 / 2}$ and $4 P_{1 / 2}$ states to the ground state are $766.41 \mathrm{nm}$ and $769.90 \mathrm{nm} .$ (a) Calculate the energies of these photons in electron volts. ( $b$ ) The difference in energies of these photons equals the difference in energy $\Delta E$ between the $4 P_{3 / 2}$ and $4 P_{1 / 2}$ states in potassium. Calculate $\Delta E .(c)$ Estimate the magnetic field that the $4 p$ electron in potassium experiences.

Suzanne W.

Numerade Educator

Problem 70

The radius of the proton is about $R_{0}=10^{-15} \mathrm{~m} .$ The probability that the electron is inside the volume occupied by the proton is given by
$$
P=\int_{0}^{R_{0}} P(r) d r
$$
where $P(r)$ is the radial probability density. Compute $P$ for the hydrogen ground state. (Hint: Show that $e^{-2 r / a_{0}} \approx 1$ for $r \ll a_{0}$ is valid for this calculation.)

Suzanne W.

Numerade Educator

Problem 71

(a) Calculate the Landé $g$ factor (Equation $7-72$ ) for the ${ }^{2} P_{1 / 2}$ and ${ }^{2} S_{1 / 2}$ levels in a one-electron atom and show that there are four different energies for the transition between these levels in a magnetic field. $(b)$ Calculate the Landé $g$ factor for the ${ }^{2} P_{3 / 2}$ level and show that there are six different energies for the transition ${ }^{2} P_{3 / 2} \rightarrow{ }^{2} S_{1 / 2}$ in a magnetic field.

Zachary Warner

Numerade Educator

Problem 72

( $a$ ) Show that the function
$$
\psi=A \frac{r}{a_{0}} e^{-r / 2 a_{0}} \cos \theta
$$
is a solution of Equation $7-9,$ where $A$ is a constant and $a_{0}$ is the Bohr radius. $(b)$ Find the constant $A$.

Suzanne W.

Numerade Educator

Problem 73

Consider a hypothetical hydrogen atom in which the electron is replaced by a $K^{-}$ particle. The $K^{-}$ is a meson with spin 0 ; hence, no intrinsic magnetic moment. The only magnetic moment for this atom is that given by Equation $7-41 .$ If this atom is placed in a magnetic field with $B_{z}=1.0 \mathrm{~T},(a)$ what is the effect on the $1 \mathrm{~s}$ and $2 p$ states?
(b) Into how many lines does the $2 p \rightarrow 1 s$ spectral line split? $(c)$ What is the fractional separation $\Delta \lambda / \lambda$ between adjacent lines? (See Problem $7-59 .$ ) The mass of the $K^{-}$ is $493.7 \mathrm{MeV} / c^{2}$.

Suzanne W.

Numerade Educator

Problem 74

If relativistic effects are ignored, the $n=3$ level for one-electron atoms consists of the $3^{2} S_{1 / 2}, 3^{2} P_{1 / 2}, 3^{2} P_{3 / 2}, 3^{2} D_{3 / 2},$ and $3^{2} D_{5 / 2}$ states. Compute the spin-orbit-effect splittings of $3 P$ and $3 D$ states for hydrogen.

Suzanne W.

Numerade Educator

Problem 75

In the anomalous Zeeman effect, the external magnetic field is much weaker than the internal field seen by the electron as a result of its orbital motion. In the vector model (Figure $7-29$ ) the vectors $\mathbf{L}$ and $\mathbf{S}$ precess rapidly around $\mathbf{J}$ because of the internal field and $\mathbf{J}$ precesses slowly around the external field. The energy splitting is found by first calculating the component of the magnetic moment $\mu_{J}$ in the direction of $\mathbf{J}$ and then finding the component of $\mu_{z}$ in the direction of $\mathbf{B}$. $(a)$ Show that $\mu_{J}=\frac{\boldsymbol{\mu} \cdot \mathbf{J}}{J}$ can be written
$$
\mu_{J}=-\frac{\mu_{B}}{\hbar J}\left(L^{2}+2 S^{2}+3 \mathbf{S} \cdot L\right)
$$
(b) From $J^{2}=(\mathbf{L}+\mathbf{S}) \cdot(\mathbf{L}+\mathbf{S})$ show that $\mathbf{S} \cdot \mathbf{L}=\frac{1}{2}\left(J^{2}-L^{2}-S^{2}\right) \cdot(c)$ Substitute your
result in part $(b)$ into that of part $(a)$ to obtain
$$
\mu_{J}=-\frac{\mu_{B}}{2 \hbar J}\left(3 J^{2}+S^{2}-L^{2}\right)
$$
( $d$ ) Multiply your result by $J_{z} / J$ to obtain
$$
\mu_{z}=-\mu_{B}\left(1+\frac{J^{2}+S^{2}-L^{2}}{2 J^{2}}\right) \frac{J_{z}}{\hbar}
$$

Suzanne W.

Numerade Educator

Problem 76

If the angular momentum of the nucleus is $\mathbf{I}$ and that of the atomic electrons is $\mathbf{J},$ the total angular momentum of the atom is $\mathbf{F}=\mathbf{I}+\mathbf{J},$ and the total angular momentum quantum number $f$ ranges from $I+J$ to $|I-J| .$ Show that the number of possible $f$ values is $2 I+1$ if $I<J$ or $2 J+1$ if $J<I$. (If you can't find a general proof, show it for enough special cases to convince yourself of its validity.) (Because of the very small interaction of the nuclear magnetic moment with that of the electrons, a hyperfine splitting of the spectral lines is observed. When $I<J,$ the value of $I$ can be determined by counting the number of lines.)

Suzanne W.

Numerade Educator

Problem 77

Because of the spin and magnetic moment of the proton, there is a very small splitting of the ground state of the hydrogen atom called hyperfine splitting. The splitting can be thought of as caused by the interaction of the electron magnetic moment with the magnetic field due to the magnetic moment of the proton or vice versa. The magnetic moment of the proton is parallel to its spin and is about $2.8 \mu_{N},$ where $\mu_{N}=e \hbar / 2 m_{p}$ is called the nuclear magneton. ( $a$ ) The magnetic field at a distance $r$ from a magnetic moment varies with angle, but it is of the order of $B \sim 2 k_{m} \mu / r^{3},$ where $k_{m}=10^{-7}$ in SI units. Find $B$ at $r=a_{0}$ if $\mu=2.8 \mu_{N} .(b)$ Calculate the order of magnitude of the hyperfine splitting energy $\Delta E \approx 2 \mu_{B} B,$ where $\mu_{B}$ is the Bohr magneton and $B$ is your result from part $(a)$
(c) Calculate the order of magnitude of the wavelength of radiation emitted if a hydrogen atom makes a "spin flip" transition between the hyperfine levels of the ground state. [Your result is greater than the actual wavelength of this transition, $21.22 \mathrm{~cm}$, because $\left\langle r^{-3}\right\rangle$ is appreciably smaller than $a_{0}^{-3}$, making the energy $\Delta E$ found in part ( $b$ ) greater. The detection of this radiation from hydrogen atoms in interstellar space is an important part of radio astronomy.

Suzanne W.

Numerade Educator