||

1. Draw a diagram similar to Figure 41.2 to show all the possible orientations of the angular momentum vector $\vec{L}$ for the case $l=3$. Label each $\vec{L}$ with the appropriate value of m.

2. What is the minimum angle between $\vec{L}$ and the $z$-axis?

| There exist subatomic particles whose spin is characterized by $s=1$, rather than the $s=\frac{1}{2}$ of electrons. These particles are said to have a spin of one.

1. What is the magnitude (as a multiple of $\text{ℏ}$) of the spin angular momentum $S$ for a particle with a spin of one?

2. What are the possible values of the spin quantum number?

3. Draw a vector diagram similar to Figure 41.13 to show the possible orientations of $\vec{S}$.

|| A hydrogen atom has $l=2$. What are the (a) minimum (as a multiple of $\text{ℏ}$) and (b) maximum values of the quantity ${\left(L_x{}^2+L_y{}^2\right)}^{1\text{/}2}?$

| A hydrogen atom in its fourth excited state emits a photon with a wavelength of 1282 nm. What is the atom’s maximum possible orbital angular momentum (as a multiple of $\text{ℏ}$) after the emission?

|| Calculate (a) the radial wave function and (b) the radial probability density at $r=\frac{1}{2}{a}_{\text{B}}$ for an electron in the $1s$ state of hydrogen. Give your answers in terms of $a_{\text{B}}$.

|| For an electron in the $1s$ state of hydrogen, what is the probability of being in a spherical shell of thickness $0.010a_{\text{B}}$ at distance (a) $\frac{1}{2}{a}_{\text{B}}$, (b) $a_{\text{B}}$, and (c) $2a_{\text{B}}$ from the proton?

|| CALC Prove that the normalization constant of the $1s$ radial wave function of the hydrogen atom is ${(\pi a_{\text{B}}{}^3)}^{-1\text{/}2}$, as given in Equations 41.7.

Hint: A useful definite integral is

|| CALC Prove that the normalization constant of the $2p$ radial wave function of the hydrogen atom is ${(24\pi a_{\text{B}}{}^3)}^{-1\text{/}2}$, as shown in Equations 41.7.

Hint: See the hint in Problem 32.

||

1. CALC Calculate and graph the hydrogen radial wave function $R_{2p}\left(r\right)$ over the interval $0\le r\le 8a_{\text{B}}$.

2. Determine the value of $r$ (in terms of $a_{\text{B}}$) for which $R_{2p}\left(r\right)$ is a maximum.

3. Example 41.3 and Figure 41.7 showed that the radial probability density for the $2p$ state is a maximum at $r=4a_{\text{B}}$. Explain why this differs from your answer to part b.

|| CALC Prove that the radial probability density peaks at $r=a_{\text{B}}$ for the $1s$ state of hydrogen.

|| CALC Suppose you have a machine that gives you pieces of candy when you push a button. Eighty percent of the time, pushing the button gets you two pieces of candy. Twenty percent of the time, pushing the button yields 10 pieces. The average number of pieces per push is $N_{\text{avg}}=2\times 0.80+10\times 0.20=3.6$. That is, 10 pushes should get you, on average, 36 pieces. Mathematically, the average value when the probabilities differ is $N_{\text{avg}}={{\displaystyle \sum }}^{\text{}}(N_i\times \text{Probability of }i)$. We can do the same thing in quantum mechanics, with the difference that the sum becomes an integral. If you measured the distance of the electron from the proton in many hydrogen atoms, you would get many values, as indicated by the radial probability density. But the average value of $r$ would be

Calculate the average value of $r$ in terms of $a_{\text{B}}$ for the electron in the $1s$ and the $2p$ states of hydrogen.

|| In general, an atom can have both orbital angular momentum and spin angular momentum. The total angular momentum is defined to be $\vec{J}=\vec{L}+\vec{S}$. The total angular momentum is quantized in the same way as $\vec{L}$ and $\vec{S}$. That is, $J=\sqrt{j(j+1)}\text{ℏ}$, where $j$ is the total angular momentum quantum number. The $z$-component of $\vec{J}$ is $J_z=L_z+S_z=m_j\text{ℏ}$, where $m_j$ goes in integer steps from $-j$ to $+j$. Consider a hydrogen atom in a $p$ state, with $l=1$.

1. $L_z$ has three possible values and $S_z$ has two. List all possible combinations of $L_z$ and $S_z$. For each, compute $J_z$ and determine the quantum number $m_j$. Put your results in a table.

2. The number of values of $J_z$ that you found in part a is too many to go with a single value of $j$. But you should be able to divide the values of $J_z$ into two groups that correspond to two values of $j$. What are the allowed values of $j$? Explain. In a classical atom, there would be no restrictions on how the two angular momenta $\vec{L}$ and $\vec{S}$ can combine. Quantum mechanics is different. You’ve now shown that there are only two allowed ways to add these two angular momenta.

|

1. What downward transitions are possible for a sodium atom in the $6s$ state? (See Figure 41.23.)

2. What are the wavelengths of the photons emitted in each of these transitions?

|| The $5d\to 3p$ transition in the emission spectrum of sodium has a wavelength of 499 nm. What is the energy of the $5d$ state?

|| A sodium atom emits a photon with wavelength 818 nm shortly after being struck by an electron. What minimum speed did the electron have before the collision?

|| The ionization energy of an atom is known to be 5.5 eV. The emission spectrum of this atom contains only the four wavelengths 310.0 nm, 354.3 nm, 826.7 nm, and 1240.0 nm. Draw an energy-level diagram with the fewest possible energy levels that agrees with these experimental data. Label each level with an appropriate $l$ quantum number.

Hint: Don’t forget about the $\text{Δ}l$ selection rule.

| FIGURE P41.42 shows the first few energy levels of the lithium atom. Make a table showing all the allowed transitions in the emission spectrum. For each transition, indicate

1. The wavelength, in nm.

2. Whether the transition is in the infrared, the visible, or the ultraviolet spectral region.

3. Whether or not the transition would be observed in the lithium absorption spectrum.

FIGURE P41.42

|| FIGURE P41.43 shows a few energy levels of the mercury atom.

1. Make a table showing all the allowed transitions in the emission spectrum. For each transition, indicate the photon wavelength, in nm.

2. What minimum speed must an electron have to excite the 492-nm-wavelength blue emission line in the Hg spectrum?

FIGURE P41.43

|| Suppose you put five electrons into a 0.50-nm-wide one-dimensional rigid box (i.e., an infinite potential well).

1. Use an energy-level diagram to show the electron configuration of the ground state.

2. What is the ground-state energy—that is, the total energy of all five electrons in the ground-state configuration?

|| Three electrons are in a one-dimensional rigid box (i.e., an infinite potential well) of length 0.50 nm. Two are in the $n=1$ state and one is in the $n=6$ state. The selection rule for the rigid box allows only those transitions for which $\text{Δ}n$ is odd.

1. Draw an energy-level diagram. On it, show the filled levels and show all transitions that could emit a photon.

2. What are all the possible wavelengths that could be emitted by this system?

|| An atom in an excited state has a 1.0% chance of emitting a photon in 0.10 ns. What is the lifetime of the excited state?

||

1. What is the decay rate for the $2p$ state of hydrogen?

2. During what interval of time will 10% of a sample of $2p$ hydrogen atoms decay?

||

1. Find an expression in terms of $\tau$ for the half-life $t_{1\text{/}2}$ of a sample of excited atoms. The half-life is the time at which half of the excited atoms have undergone a quantum jump and emitted a photon.

2. What is the half-life of the $3p$ state of sodium?

|| A ruby laser emits a 100 MW, 10-ns-long pulse of light with a wavelength of 690 nm. How many chromium atoms undergo stimulated emission to generate this pulse?

| An electrical discharge in a neon-filled tube maintains a steady population of $1.0\times {10}^9$ atoms in an excited state with $\tau =20\text{ns}$. How many photons are emitted per second from atoms in this state?

|| The 1997 Nobel Prize in physics went to Steven Chu, Claude Cohen-Tannoudji, and William Phillips for their development of techniques to slow, stop, and “trap” atoms with laser light. To see how this works, consider a beam of rubidium atoms $(\text{mass}1.4\times {10}^{-25}\text{kg})$ traveling at $500\text{m}\text{/}\text{s}$ after being evaporated out of an oven. A laser beam with a wavelength of 780 nm is directed against the atoms. This is the wavelength of the $5s\to 5p$ transition in rubidium, with $5s$ being the ground state, so the photons in the laser beam are easily absorbed by the atoms. After an average time of 15 ns, an excited atom spontaneously emits a 780-nm-wavelength photon and returns to the ground state.

1. The energy-momentum-mass relationship of Einstein’s theory of relativity is $E^2=p^2{c}^2+m^2{c}^4$. A photon is massless, so the momentum of a photon is $p=E_{\text{photon}}\text{/}c$. Assume that the atoms are traveling in the positive $x$-direction and the laser beam in the negative $x$-direction. What is the initial momentum of an atom leaving the oven? What is the momentum of a photon of light?

2. The total momentum of the atom and the photon must be conserved in the absorption processes. As a consequence, how many photons must be absorbed to bring the atom to a halt?

   NOTE   Momentum is also conserved in the emission processes. However, spontaneously emitted photons are emitted in random directions. Averaged over many absorption/emission cycles, the net recoil of the atom due to emission is zero and can be ignored.

3. Assume that the laser beam is so intense that a ground-state atom absorbs a photon instantly. How much time is required to stop the atoms?

4. Use Newton’s second law in the form $F=\text{Δ}p\text{/}\text{Δ}t$ to calculate the force exerted on the atoms by the photons. From this, calculate the atoms’ acceleration as they slow.

5. Over what distance is the beam of atoms brought to a halt?

||| BIO In fluorescence microscopy, an important tool in biology, a laser beam is absorbed by target molecules in a sample. These molecules are then imaged by a microscope as they emit longer-wavelength photons in quantum jumps back to lower energy levels, a process known as fluorescence. A variation on this technique is two-photon excitation. If two photons are absorbed simultaneously, their energies add. Consequently, a molecule that is normally excited by a photon of energy $E_{\text{photon}}$ can be excited by the simultaneous absorption of two photons having half as much energy. For this process to be useful, the sample must be irradiated at the very high intensity of at least ${10}^{32}\text{photons}\text{/}{m}^2\text{s}$. This is achieved by concentrating the laser power into very short pulses (100 fs pulse length) and then focusing the laser beam to a small spot. The laser is fired at the rate of ${10}^8$ pulses each second. Suppose a biologist wants to use two-photon excitation to excite a molecule that in normal fluorescence microscopy would be excited by a laser with a wavelength of 420 nm. If she focuses the laser beam to a $2.0$-$\mu \text{m}$-$\text{diameter}$ spot, what minimum energy must each pulse have?

||| CALC Two excited energy levels are separated by the very small energy difference $\text{Δ}E$. As atoms in these levels undergo quantum jumps to the ground state, the photons they emit have nearly identical wavelengths $\lambda$.

1. Show that the wavelengths differ by

   $$\text{Δ}\lambda ={\displaystyle \frac{{\lambda }^2}{hc}}\text{Δ}E$$

2. In the Lyman series of hydrogen, what is the wavelength difference between photons emitted in the $n=20$ to $n=1$ transition and photons emitted in the $n=21$ to $n=1$ transition?

||| CALC What is the probability of finding a $1s$ hydrogen electron at distance $r>a_{\text{B}}$ from the proton?

||| CALC Prove that the most probable distance from the proton of an electron in the $2s$ state of hydrogen is $5.236a_{\text{B}}$.

||| CALC Find the distance, in terms of $a_{\text{B}}$, between the two peaks in the radial probability density of the $2s$ state of hydrogen.

Hint: This problem requires a numerical solution.

||| An atom in an excited state has a 1.0% chance of emitting a photon in 0.20 ns. How long will it take for 25% of a sample of excited atoms to decay?