Chapter 39

(I) For \(n=6, \quad \ell=3,\) what are the possible values of \(m_{\ell}\) and \(m_{s} ?\)

(1) For \(n=6, \quad \ell=3,\) what are the possible values of \(m_{l}\) and \(m_{s} ?\)

(I) How many different states are possible for an electron whose principal quantum number is \(n=5 ?\) Write down the quantum numbers for each state.

(I) A hydrogen atom has \(\ell=5 .\) What are the possible values for \(n, m_{\ell},\) and \(m_{s} ?\)

(I) Calculate the magnitude of the angular momentum of an electron in the \(n=5, \ell=3\) state of hydrogen.

(II) A hydrogen atom is in the \(7 g\) state. Determine \((a)\) the principal quantum number, \((b)\) the energy of the state, (c) the orbital angular momentum and its quantum number \(\ell\), and \((d)\) the possible values for the magnetic quantum number.

(II) A hydrogen atom is in the 7\(g\) state. Determine \((a)\) the principal quantum number, (b) the energy of the state, (c) the orbital angular momentum and its quantum number \(\ell,\) and \((d)\) the possible values for the magnetic quantum number.

(II) An excited \(\mathrm{H}\) atom is in a \(5 d\) state. \((a)\) Name all the states to which the atom is "allowed" to jump with the emission of a photon. (b) How many different wavelengths are there (ignoring fine structure)?

(II) An excited \(\mathrm{H}\) atom is in a 5\(d\) state. (a) Name all the states to which the atom is "allowed" to jump with the emission of a photon. (b) How many different wavelengths are there (ignoring fine structure)?

(II) The magnitude of the orbital angular momentum in an excited state of hydrogen is \(6.84 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}\) and the \(z\) component is \(2.11 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s} .\) What are all the possible values of \(n, \ell,\) and \(m_{\ell}\) for this state?

(II) By what factor is it more likely to find the electron in the ground state of hydrogen at the Bohr radius \(\left(r_{0}\right)\) than at twice the Bohr radius \(\left(2 r_{0}\right) ?\)

(II) Show that the mean value of \(r\) for an electron in the ground state of hydrogen is \(\bar{r}=\frac{3}{2} r_{0},\) by calculating $$ \overline{\boldsymbol{r}}=\int_{\text {all space }} \boldsymbol{r}\left|\psi_{100}\right|^{2} d \boldsymbol{V}=\int_{0}^{\infty} \boldsymbol{r}\left|\psi_{100}\right|^{2} 4 \boldsymbol{\pi} r^{2} d r $$

(II) Determine the average radial probability distribution \(P_{\mathrm{r}}\) for the \(n=2, \ell=1\) state in hydrogen by calculating $$ P_{\mathrm{r}}=4 \pi r^{2}\left[\frac{1}{3}\left|\psi_{210}\right|^{2}+\frac{1}{3}\left|\psi_{211}\right|^{2}+\frac{1}{3}\left|\psi_{21-1}\right|^{2}\right] . $$

(III) The wave function for the \(n=3, \quad \ell=0\) state in hydrogen is $$ \psi_{300}=\frac{1}{\sqrt{27 \pi r_{0}^{3}}}\left(1-\frac{2 r}{3 r_{0}}+\frac{2 r^{2}}{27 r_{0}^{2}}\right) e^{-\frac{r}{3 r_{0}}} $$ (a) Determine the radial probability distribution \(P_{\mathrm{r}}\) for this state, and (b) draw the curve for it on a graph. (c) Determine the most probable distance from the nucleus for an electron in this state.

(I) List the quantum numbers for each electron in the ground state of oxygen \((Z=8)\).

(I) List the quantum numbers for each electron in the ground state of \((a)\) carbon \((Z=6)\) (b) aluminum \((Z=13)\).

(I) How many electrons can be in the \(n=6, \ell=4\) subshell?

(II) If the principal quantum number \(n\) were limited to the range from 1 to \(6,\) how many elements would we find in nature?

(II) What is the full electron configuration for ( \(a\) ) nickel (Ni), (b) silver (Ag), (c) uranium (U)?

(II) Estimate the binding energy of the third electron in lithium using Bohr theory. [Hint: This electron has \(n=2\) and "sees" a net charge of approximately \(+1 e .]\) The measured value is \(5.36 \mathrm{eV}\).

(II) Let us apply the exclusion principle to an infinitely high square well (Section \(38-8\) ). Let there be five electrons confined to this rigid box whose width is \(\ell\). Find the lowest energy state of this system, by placing the electrons in the lowest available levels, consistent with the Pauli exclusion principle.

(II) Show that the total angular momentum is zero for a filled subshell.

(I) If the shortest-wavelength bremsstrahlung X-rays emitted from an X-ray tube have \(\lambda=0.027 \mathrm{nm}\), what is the voltage across the tube?

(I) If the shortest-wavelength bremstrahlung X-rays emitted from an X-ray tube have \(\lambda=0.027 \mathrm{nm},\) what is the voltage across the tube?

(I) What are the shortest-wavelength X-rays emitted by electrons striking the face of a \(32.5-\mathrm{kV}\) TV picture tube? What are the longest wavelengths?

(I) Show that the cutoff wavelength \(\lambda_{0}\) in an \(X\) -ray spectrum is given by $$\lambda_{0}=\frac{1240}{V} \mathrm{nm}$$ where \(V\) is the \(X\) -ray tube voltage in volts.

(II) Estimate the wavelength for an \(n=2\) to \(n=1\) transition in iron \((Z=26)\).

(II) A mixture of iron and an unknown material are bombarded with electrons. The wavelength of the \(\mathrm{K}_{\alpha}\) lines are \(194 \mathrm{pm}\) for iron and \(229 \mathrm{pm}\) for the unknown. What is the unknown material?

(II) Use conservation of energy and momentum to show that a moving electron cannot give off an X-ray photon unless there is a third object present, such as an atom or nucleus.

(I) If the quantum state of an electron is specified by \(\left(n, \ell, m_{\ell}, m_{s}\right),\) estimate the energy difference between the states \(\left(1,0,0,-\frac{1}{2}\right)\) and \(\left(1,0,0,+\frac{1}{2}\right)\) of an electron in the \(1 s\) state of helium in an external magnetic field of \(2.5 \mathrm{~T}\)

(II) Silver atoms \(\left(\right.\) spin \(\left.=\frac{1}{2}\right)\) are placed in a 1.0-T magnetic field which splits the ground state into two close levels. (a) What is the difference in energy between these two levels, and \((b)\) what wavelength photon could cause a transition from the lower level to the upper one? (c) How would your answer differ if the atoms were hydrogen?

(II) Silver atoms \(\left(\operatorname{spin}=\frac{1}{2}\right)\) are placed in a \(1.0-\) T magnetic field which splits the ground state into two close levels. (a) What is the difference in energy between these two levels, and \((b)\) what wavelength photon could cause a transition from the lower level to the upper one? (c) How would your answer differ if the atoms were hydrogen?

(II) In a Stern-Gerlach experiment, Ag atoms exit the oven with an average speed of \(780 \mathrm{~m} / \mathrm{s}\) and pass through a magnetic field gradient \(d B / d z=1.8 \times 10^{3} \mathrm{~T} / \mathrm{m}\) for a distance of \(5.0 \mathrm{~cm} .(a)\) What is the separation of the two beams as they emerge from the magnet? \((b)\) What would the separation be if the \(g\) -factor were 1 for electron spin?

(II) For an electron in a \(5 g\) state, what are all the possible values of \(j, m_{j}, J,\) and \(J_{z} ?\)

(II) For an electron in a 5\(g\) state, what are all the possible values of \(j, m_{i}, J,\) and \(J_{z} ?\)

(II) What are the possible values of \(j\) for an electron in (a) the \(4 p,\) (b) the \(4 f,\) and \((c)\) the \(3 d\) state of hydrogen? (d) What is \(J\) in each case?

(II) (a) Write down the quantum numbers for each electron in the gallium atom. (b) Which subshells are filled? (c) The last electron is in the \(4 p\) state; what are the possible values of the total angular momentum quantum number, \(j,\) for this electron? \((d)\) Explain why the angular momentum of this last electron also represents the total angular momentum for the entire atom (ignoring any angular momentum of the nucleus). \((e)\) How could you use \(a\) Stern-Gerlach experiment to determine which value of \(j\) the atom has?

(II) A laser used to weld detached retinas puts out 23 -mslong pulses of 640 -nm light which average 0.63 -W output during a pulse. How much energy can be deposited per pulse and how many photons does each pulse contain?

(II) Estimate the angular spread of a laser beam due to diffraction if the beam emerges through a 3.6-mm-diameter mirror. Assume that \(\lambda=694 \mathrm{nm} .\) What would be the diameter of this beam if it struck \((a)\) a satellite \(380 \mathrm{~km}\) above the Earth, (b) the Moon?

(II) A low-power laser used in a physics lab might have a power of 0.50 \(\mathrm{mW}\) and a beam diameter of 3.0 \(\mathrm{mm}\) . Calculate \((a)\) the average light intensity of the laser beam, and \((b)\) compare it to the intensity of a lightbulb emitting15 \(\mathrm{W}\) of light viewed from a distance of 2.0 \(\mathrm{m} .\)

(II) Show that a population inversion for two levels (as in a pumped laser) corresponds to a negative Kelvin temperature in the Boltzmann distribution. Explain why such a situation does not contradict the idea that negative Kelvin temperatures cannot be reached in the normal sense of temperature.

The ionization (binding) energy of the outermost electron in boron is \(8.26 \mathrm{eV}\). \((a)\) Use the Bohr model to estimate the "effective charge," \(Z_{\text {eff }}\), seen by this electron. (b) Estimate the average orbital radius.

How many electrons can there be in an " \(h\) " subshell?

What is the full electron configuration in the ground state for elements with \(Z\) equal to \((a) 25,(b) 34,(c) 39 ?[\) Hint: See the Periodic Table inside the back cover.]

What are the largest and smallest possible values for the angular momentum \(L\) of an electron in the \(n=6\) shell?

Determine the most probable distance from the nucleus of an electron in the \(n=2, \ell=0\) state of hydrogen.

Show that the diffractive spread of a laser beam, \(\approx \lambda / D\) as described in Section \(39-9,\) is precisely what you might expect from the uncertainty principle. [Hint: Since the beam's width is constrained by the dimension of the aperture \(D\), the component of the light's momentum perpendicular to the laser axis is uncertain.]

In the so-called vector model of the atom, space quantization of angular momentum (Fig. 3\()\) is illustrated as shown in Fig. 28. The angular momentum vector of magnitude \(L=\sqrt{\ell(\ell+1)} \hbar\) is thought of as processing around the \(z\) axis (like a spinning top or gyroscope) in such a way that the \(z\) component of angular momentum, \(L_{z}=m_{\ell} \hbar,\) also stays constant. Calculate the possible values for the angle \(\theta\) between \(\vec{\mathbf{L}}\) and the \(z\) axis \((a)\) for \(\ell=1,\) (b) \(\ell=2,\) and (c) \(\ell=3 .\) (d) Determine the minimum value of \(\theta\) for \(\ell=100\) and \(\ell=10^{6} .\) Is this consistent with the correspondence principle?

The angular momentum in the hydrogen atom is given both by the Bohr model and by quantum mechanics. Compare the results for \(n=2\)

For each of the following atomic transitions, state whether the transition is allowed or forbidden, and why: \((a) 4 p \rightarrow 3 p\); (b) \(3 p \rightarrow 1 s ;(c) 4 d \rightarrow 3 d ;(d) 4 d \rightarrow 3 s\) (e) \(4 s \rightarrow 2 p\).