Chapter 41

An electron is in the hydrogen atom with \(n=3 .\) (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 \(L\) and the \(z\) -axis.

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 \(L\) and the \(z\) -axis. (c) What are the maximum and minimum values of the magnitude of the angle between \(L\) and the \(z\) -axis?

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 \(I\) for this electron?

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 \(L_{v}\) 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 41.2\(?\)

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 \(n\) h postulated in the Bohr model. What trend do you see?

Problem-Solving Strategy 41.1 claims that the electric potential energy of a proton and an electron 0.10 \(\mathrm{nm}\) apart has magnitude 15 eV. Verify this claim.

An electron in a hydrogen atom is in an \(s\) level, and the atom is in a magnetic field \(\vec{B}=B \hat{k} .\) Explain why the "spin up" state \(\left(m_{s}=+\frac{1}{2}\right)\) has a higher energy than the "spin down" state \(\left(m_{s}=-\frac{1}{2}\right)\)

Show that \(\Phi(\phi)=e^{i m \phi}=\Phi(\phi+2 \pi)(\text { that is, show that }\) \(\Phi(\phi)\) 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 \(e^{i \phi}=\cos \phi+i \sin \phi_{\cdot} )\).

A hydrogen atom is in a \(d\) state. In the absence of an external magnetic field the states with different \(m_{I}\) 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 - 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.

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?

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 \(\mathrm{nm}\) . 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.

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 \(\mathrm{T}\) 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 \(n \neq 1 ?\)

Calculate the energy difference between the \(m_{s}=\frac{1}{2}\left(^{u} \text { spin }\right.\) \(\mathrm{up}^{\prime \prime}\) and \(m_{s}=-\frac{1}{2}\) ("spin down") levels of a hydrogen atom in the 1 \(s\) 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?

List the different possible combinations of \(l\) and \(j\) for a hydrogen atom in the \(n=3\) level.

A hydrogen atom in a particular orbital angular momentum state is found to have \(j\) quantum numbers \(\frac{7}{2}\) and \(\frac{9}{2} .\) What is the letter that labels the value of \(l\) for the state?

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 \(\sqrt{\frac{3}{4}} \hbar ?(\mathrm{b})\) Use \(v=r \omega\) and the result of part \((\mathrm{a})\) to calculate the speed \(v\) of a point at the electron's equator. What does your result suggest about the validity of this model?

For germanium (Ge, \(Z=32 )\) , 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 .

Make a list of the four quantum numbers \(n, l, m_{b},\) 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 .

For magnesium, the first ionization potential is 7.6 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?

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

(a) What is the lowest possible energy (in electron volts) of an electron in hydrogen if its orbital angular momentum is \(\sqrt{12} \mathrm{k}\) ? (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?

(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?

(a) The energy of an electron in the 4\(s\) state of sodium is \(-1.947 \mathrm{eV} .\) 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?

(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 \(l, m_{1}\) and \(\hbar .\) (b) What is the meaning of \(\sqrt{L_{x}^{2}+L_{y}^{2}} ?\) (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.

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 \(m_{I}\) 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.

Effective Magnetic Field. An electron in a hydrogen atom 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 \(L=m v r\) 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.

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?

For an ion with nuclear charge \(Z\) and a single electron, the electric potential energy is \(-\mathrm{Ze}^{2} / 4 \pi \epsilon_{0} r\) 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 \(\mathrm{Ze}^{2}\) . Consider the \(\mathrm{N}^{6+}\) ion, with seven protons and one electron. (a) What is the wavelength of the photon emitted when the \(\mathrm{N}^{6+}\) ion makes a transition from the \(n=2\) state to the \(n=1\) ground state?

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 wave-length 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? Docs the magnetic field increase or decrease the wavelength? Disregard the effect of electron spin. [Hint: Use the result of Problem \(39.56(\mathrm{c}) . ]\)

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.

(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 \(N(N+1) / 2 . ](\mathrm{b})\) Which shell has 50 states?

(a) If the intrinsic spin angular momentum \(S\) of the earth had the same limitations as that of the electron, what would be the angular velocity of our planet's spin on its axis? To get a reasonable answer but simplify the calculations, assume that the earth is uniform throughout. (b) Could we, in principle, use the method of part (a) to determine the angular velocity of the electron's spin? Why?

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\(N\) elecurons is \(\hbar N^{2} \sqrt{k^{\prime} / m}\) (Hint: See Section 40.4 and the hint given in Problem \(41.59 .\) .)