Chapter 41

For a particle in a three-dimensional box, what is the degeneracy (number of different quantum states with the same energy) of the following energy levels: (a) 3\(\pi^{2} \hbar^{2} / 2 m L^{2}\) and \((b)\) 9\(\pi^{2} \hbar^{2} / 2 m L^{2 n} ?\)

CP 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, the box, the equals the volume of a sphere of radius \(a=5.29 \times 10^{-11} \mathrm{m},\) 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.

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 \(\hbar\) 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 \(\hbar\) 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 \(\hbar\) 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?

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 \(\vec{L}\) and the \(z\) -axis. (c) What are the maximum and minimum values of the magnitude of the angle between \(\vec{L}\) 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 \(/\) 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 \(L_{z}\) , what angle does the vector \(\vec{\boldsymbol{L}}\) 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\(?\)

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

(a) Make a chart showing all the possible sets of quantum numbers \(l\) and \(m_{l}\) for the states of the electron in the hydrogen atom when \(n=5 .\) How many combinations are there? (b) What are the energies of these states?

(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?

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 \(0.400-\mathrm{T}\) 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?

CP 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 \(\sqrt{\frac{3}{4}} \hbar ?\) (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?

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 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 \(m_{s}=\frac{1}{2}\) \(\left(\) "spin up \(^{\prime \prime}\right)\) and \(m_{s}=-\frac{1}{2}(\) "spin down") levels of a hydrogen atom in the 1 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?

A hydrogen atom in a particular orbital angular momentum state is found to have \(j\) quantum numbers \(\frac{7}{2}\) and \(\frac{4}{2} .\) What is the letter that labels the value of \(l\) for the state?

For germanium \((\mathrm{Ge}, \mathrm{Z}=32),\) 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 \(n, l, m_{l},\) 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 \(41.3 .\)

(a) Write out the ground-state electron configuration \(\left(1 s^{2}, 2 s^{2}, \ldots\right)\) for the carbon atom. (b) What element of next-larger \(Z\) has chemical properties similar to those of carbon? Give the ground-state electron configuration for this element.

(a) Write out the ground-state electron configuration \(\left(1 s^{2},\right.\) \(2 s^{2}, \ldots .\) for the beryllium atom. (b) What element of next-larger \(Z\) 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 next-larger \(Z\) 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.

For magnesium, the first ionization potential is 7.6 \(\mathrm{eV}\) 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 \(N^{2+}\) ion? (b) Estimate the energy of the least strongly bound level in the \(L\) shell of \(N^{2+} .(c)\) The doubly charged ion \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\( shell of \)\mathrm{P}^{2+}$ .

The energies for an electron in the \(K, L,\) and \(M\) shells of the tungsten atom are \(-69,500 \mathrm{eV},-12,000 \mathrm{eV},\) and \(-2200 \mathrm{eV}\) respectively. Calculate the wavelengths of the \(K_{\alpha}\) and \(K_{\beta} \mathrm{x}\) rays of tungsten.

CALC A particle is described by the normalized wave function \(\psi(x, y, z)=A x e^{-\alpha x^{2}} e^{-\beta y^{2}} e^{-\gamma y^{2}},\) where \(A, \alpha, \beta,\) and \(\gamma\) are all real, positive constants. The probability that the particle will be found in the infinitesimal volume \(d x d y d z\) centered at the point \(\left(x_{0}, y_{0}, z_{0}\right)\) is \(\left|\psi\left(x_{0}, y_{0}, z_{0}\right)\right|^{2} d x d y d z z\) (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 \(x_{0} ?\)

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

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

(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?

CALC The wave function for a hydrogen atom in the 2\(s\) state is $$\psi_{2 s}(r)=\frac{1}{\sqrt{32 \pi a^{3}}}\left(2-\frac{r}{a}\right) e^{-r / 2 a}$$ (a) Verify that this function is normalized. (b) In the Bohr model, the distance between the electron and the nucleus in the \(n=2\) state is exactly 4\(a\) . Calculate the probability that an electron in the 2\(s\) state will be found at a distance less than 4\(a\) from the nucleus.

(a) For an excited state of hydrogen, show that the smallest angle that the orbital angular momentum vector \(\vec{L}\) can have with the \(z\) -axis is $$\left(\theta_{L}\right)_{\min }=\arccos \left(\frac{n-1}{\sqrt{n(n-1)}}\right)$$ (b) What is the corresponding expression for \(\left(\theta_{L}\right)_{\max },\) the largest possible angle between \(\vec{L}\) and the \(z\) -axis?

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

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

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?

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 magnetic-field 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 3.50 \(\mathrm{cm} .\)

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