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 equals the volume of a sphere of radius $a=5.29\times {10}^{-11}$ 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.

A photon is emitted when an electron in a three-dimensional cubical box of side length $8.00\times {10}^{-11}$ m makes a transition from the $n_X=2$, $n_Y=2$, $n_Z=1$ state to the $n_X=1$, $n_Y=1$, $n_Z=1$ state. What is the wavelength of this photon?

Consider an electron in the $N$ shell. What is the smallest orbital angular momentum it could have?

Consider an electron in the $N$ shell. What is the largest orbital angular momentum it could have? Express your answers in terms of $\hslash$ and in SI units.

Consider an electron in the $N$ shell. What is the largest orbital angular momentum this electron could have in any chosen direction? Express your answers in terms of $\hslash$ and in SI units.

Consider an electron in the $N$ shell. What is the largest spin angular momentum this electron could have in any chosen direction? Express your answers in terms of $\hslash$ and in SI units.

Consider an electron in the $N$ shell. 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? Note: Part (c) asked for the largest orbital angular momentum this electron could have in any chosen direction.

The orbital angular momentum of an electron has a magnitude of $4.716\times {10}^{-34}$ kg-m²/s. What is the angular momentum quantum number $l$ for this electron?

In a particular state of the hydrogen atom, the angle between the angular momentum vector $\overrightarrow{L}$ and the $z$-axis is $u=26.6$°. If this is the smallest angle for this particular value of the orbital quantum number $l$, what is $l$?

Calculate, in units of $U$, 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 $nh$ postulated in the Bohr model. What trend do you see?

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. Calculate the splitting (in electron volts) of the ml levels when the atom is put in a $0.800$ T magnetic field that is in the $+z$-direction

A hydrogen atom in a $3p$ state is placed in a uniform external magnetic field $\overrightarrow{B}$. Consider the interaction of the magnetic field with the atom's orbital magnetic dipole moment. What field magnitude $B$ is required to split the $3p$ state into multiple levels with an energy difference of $2.71\times {10}^{-5}$ eV between adjacent levels?

A hydrogen atom undergoes a transition from a $2p$ state to the $1s$ ground state. In the absence of a magnetic field, the energy of the photon emitted is $122$ 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. How many different photon wavelengths are observed for the $2p\to 1s$ transition? What are the $m_l$ values for the initial and final states for the transition that leads to each photon wavelength?

A hydrogen atom in the $5g$ state is placed in a magnetic field of $0.600$ T that is in the $z$-direction. Into how many levels is this state split by the interaction of the atom's orbital magnetic dipole moment with the magnetic field?

(a) If you treat an electron as a classical spherical object with a radius of $1.0\times {10}^{-17}$ m, what angular speed is necessary to produce a spin angular momentum of magnitude $\sqrt{\frac{3}{4}}h$?

(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?

The hyperfine interaction in a hydrogen atom between the magnetic dipole moment of the proton and the spin magnetic dipole moment of the electron splits the ground level into two levels separated by $5.9\times {10}^{-6}$ eV. Calculate the wavelength and frequency of the photon emitted when the atom makes a transition between these states, and compare your answer to the value given at the end of Section $41.5$. In what part of the electromagnetic spectrum does this lie? Such photons are emitted by cold hydrogen clouds in interstellar space; by detecting these photons, astronomers can learn about the number and density of such clouds.

Calculate the energy difference between the $m_s=\frac{1}{2}$ ('spin up') and $m_s=-\frac{1}{2}$ ('spin down') levels of a hydrogen atom in the $1s$ 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 $\frac{9}{2}$. If $n=5$, what is the energy difference between the $j=\frac{7}{2}$ and $j=\frac{9}{2}$ levels?

The $5s$ electron in rubidium (Rb) sees an effective charge of $2.771e$. Calculate the ionization energy of this electron.

The energies of the $4s$, $4p$, and $4d$ states of potassium are given in Example $41.10$. Calculate $Z_{eff}$ for each state. What trend do your results show? How can you explain this trend?

The doubly charged ion N²⁺ is formed by removing two electrons from a nitrogen atom. What is the ground-state electron configuration for the N²⁺ ion?

Estimate the energy of the least strongly bound level in the $L$ shell of N²⁺.

The energy of the $2s$ state of lithium is $-5.391$ eV. Calculate the value of $Z_{eff}$ for this state.

Estimate the energy of the highest-$l$ state for (a) the $L$ shell of Be⁺ and (b) the $N$ shell of Ca⁺.

The energies for an electron in the $K$, $L$, and $M$ shells of the tungsten atom are $-69,500$ eV, $-12,000$ eV, and $-2,200$ eV, respectively. Calculate the wavelengths of the $K_{\alpha }$ and $K_{\beta }$ x rays of tungsten.