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 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.

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

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?

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.

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

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?