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

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?

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) 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?

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 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?

Pure germanium has a band gap of $0.67$ eV. The Fermi energy is in the middle of the gap. For temperatures of $250$ K, $300$ K, and $350$ K, calculate the probability $f\left(E\right)$ that a state at the bottom of the conduction band is occupied.