Our analysis of the hydrogen atom simply extends to any element which has been ionized to have a single electron orbiting the nucleus. In this problem, we will consider such an atom, whose nucleus consists of \(Z\) protons and a single electron.

(a) What are the energy eigenstates of this ionized atom? What is its "Bohr radius" and how does it compare to that of hydrogen?

(b) What is the minimal value of \(Z\) such that the speed of the orbiting electron in the ground state is comparable to the speed of light, say, \(v \sim c / 2\) ? What element does this correspond to?

(c) The first corrections to the description of the hydrogen atom from the speed of the electron can be accounted for by the expression for the kinetic energy of a relativistic particle. Classically, this is

\[\begin{equation*}K=\sqrt{m_{e}^{2} c^{4}+|\vec{p}|^{2} c^{2}}-m_{e} c^{2}, \tag{9.169}\end{equation*}\]

where \(\vec{p}\) is the momentum of the particle. By Taylor expanding this expression in the non-relativistic limit where \(|\vec{p}| \ll m_{e} c\), show that the first non-zero term is independent of the speed of light \(c\), and is the non-relativistic kinetic energy with which we are familiar. What is the next term in this Taylor expansion? Do relativistic corrections tend to increase or decrease the kinetic energy, as the speed of the electron becomes comparable to the speed of light?