Chapter 13 The Hydrogen Atom

13.1 The Eigenvalue Problem

Exercise 13.1.1 Derive Eqs. (13.1.11) and (13.1.14) starting from Eqs. (13.1.8)-(13.1.10).

Exercise 13.1.2 Derive the degeneracy formula, Eq. (13.1.18).

Exercise 13.1.3 Starting from the recursion relation, obtain $\psi_{210}$ (normalized).

Exercise 13.1.4 Recall from the last chapter [Eq. (12.6.19)] that as $r \rightarrow \infty$, $U_E \sim(r)^{m e^2 / \kappa \hbar^2} \mathrm{e}^{-\kappa r}$ in a Coulomb potential $V=-\mathrm{e}^2 / r$ $\left[\kappa=\left(2 m W / \hbar^2\right)^{1 / 2}\right]$. Show that this agrees with Eq. (13.1.26).

Exercise 13.1.5 (Virial Theorem) Since $|n, l, m\rangle$ is a stationary state, $\langle\dot{\mathbf{\Omega}}\rangle=0$ for any $\mathbf{\Omega}$.\ Consider $\mathbf{\Omega}=\mathbf{R} \cdot \mathbf{P}$ and use Ehrenfest's theorem to show that $\langle T\rangle=(-1 / 2)\langle V\rangle$ in the state $|n, l, m\rangle$.

13.2 The Degeneracy of the Hydrogen Spectrum

Exercise 13.2.1 Let us see why the conservation of the Runge-Lenz vector $\mathbf{n}$ implies closed orbits.

(1) Express $\mathbf{n}$ in terms of $\mathbf{r}$ and $\mathbf{p}$ alone (get rid of $\mathbf{1}$).

(2) Since the particle is bound, it cannot escape to infinity. So, as we follow it from some arbitrary time onward, it must reach a point $\mathbf{r}_{\max}$ where its distance from the origin stops growing. Show that

13.3 Numerical Estimates and Comparison with Experiment

Exercise 13.3.1 The pion has a range of $1~\mathrm{Fermi}=10^{-5} \text{\AA}$ as a mediator of nuclear force. Estimate its rest energy.

Exercise 13.3.2 Estimate the de Broglie wavelength of an electron of kinetic energy $200~\mathrm{eV}$. (Recall $\lambda=2 \pi \hbar / p$.)

13.4 Multielectron Atoms and the Periodic Table

Exercise 13.4.1 Show that if we ignore interelectron interactions, the energy levels of a multielectron atom go as $Z^2$. Since the Coulomb potential is $Z e / r$, why is the energy $\propto Z^2$?

Exercise 13.4.2 Compare (roughly) the sizes of the uranium atom and the hydrogen atom. Assume levels fill in the order of increasing $n$, and that the nonrelativistic description holds. Ignore interelectron effects.

Exercise 13.4.3 Visible light has a wavelength of approximately $5000 \text{\AA}$. Which of the series-Lyman, Balmer, Paschen-do you think was discovered first?