Chapter 38 Exercises: Capstone — Hydrogen Atom from First Principles

Part A: Conceptual Questions (*)

These questions test your understanding of the core synthesis ideas. No calculations required.

A.1 Explain why the hydrogen atom has an "accidental" degeneracy in $l$ that other atoms (helium, lithium, etc.) do not share. What symmetry is responsible, and what breaks it in multi-electron atoms?

A.2 The fine-structure formula $E_{\text{fs}} \propto \alpha^2 E_n (n/(j+1/2) - 3/4)$ depends on $j$ but not on $l$ separately. Explain why this is physically significant — what does it tell us about the relationship between perturbation theory and the Dirac equation?

A.3 A friend claims: "The variational method is always less accurate than the exact solution, so it's useless for hydrogen." Respond to this claim. When might the variational method be preferred even when an exact solution exists?

A.4 The Lamb shift breaks the degeneracy between $2S_{1/2}$ and $2P_{1/2}$ that the Dirac equation preserves. What is the physical origin of this splitting? Why does the $2S_{1/2}$ state shift more than the $2P_{1/2}$ state?

A.5 Explain why the $2S_{1/2}$ state of hydrogen is metastable (lifetime $\sim 0.14$ s) while the $2P_{1/2}$ state has a lifetime of only $\sim 1.6$ ns. What selection rule is responsible? What decay mechanism eventually allows the $2S$ state to reach the ground state?

A.6 The 21-cm hydrogen line has been proposed as a universal standard for interstellar communication. Why is this particular transition significant? What are its advantages over, say, the Lyman-$\alpha$ line?

A.7 Why do numerical methods on a uniform radial grid perform poorly for the hydrogen atom? What property of the Coulomb potential causes the difficulty, and how does a logarithmic grid address it?

A.8 In the hierarchy of corrections (gross → fine → Lamb → hyperfine), each layer is smaller than the previous by roughly $\alpha^2$. Why does $\alpha$ control this hierarchy? What would atomic physics look like if $\alpha$ were of order unity?

Part B: Applied Problems (**)

These problems require direct application of the chapter's key results.

B.1: Exact Wavefunctions

(a) Write out the complete wavefunction $\psi_{210}(r,\theta,\phi)$ for hydrogen, including the radial part, the spherical harmonic, and the normalization constant.

(b) Compute $\langle r \rangle$, $\langle r^2 \rangle$, and $\Delta r = \sqrt{\langle r^2\rangle - \langle r\rangle^2}$ for the $\psi_{210}$ state.

(c) At what radius $r$ is the radial probability density $|R_{21}(r)|^2 r^2$ maximized? Compare to the Bohr model prediction for the $n = 2, l = 1$ orbit.

(d) Compute $|\psi_{200}(0)|^2$ and $|\psi_{210}(0)|^2$. Explain the physical significance of the difference.

B.2: Fine-Structure Splittings

(a) Compute the fine-structure energy corrections for all states with $n = 3$: $3S_{1/2}$, $3P_{1/2}$, $3P_{3/2}$, $3D_{3/2}$, $3D_{5/2}$.

(b) Construct the complete energy-level diagram for $n = 3$, showing the gross structure and fine structure on appropriate scales.

(c) Determine which of the following transitions are allowed by electric dipole selection rules: - $3D_{5/2} \to 2P_{3/2}$ - $3D_{5/2} \to 2S_{1/2}$ - $3S_{1/2} \to 2S_{1/2}$ - $3P_{3/2} \to 2P_{1/2}$ - $3D_{3/2} \to 2P_{1/2}$

(d) For each allowed transition in part (c), compute the transition wavelength including fine-structure corrections.

B.3: Variational Calculations

(a) Use the trial wavefunction $\psi(r) = A(1 + \beta r)e^{-\alpha r}$ with two variational parameters to compute the ground-state energy of hydrogen. Show that the optimal parameters give $\beta = 0$ and $\alpha = 1/a_0$, recovering the exact result.

(b) Now try $\psi(r) = A(c_1 e^{-\alpha_1 r} + c_2 r e^{-\alpha_2 r})$ with the constraint that $\psi$ is orthogonal to the ground state. Minimize the energy to obtain a variational estimate for the $2S$ energy. How does your result compare to $E_2 = -3.40$ eV?

(c) Using a Gaussian trial $\psi(r) = Ae^{-\alpha r^2}$, compute the variational ground-state energy. Express your answer in terms of $E_1$ and evaluate numerically. Explain why the Gaussian gives a worse bound than the exponential.

B.4: Numerical Solutions

(a) Set up the finite-difference radial Schrödinger equation for $l = 0$ on a uniform grid with $N = 50$ points and $r_{\max} = 20a_0$. Write the Hamiltonian matrix explicitly.

(b) Diagonalize your matrix and extract the three lowest eigenvalues. Compare to $E_1, E_2, E_3$.

(c) Repeat with $N = 100, 200, 500$. Plot the error in $E_1$ as a function of $N$. What power law does the error follow?

(d) Now switch to a logarithmic grid and repeat with $N = 50$. Compare the accuracy to the uniform grid with $N = 500$.

B.5: Hyperfine Structure

(a) Compute the hyperfine splitting frequency for the ground state of hydrogen ($n = 1$). Express your answer in MHz and in wavelength (cm).

(b) For deuterium (proton replaced by deuteron with $I = 1$, $g_D = 0.857$), compute the hyperfine splitting of the ground state. What are the possible $F$ values?

(c) For the $n = 2, l = 0, j = 1/2$ state of hydrogen, compute the hyperfine splitting. Compare to the ground-state splitting — why is it smaller?

Part C: Synthesis and Analysis (***)

These problems require integrating multiple concepts and making connections across the chapter.

C.1: Complete Energy Accounting

Compute the total energy of the hydrogen $1S_{1/2}$ ground state including: - Gross structure ($E_1$) - Fine-structure correction - Lamb shift (use 8172 MHz for $1S$) - Hyperfine shift (for $F = 1$ and $F = 0$ separately) - Reduced mass correction

Present your final answer to 8 significant figures and compare to the experimental ionization energy of $13.598434$ eV.

C.2: Method Comparison

For the $3D$ states of hydrogen:

(a) Compute the energies analytically (exact Coulomb + fine structure).

(b) Compute the energies numerically using a 500-point logarithmic grid. Evaluate the expectation values $\langle 1/r^3 \rangle$ numerically and use them to compute the spin-orbit splitting.

(c) Set up a variational calculation using the trial wavefunction $\psi(r) = Ar^2 e^{-\beta r}Y_2^0(\theta,\phi)$ and optimize $\beta$.

(d) Compare all three results in a table. Which method gives the best accuracy for each quantity (energy, splitting, expectation values)?

C.3: The Hydrogen Spectrum

Write a Python script that computes all allowed electric dipole transitions between hydrogen states with $n \leq 5$, including fine-structure corrections. Output a table of wavelengths and classify each transition into its spectral series. Plot a stick spectrum with the Lyman, Balmer, and Paschen series in different colors.

C.4: Convergence Analysis

Using the finite-difference method with a logarithmic grid:

(a) Compute the ground-state energy for $N = 10, 20, 50, 100, 200, 500, 1000, 2000$.

(b) Plot $|E_{\text{numerical}} - E_{\text{exact}}|$ vs. $N$ on a log-log scale.

(c) Fit the convergence rate: $|E - E_0| \propto N^{-p}$. What is $p$? Does it agree with the theoretical convergence order of the finite-difference method?

(d) Repeat for the $n = 3, l = 2$ state. Is the convergence rate the same?

(e) Now use Richardson extrapolation to accelerate convergence. How much does this improve the accuracy for a given $N$?

C.5: Muonic Hydrogen

Muonic hydrogen replaces the electron with a muon ($m_\mu = 207 m_e$).

(a) Compute the Bohr radius, ground-state energy, and $2P-2S$ fine-structure splitting for muonic hydrogen. By what factor do these differ from ordinary hydrogen?

(b) Compute the Lamb shift for the $2S_{1/2} - 2P_{1/2}$ splitting in muonic hydrogen. Why is the Lamb shift relatively larger in muonic hydrogen than in electronic hydrogen?

(c) Explain why muonic hydrogen provides a more sensitive probe of the proton charge radius than electronic hydrogen.

Part D: Capstone Challenge Problems (****)

These problems require original thinking and may take several hours each. They are appropriate for a final course project.

D.1: Build Your Own Hydrogen Simulator

Without looking at the provided code, write a Python program from scratch that: 1. Solves the radial Schrödinger equation numerically for $l = 0, 1, 2$ and $n$ up to 5 2. Computes fine-structure corrections using numerical expectation values 3. Generates a Grotrian diagram showing all energy levels 4. Computes and plots all allowed transition wavelengths

Validate your results against the analytical formulas and NIST data.

D.2: Hydrogen in a Magnetic Field

Extend the hydrogen simulation to include a uniform magnetic field (the Zeeman effect):

$$\hat{H}_Z = \frac{eB}{2m_e}(\hat{L}_z + 2\hat{S}_z)$$

(a) For weak fields ($\mu_B B \ll E_{\text{fs}}$), compute the Zeeman splitting of the $n = 2$ fine-structure levels using degenerate perturbation theory.

(b) For strong fields ($\mu_B B \gg E_{\text{fs}}$), compute the Paschen-Back effect for $n = 2$.

(c) Numerically diagonalize the complete Hamiltonian ($H_0 + H_{\text{fs}} + H_Z$) for intermediate field strengths and plot the energy levels as a function of $B$ from 0 to 10 T. The resulting Breit-Rabi diagram should show the smooth transition from weak-field to strong-field limits.

D.3: Helium Ground State

Using the variational method with a hydrogen-like trial wavefunction $\psi(\mathbf{r}_1, \mathbf{r}_2) = \phi(\mathbf{r}_1)\phi(\mathbf{r}_2)$ where $\phi(\mathbf{r}) = (Z_{\text{eff}}^3/\pi a_0^3)^{1/2}e^{-Z_{\text{eff}}r/a_0}$:

(a) Compute the ground-state energy of helium as a function of $Z_{\text{eff}}$.

(b) Optimize $Z_{\text{eff}}$ and compare to the experimental ionization energy.

(c) Interpret the optimal $Z_{\text{eff}}$ physically — why is it less than 2?

(d) Estimate the error in this simple variational calculation and describe how you would improve it (Hylleraas coordinates, configuration interaction, etc.).