Chapter 18 Exercises: Degenerate Perturbation Theory and Fine Structure

Part A: Conceptual Questions (No Calculation Required)

A.1 Explain in your own words why non-degenerate perturbation theory fails when the unperturbed spectrum has degeneracies. What exactly diverges, and why?

A.2 What does it mean to say that $j$ and $m_j$ are "good quantum numbers" for the hydrogen fine structure? Why are $m_l$ and $m_s$ not good quantum numbers in this context?

A.3 The Darwin term affects only $s$-states, while spin-orbit coupling affects only states with $l \geq 1$. Explain physically why each correction vanishes in the other's domain.

A.4 The fine structure formula depends on $n$ and $j$ but not on $l$. The $2s_{1/2}$ and $2p_{1/2}$ states have the same fine structure correction despite very different spatial distributions. What is the deep reason for this $l$-degeneracy? Under what circumstances is it lifted?

A.5 The spontaneous emission rate for the 21 cm transition is $A \approx 2.9 \times 10^{-15}$ s$^{-1}$, corresponding to a lifetime of about 11 million years. How is it possible that the 21 cm line is one of the strongest observed radio lines in astronomy?

A.6 In the strong-field (Paschen-Back) Zeeman effect, the splitting pattern collapses to a simple triplet. Explain physically why the strong field "undoes" the complexity of the anomalous Zeeman effect.

A.7 A student claims: "The Thomas precession factor of 1/2 in the spin-orbit coupling is just a relativistic correction. If we do the calculation in a fully relativistic framework (the Dirac equation), we do not need it." Is this correct? Explain.

A.8 Why is the fine structure constant $\alpha$ dimensionless? What would change about atomic physics if $\alpha$ were twice as large?

Part B: Degenerate Perturbation Theory — Mechanics

B.1: Two-Fold Degeneracy — Complete Workout

Consider a system with two degenerate states $|a\rangle$ and $|b\rangle$ at energy $E^{(0)}$. The perturbation matrix in this subspace is:

$$W = \begin{pmatrix} W_{aa} & W_{ab} \\ W_{ab}^* & W_{bb} \end{pmatrix}$$

(a) Find the eigenvalues of $W$ (the first-order energy corrections). Express them in terms of $W_{aa}$, $W_{bb}$, and $|W_{ab}|$.

(b) Find the "good" zeroth-order states.

(c) Under what condition is the degeneracy not lifted at first order?

(d) Apply your result to the specific case $W_{aa} = 3\epsilon$, $W_{bb} = \epsilon$, $W_{ab} = 2\epsilon$, where $\epsilon > 0$ is small. What are the corrected energies?

B.2: Three-Fold Degeneracy

A perturbation acts on a three-fold degenerate subspace. In the original basis $\{|1\rangle, |2\rangle, |3\rangle\}$, the perturbation matrix is:

$$W = V_0 \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}$$

(a) Find the eigenvalues and eigenvectors of $W$.

(b) Which degeneracies are lifted, and which remain?

(c) Write down the correct zeroth-order states.

B.3: Real vs. Degenerate

A particle in a box has energy levels $E_n = n^2 E_1$. Consider the perturbation $\hat{H}' = \lambda \delta(x - L/3)$, where $L$ is the box length.

(a) For the non-degenerate state $n = 1$, calculate $E_1^{(1)}$ using first-order non-degenerate perturbation theory.

(b) Now consider a two-dimensional square box ($L \times L$). The states $|n_x, n_y\rangle = |1, 2\rangle$ and $|2, 1\rangle$ are degenerate (both have energy $5E_1$). If the perturbation is $\hat{H}' = \lambda \delta(x - L/3)\delta(y - L/3)$, apply degenerate perturbation theory to find the first-order corrected energies.

(c) What are the "good" states?

B.4: Near-Degeneracy

Two states $|a\rangle$ and $|b\rangle$ have unperturbed energies $E_a^{(0)}$ and $E_b^{(0)}$ that are close but not exactly equal: $\Delta \equiv E_b^{(0)} - E_a^{(0)}$ is small compared to the perturbation matrix elements. The perturbation matrix in this two-state subspace is:

$$H = \begin{pmatrix} E_a^{(0)} + W_{aa} & W_{ab} \\ W_{ab}^* & E_b^{(0)} + W_{bb} \end{pmatrix}$$

(a) Find the exact eigenvalues by diagonalizing $H$.

(b) Show that in the limit $\Delta \to 0$ (exact degeneracy), you recover the degenerate perturbation theory result.

(c) Show that in the limit $|W_{ab}| \ll \Delta$ (no effective degeneracy), you recover the non-degenerate perturbation theory result (including the second-order correction).

(d) Estimate the critical value of $\Delta$ below which the degenerate formalism must be used.

Part C: Relativistic Correction

C.1: Ground State Relativistic Correction

Calculate the relativistic kinetic energy correction for the hydrogen ground state ($n = 1$, $l = 0$).

(a) Using the formula $E_{\text{rel}}^{(1)} = -\frac{(E_n^{(0)})^2}{2m_e c^2}(4n/(l+1/2) - 3)$, compute the correction in eV.

(b) Express the correction as a fraction of the unperturbed energy $E_1^{(0)}$.

(c) Compute $\langle v^2/c^2 \rangle$ for the ground state of hydrogen using $\langle p^2 \rangle = 2m_e(E_n^{(0)} - \langle V \rangle)$. Does the expansion in powers of $v/c$ seem justified?

C.2: Comparing $l$ Values

For $n = 3$, calculate the relativistic correction for $l = 0$, $l = 1$, and $l = 2$.

(a) Which state has the largest relativistic correction? Explain physically.

(b) Compute the ratios $E_{\text{rel}}^{(1)}(l=0) / E_{\text{rel}}^{(1)}(l=2)$ and $E_{\text{rel}}^{(1)}(l=1) / E_{\text{rel}}^{(1)}(l=2)$.

C.3: Anharmonic Oscillator — Degenerate Version

The two-dimensional isotropic harmonic oscillator has energy levels $E_{n_x, n_y} = (n_x + n_y + 1)\hbar\omega$. The first excited level ($n_x + n_y = 1$) is two-fold degenerate: $|1, 0\rangle$ and $|0, 1\rangle$.

(a) If the perturbation is $\hat{H}' = \lambda m\omega^2 xy$, construct the $2 \times 2$ perturbation matrix $W$ in the degenerate subspace. You will need $\langle n | \hat{x} | n' \rangle$ matrix elements from the ladder operator formalism.

(b) Find the first-order corrected energies.

(c) What are the "good" states? Interpret them physically.

Part D: Spin-Orbit Coupling

D.1: Spin-Orbit for $n = 3$

For hydrogen with $n = 3$:

(a) List all allowed combinations of $l$ and $j$. How many distinct fine structure levels are there?

(b) Calculate $\langle \hat{\mathbf{L}} \cdot \hat{\mathbf{S}} \rangle$ for each $(l, j)$ combination.

(c) Calculate the spin-orbit correction $E_{\text{SO}}^{(1)}$ for each state with $l \geq 1$.

(d) Verify that adding the relativistic and Darwin corrections gives a result that depends only on $n$ and $j$.

D.2: Spin-Orbit in Alkali Atoms

In sodium, the valence electron in the $3p$ state experiences an effective potential $V_{\text{eff}}(r)$ that is not purely Coulombic but still spherically symmetric. The spin-orbit interaction is still $\hat{H}'_{\text{SO}} \propto \hat{\mathbf{L}} \cdot \hat{\mathbf{S}} / r^3$, but $\langle 1/r^3 \rangle$ must be evaluated with the actual sodium $3p$ radial wavefunction.

(a) Without knowing $\langle 1/r^3 \rangle$ explicitly, express the ratio $E_{\text{SO}}(3p_{3/2}) / E_{\text{SO}}(3p_{1/2})$ in terms of the quantum numbers only.

(b) The measured sodium D-line doublet has wavelengths 589.0 nm and 589.6 nm. Calculate the spin-orbit splitting energy $\Delta E_{\text{SO}}$ in eV and cm$^{-1}$.

(c) From this splitting, estimate $\langle 1/r^3 \rangle$ for the sodium $3p$ state (in units of $a_0^{-3}$).

D.3: Thomas Precession

(a) An electron in the Bohr model orbits the nucleus at speed $v = \alpha c / n$. Calculate the Thomas precession frequency $\omega_T = a \gamma v / (2c^2)$ for the $n = 2$ orbit, where $a = v^2/r$ is the centripetal acceleration and $\gamma$ is the Lorentz factor.

(b) Compare this with the Larmor precession frequency of the spin in the magnetic field seen in the electron's rest frame. Show that the Thomas precession is exactly half the spin precession, which is why it reduces the spin-orbit coupling by a factor of 2.

Part E: Combined Fine Structure

E.1: Fine Structure for $n = 4$

(a) List all fine structure levels for $n = 4$ hydrogen, specifying $(l, j)$ for each.

(b) Calculate $E_{\text{FS}}^{(1)}$ for each level.

(c) What is the total fine structure splitting (energy difference between the highest and lowest levels)?

(d) How many distinct energy levels are there? What is the remaining degeneracy of each?

E.2: Lyman-$\alpha$ Fine Structure

The Lyman-$\alpha$ line corresponds to the $n = 2 \to n = 1$ transition.

(a) Including fine structure, into how many distinct spectral lines does Lyman-$\alpha$ split? List all allowed transitions (selection rules: $\Delta l = \pm 1$, $\Delta j = 0, \pm 1$, but $j = 0 \to j = 0$ is forbidden).

(b) Calculate the energy of each component.

(c) What is the frequency separation (in GHz) between the outermost components?

E.3: Testing the Dirac Formula

(a) Using the exact Dirac energy formula, compute $E_{2,1/2}$ and $E_{2,3/2}$ to 8 significant figures (keep $\alpha = 1/137.036$ and $m_e c^2 = 0.511\,003\,4$ MeV).

(b) Verify that the difference agrees with the perturbative fine structure formula to the expected accuracy.

(c) Compute the next-order (order $\alpha^6$) correction from the Dirac formula for the $n = 2$, $j = 1/2$ state. How large is it compared to the $\alpha^4$ fine structure?

Part F: Hyperfine Structure

F.1: Ground State Hyperfine

(a) Calculate the ground-state hyperfine splitting $\Delta E_{\text{HF}}$ for hydrogen using:

$$\Delta E_{\text{HF}} = \frac{4}{3} g_p \alpha^4 \frac{m_e}{m_p} m_e c^2$$

Use $g_p = 5.586$, $\alpha = 1/137.036$, $m_e/m_p = 1/1836.15$, $m_e c^2 = 0.511$ MeV.

(b) Convert to frequency (MHz) and wavelength (cm). Compare with the measured value of 1420.405 MHz.

(c) Why is the agreement not perfect? What corrections are needed?

F.2: Deuterium Hyperfine

Deuterium has a nuclear spin $I = 1$ (instead of $I = 1/2$ for hydrogen) and nuclear $g$-factor $g_d = 0.857$.

(a) What are the possible values of $F$ for the ground state of deuterium?

(b) Calculate the hyperfine splitting $\Delta E_{\text{HF}}$ for deuterium. How does it compare to hydrogen?

(c) At what wavelength would you observe the deuterium hyperfine transition?

F.3: Muonium Hyperfine

Muonium is a bound state of a positive muon $\mu^+$ and an electron $e^-$. The muon has spin 1/2, mass $m_\mu = 207 m_e$, and $g$-factor $g_\mu \approx 2$.

(a) Calculate the ground-state hyperfine splitting of muonium.

(b) How does it compare to hydrogen? Why is the muonium hyperfine splitting much larger than hydrogen's?

(c) Muonium hyperfine spectroscopy provides one of the most precise tests of QED. What makes it cleaner than hydrogen for this purpose?

Part G: Zeeman Effect

G.1: Lande $g$-Factor Practice

Calculate the Lande $g$-factor for the following states:

(a) $^2S_{1/2}$, (b) $^2P_{1/2}$, (c) $^2P_{3/2}$, (d) $^2D_{3/2}$, (e) $^2D_{5/2}$, (f) $^4F_{3/2}$ (this one has $s = 3/2$).

G.2: Weak-Field Zeeman Splitting of Sodium D Lines

The sodium D lines arise from $3p \to 3s$ transitions.

(a) In a magnetic field of $B = 0.1$ T, how many Zeeman components does the $D_1$ line ($3p_{1/2} \to 3s_{1/2}$) split into? List the transition energies relative to the zero-field line center, using the selection rule $\Delta m_j = 0, \pm 1$.

(b) Repeat for the $D_2$ line ($3p_{3/2} \to 3s_{1/2}$).

(c) Draw an energy level diagram showing all transitions and label each as $\sigma^+$, $\pi$, or $\sigma^-$ (corresponding to $\Delta m_j = +1, 0, -1$).

G.3: Paschen-Back Limit

For hydrogen in a magnetic field of $B = 10$ T:

(a) Calculate $\mu_B B$ in eV. Compare with the $n = 2$ fine structure splitting. Are we in the weak-field, intermediate, or strong-field regime?

(b) List all $n = 2$ Paschen-Back energy levels, labeling each by $(m_l, m_s)$. What is the Paschen-Back splitting pattern?

(c) How many spectral lines do you expect for the $n = 2 \to n = 1$ Paschen-Back transition (using selection rules $\Delta m_l = 0, \pm 1$, $\Delta m_s = 0$)?

G.4: Breit-Rabi Formula

For the hydrogen ground state ($n = 1$, $F = 0, 1$):

(a) Plot the four energy levels as a function of magnetic field from $B = 0$ to $B = 0.2$ T using the Breit-Rabi formula. Identify the weak-field and strong-field limits.

(b) At what field strength do the $|F = 1, m_F = 0\rangle$ and $|F = 0, m_F = 0\rangle$ states have their closest approach? What is the minimum energy gap?

(c) In a hydrogen maser, the transition between $|F = 1, m_F = 0\rangle$ and $|F = 0, m_F = 0\rangle$ is used as a frequency standard. Using the Breit-Rabi formula, show that this transition frequency is insensitive to magnetic field to first order (i.e., $dE/dB = 0$ at $B = 0$ for both states). Why is this property essential for a frequency standard?

Part H: Synthesis and Advanced Problems

H.1: Positronium Fine Structure

Positronium is an electron-positron bound state. It has the same Bohr energy levels as hydrogen (with reduced mass $\mu = m_e/2$) but very different fine structure because both particles have the same mass.

(a) By what factor does the Bohr energy change compared to hydrogen?

(b) The fine structure of positronium is of order $\alpha^4 \mu c^2$, but there is no spin-orbit coupling in the same sense as hydrogen (why?). What corrections contribute?

(c) In positronium, the total spin $S$ of the $e^- e^+$ pair can be $S = 0$ (para-positronium) or $S = 1$ (ortho-positronium). The ground-state splitting between these is about $8.4 \times 10^{-4}$ eV. Which has higher energy? (Hint: the dominant interaction is the spin-spin contact interaction, similar to hyperfine structure but with $m_e$ instead of $m_p$.)

H.2: Perturbation Theory for a Real Atom — Beyond Hydrogen

Lithium has electron configuration $1s^2 2s^1$. The outermost electron sees an effective potential that is not exactly Coulombic (due to shielding by the $1s^2$ core).

(a) Explain qualitatively why the $2s$ and $2p$ states of lithium are not degenerate (unlike hydrogen). Which has lower energy?

(b) The spin-orbit splitting of the lithium $2p$ level is 0.337 cm$^{-1}$. Calculate the spin-orbit parameter $\zeta_{2p}$ defined by $E_{\text{SO}} = \frac{1}{2}\zeta_{2p}[j(j+1) - l(l+1) - s(s+1)]$.

(c) Compare $\zeta_{2p}$ for lithium with the corresponding hydrogen value. Why is it larger?

H.3: Quadratic Zeeman Effect

For very strong magnetic fields or highly excited states, the quadratic (diamagnetic) Zeeman effect becomes important. The quadratic term in the Hamiltonian is:

$$\hat{H}'_{\text{dia}} = \frac{e^2 B^2}{8m_e}(x^2 + y^2) = \frac{e^2 B^2}{8m_e} r^2 \sin^2\theta$$

(a) Calculate $\langle \hat{H}'_{\text{dia}} \rangle$ for the hydrogen ground state ($n = 1$, $l = 0$). You will need $\langle r^2 \rangle_{10} = 3a_0^2$.

(b) At what magnetic field strength does the quadratic Zeeman shift equal the linear Zeeman shift for the ground state? Express your answer in Tesla.

(c) For a Rydberg state with $n = 50$, estimate the field strength at which the quadratic term becomes comparable to the linear term. (Use $\langle r^2 \rangle \sim n^4 a_0^2$.)

H.4: The Stark Effect in Hydrogen — Degenerate Perturbation Theory Application

An external electric field $\mathbf{E} = E_0 \hat{\mathbf{z}}$ acts on a hydrogen atom. The perturbation is $\hat{H}' = eE_0 z = eE_0 r\cos\theta$.

(a) Show that for $n = 1$, the first-order Stark effect vanishes: $E_1^{(1)} = 0$. (Hint: parity.)

(b) For $n = 2$, construct the $4 \times 4$ perturbation matrix in the basis $\{|2,0,0\rangle, |2,1,0\rangle, |2,1,1\rangle, |2,1,-1\rangle\}$. Show that only one off-diagonal element is nonzero.

(c) Diagonalize to find the first-order energy corrections. Show that $E^{(1)} = 0, 0, \pm 3eE_0 a_0$.

(d) What are the "good" states? Express them in terms of the original hydrogen eigenstates. What is their physical interpretation?

Part I: Computational Exercises

I.1: Fine Structure Level Diagram

Write a Python script that plots the hydrogen energy levels for $n = 1, 2, 3$ including fine structure. The plot should show: - The gross structure (Bohr energies) on the left - The fine-structure-corrected energies on the right - Lines connecting corresponding levels - Labels with spectroscopic notation ($nL_J$)

I.2: Breit-Rabi Diagram

Using the Breit-Rabi formula, generate a publication-quality plot of all four ground-state hydrogen hyperfine levels as a function of $B$ from 0 to 0.15 T. Label the weak-field quantum numbers ($F$, $m_F$) and the strong-field quantum numbers ($m_I$, $m_S$).

I.3: Zeeman Transition Simulator

Write a program that, given a transition (e.g., $3p_{3/2} \to 3s_{1/2}$), a magnetic field strength $B$, and the Zeeman regime (weak/strong), outputs all allowed transition energies and their polarizations.

I.4: Convergence of Perturbation Theory

For hydrogen fine structure, compare the first-order perturbative result with the exact Dirac formula for $n = 1$ through $n = 10$. Plot the fractional error $(E_{\text{pert}} - E_{\text{Dirac}})/E_{\text{Dirac}}$ as a function of $n$. For which $n$ does the perturbative result become unreliable?