Chapter 18 Key Takeaways: Degenerate Perturbation Theory and Fine Structure
Core Message
When energy levels of $\hat{H}_0$ are degenerate, standard perturbation theory breaks down because the "correct" zeroth-order states are not uniquely determined. The cure is to diagonalize the perturbation $\hat{H}'$ within each degenerate subspace, finding the basis in which perturbation theory can proceed. Applied to hydrogen, this technique reveals the fine structure — three corrections of order $\alpha^2 E_n$ that depend on $n$ and $j$ alone — the hyperfine structure and its astrophysically important 21 cm line, and the rich phenomenology of the Zeeman effect.
Key Concepts
1. Degenerate Perturbation Theory
Non-degenerate perturbation theory fails when $E_n^{(0)} = E_m^{(0)}$ for $n \neq m$ because the second-order energy correction contains terms with vanishing denominators. The resolution: within the $g$-fold degenerate subspace, construct the $g \times g$ matrix $W_{ij} = \langle i | \hat{H}' | j \rangle$ and diagonalize it. The eigenvalues are the first-order energy corrections, and the eigenvectors are the "correct" zeroth-order states from which standard perturbation theory may proceed.
2. Good Quantum Numbers
A quantum number is "good" if it labels eigenstates of operators that commute with both $\hat{H}_0$ and $\hat{H}'$. Finding good quantum numbers is equivalent to finding the correct basis without explicit diagonalization. For hydrogen fine structure: $n$, $l$, $j$, $m_j$ are good. For the strong-field Zeeman effect: $n$, $l$, $m_l$, $m_s$ are good.
3. Fine Structure of Hydrogen
Three corrections of order $\alpha^2 E_n$ collectively produce the fine structure:
- Relativistic kinetic energy: $\hat{H}'_{\text{rel}} = -\hat{p}^4/(8m_e^3 c^2)$ — lowers all levels, largest for low $l$
- Spin-orbit coupling: $\hat{H}'_{\text{SO}} \propto \hat{\mathbf{L}} \cdot \hat{\mathbf{S}} / r^3$ — splits levels by $j$ value, zero for $l = 0$
- Darwin term: $\hat{H}'_{\text{Darwin}} \propto \delta^3(\mathbf{r})$ — affects only $l = 0$ states
4. The Fine Structure Constant
$\alpha = e^2/(4\pi\epsilon_0\hbar c) \approx 1/137.036$ — the dimensionless coupling constant of electromagnetism. It sets the scale of fine structure corrections relative to the gross structure: $E_{\text{FS}} \sim \alpha^2 E_n$.
5. Hyperfine Structure
The magnetic interaction between electron and nuclear spins splits energy levels by $\sim (m_e/m_p)\alpha^2 E_n$ — roughly 1000 times smaller than fine structure. The ground-state hydrogen hyperfine splitting produces the 21 cm line ($\nu = 1420$ MHz).
6. The Zeeman Effect
An external magnetic field $B$ adds $\hat{H}'_Z = \mu_B(\hat{L}_z + 2\hat{S}_z)B/\hbar$. The behavior depends on the ratio of $\mu_B B$ to the fine structure splitting: - Weak field: $j$, $m_j$ good; splitting $= g_j m_j \mu_B B$ - Strong field (Paschen-Back): $m_l$, $m_s$ good; splitting $= (m_l + 2m_s)\mu_B B$ - Intermediate field: must diagonalize full Hamiltonian
Key Equations
| Equation | Name | When to Use |
|---|---|---|
| $W_{ij} = \langle i \| \hat{H}' \| j \rangle$ | Perturbation matrix | Degenerate perturbation: construct in degenerate subspace |
| $\det(\mathbf{W} - E^{(1)}\mathbf{I}) = 0$ | Secular equation | Find first-order energies by diagonalizing $\mathbf{W}$ |
| $E_{\text{FS}}^{(1)} = -\frac{(E_n^{(0)})^2}{2m_ec^2}\left(\frac{4n}{j+1/2} - 3\right)$ | Fine structure formula | Total fine structure correction for hydrogen |
| $\langle \hat{\mathbf{L}}\cdot\hat{\mathbf{S}} \rangle = \frac{\hbar^2}{2}[j(j+1) - l(l+1) - s(s+1)]$ | $\hat{\mathbf{L}}\cdot\hat{\mathbf{S}}$ expectation | Spin-orbit coupling in coupled basis |
| $g_j = 1 + \frac{j(j+1) - l(l+1) + s(s+1)}{2j(j+1)}$ | Lande $g$-factor | Weak-field Zeeman splitting |
| $E_Z^{(1)} = g_j m_j \mu_B B$ | Weak-field Zeeman energy | When $\mu_B B \ll \Delta E_{\text{FS}}$ |
| $\Delta E_{\text{HF}} = \frac{4}{3}g_p\alpha^2(m_e/m_p)\|E_1^{(0)}\|$ | Ground-state hyperfine | 21 cm line frequency |
Key Expectation Values in Hydrogen
| Quantity | Value | Used For |
|---|---|---|
| $\langle 1/r \rangle_{nl}$ | $1/(n^2 a_0)$ | Relativistic correction |
| $\langle 1/r^2 \rangle_{nl}$ | $1/[n^3(l+1/2)a_0^2]$ | Relativistic correction |
| $\langle 1/r^3 \rangle_{nl}$ | $1/[n^3 l(l+1/2)(l+1)a_0^3]$ ($l \geq 1$) | Spin-orbit coupling |
| $\|\psi_{n00}(0)\|^2$ | $1/(\pi n^3 a_0^3)$ | Darwin term, hyperfine |
Decision Framework: Which Perturbation Theory to Use?
Is the unperturbed level degenerate?
├── NO → Use non-degenerate perturbation theory (Chapter 17)
└── YES → Is the perturbation diagonal in your chosen basis?
├── YES → Each diagonal element is E^(1); proceed as non-degenerate
└── NO → Diagonalize H' within the degenerate subspace
├── All eigenvalues distinct → Degeneracy fully lifted; proceed
└── Some eigenvalues equal → Residual degeneracy; go to 2nd order
within remaining degenerate subspace
Common Misconceptions
| Misconception | Correction |
|---|---|
| "Degenerate perturbation theory is a fundamentally different formalism" | It is the same perturbation theory, preceded by a basis change within the degenerate subspace. |
| "The Darwin term is a quantum gravity effect" | It is a purely relativistic quantum effect (Zitterbewegung), unrelated to gravity. |
| "Spin-orbit coupling is always the dominant fine structure correction" | For $s$-states ($l = 0$), spin-orbit vanishes and the relativistic + Darwin corrections dominate. |
| "The anomalous Zeeman effect violates theory" | It is perfectly explained by spin and the Lande $g$-factor; "anomalous" is a historical misnomer. |
| "The 21 cm line is unobservable because the transition is too slow" | The enormous number of hydrogen atoms in the universe compensates for the slow rate. |
| "Fine structure depends on $l$" | The three corrections individually depend on $l$, but they combine to give a result depending only on $n$ and $j$. |
Numerical Reference
| Constant | Symbol | Value |
|---|---|---|
| Fine structure constant | $\alpha$ | $1/137.036$ |
| Bohr magneton | $\mu_B$ | $5.788 \times 10^{-5}$ eV/T |
| Nuclear magneton | $\mu_N$ | $3.152 \times 10^{-8}$ eV/T |
| Proton $g$-factor | $g_p$ | $5.586$ |
| Electron rest energy | $m_ec^2$ | $0.511$ MeV |
| Proton/electron mass ratio | $m_p/m_e$ | $1836.15$ |
| Bohr radius | $a_0$ | $5.292 \times 10^{-11}$ m |
| 21 cm frequency | $\nu_{\text{HF}}$ | $1420.405$ MHz |
Energy Scale Hierarchy for Hydrogen
| Level | Scale | Example ($n = 2$) |
|---|---|---|
| Gross structure (Bohr) | $\alpha^2 m_e c^2 \sim 10$ eV | $-3.4$ eV |
| Fine structure | $\alpha^4 m_e c^2 \sim 10^{-4}$ eV | $4.5 \times 10^{-5}$ eV |
| Lamb shift (QED) | $\alpha^5 m_e c^2 \sim 10^{-6}$ eV | $4.4 \times 10^{-6}$ eV |
| Hyperfine ($n = 1$) | $\alpha^4 (m_e/m_p) m_e c^2 \sim 10^{-6}$ eV | $5.9 \times 10^{-6}$ eV |
Looking Ahead
- Chapter 19 (Variational Principle): A complementary approximation method that does not require a solvable $\hat{H}_0$.
- Chapter 21 (Time-Dependent Perturbation Theory): Transition rates between fine structure levels; selection rules for spectral lines.
- Chapter 22 (Scattering): Cross-sections for scattering off hydrogen-like potentials.
- Chapter 29 (Relativistic QM): The Dirac equation derives all three fine structure corrections at once.
- Chapter 38 (Capstone): Full hydrogen simulation including fine structure, hyperfine, and Zeeman effects.