Quantum Mechanics Course

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← Part VI/Fine Structure

3. Fine Structure of Hydrogen

Reading time: ~36 minutes | Pages: 9

Relativistic corrections to hydrogen spectrum: small splittings revealing deeper physics.

The Fine Structure Constant

$$\alpha = \frac{e^2}{4\pi\epsilon_0\hbar c} \approx \frac{1}{137.036}$$

Dimensionless constant characterizing electromagnetic interaction strength

Fine structure corrections are of order $\alpha^2 \approx 5 \times 10^{-5}$

Three Relativistic Corrections

Perturbations to non-relativistic hydrogen:

  1. Relativistic kinetic energy correction
  2. Spin-orbit coupling
  3. Darwin term (contact interaction)

1. Relativistic Kinetic Energy

Relativistic energy-momentum relation:

$$E = \sqrt{p^2c^2 + m^2c^4} \approx mc^2 + \frac{p^2}{2m} - \frac{p^4}{8m^3c^2} + \cdots$$

Perturbation Hamiltonian:

$$\hat{H}'_r = -\frac{\hat{p}^4}{8m^3c^2}$$

Energy correction:

$$E_r^{(1)} = -\frac{(E_n)^2}{2mc^2}\left[\frac{4n}{\ell + 1/2} - 3\right]$$

2. Spin-Orbit Coupling

Interaction between electron spin and orbital angular momentum:

$$\hat{H}'_{so} = \frac{1}{2m^2c^2}\frac{1}{r}\frac{dV}{dr}\vec{L}\cdot\vec{S}$$

For Coulomb potential $V = -ke^2/r$:

$$\hat{H}'_{so} = \frac{ke^2}{2m^2c^2r^3}\vec{L}\cdot\vec{S}$$

Using total angular momentum $\vec{J} = \vec{L} + \vec{S}$:

$$\vec{L}\cdot\vec{S} = \frac{1}{2}(\hat{J}^2 - \hat{L}^2 - \hat{S}^2) = \frac{\hbar^2}{2}[j(j+1) - \ell(\ell+1) - s(s+1)]$$

Total Angular Momentum

Quantum number $j$ labels total angular momentum:

$$j = \ell \pm \frac{1}{2}$$
  • For $\ell = 0$: only $j = 1/2$
  • For $\ell > 0$: both $j = \ell + 1/2$ and $j = \ell - 1/2$

Spin-orbit energy correction:

$$E_{so}^{(1)} = \frac{(E_n)^2}{mc^2}\frac{n[j(j+1) - \ell(\ell+1) - 3/4]}{\ell(\ell+1/2)(\ell+1)}$$

3. Darwin Term

Correction for s-waves ($\ell = 0$) only:

$$\hat{H}'_D = \frac{\pi\hbar^2 ke^2}{2m^2c^2}\delta^3(\vec{r})$$

Energy correction:

$$E_D^{(1)} = \frac{(E_n)^2}{mc^2}\frac{4n}{\ell + 1/2}\delta_{\ell,0}$$

Origin: "Zitterbewegung" - rapid quantum fluctuations of electron position

Combined Fine Structure Formula

Total fine structure correction:

$$E_{fs} = E_r^{(1)} + E_{so}^{(1)} + E_D^{(1)} = \frac{(E_n)^2\alpha^2}{n^2}\left[\frac{n}{j+1/2} - \frac{3}{4}\right]$$

Remarkably, depends only on $n$ and $j$, not on $\ell$ separately!

Energy Levels with Fine Structure

$$E_{n,j} = -\frac{13.6\text{ eV}}{n^2}\left[1 + \frac{\alpha^2}{n^2}\left(\frac{n}{j+1/2} - \frac{3}{4}\right)\right]$$

Spectroscopic notation: $n^{2s+1}L_j$

  • $n = 2, \ell = 0, j = 1/2$: $2^2S_{1/2}$
  • $n = 2, \ell = 1, j = 1/2$: $2^2P_{1/2}$
  • $n = 2, \ell = 1, j = 3/2$: $2^2P_{3/2}$

Example: n=2 Level Splitting

Without fine structure: all 4 states at same energy

With fine structure:

  • $2^2S_{1/2}$ and $2^2P_{1/2}$: degenerate (same $j = 1/2$)
  • $2^2P_{3/2}$: split by $\Delta E \approx 4.5 \times 10^{-5}$ eV

This is the sodium D-line doublet splitting!

Lamb Shift

Fine structure predicts $2^2S_{1/2}$ = $2^2P_{1/2}$, but experiment shows tiny splitting:

$$\Delta E_{Lamb} \approx 1057 \text{ MHz} \approx 4.4 \times 10^{-6} \text{ eV}$$

Requires quantum electrodynamics (QED) - vacuum fluctuations of EM field

Hyperfine Structure

Even smaller splitting from nuclear spin $\vec{I}$ coupling to electron:

$$\hat{H}_{hf} \propto \vec{I}\cdot\vec{J}$$

Famous example: 21 cm line of hydrogen ($F = 1 \leftrightarrow F = 0$ transition)

Critical for radio astronomy and mapping galactic structure

← Degenerate PerturbationTime-Dependent Perturbation →