Fine structure

The end of the last section planted a landmine: in 1887 Michelson turned the interferometer that would later test the aether onto $\mathrm H\alpha$ (656.3 nm) and found it to be a doublet — two lines squeezed into about 0.016 nm. That was 39 years before the Schrödinger equation; and when the equation delivered $E_n=-13.6/n^2$ eV in 1926, this beautiful answer still could not produce the doublet: the energy depends only on $n$, so $3\to2$ should be a single line.

The same symptom was long familiar elsewhere: sodium’s yellow D line is a doublet at 589.0 and 589.6 nm, resolvable by eye. Spectral lines keep coming in pairs, as if every level were being gently snapped in two by something.

Every force needed to snap them, we have already met in Chapter 5 — all that remains is to settle the accounts.

First, the order of magnitude: why α²

There are three corrections. Before any details, let us get the ruler out.

Order-of-magnitude estimate: how big is v/cbasic~5 minexpand

How fast does the electron move? By the virial theorem (it appeared in the derivation of section 1.4: in a Coulomb field, kinetic energy = magnitude of the binding energy), the ground-state kinetic energy is 13.6 eV, so

$$\frac12\mu v^2=13.6\ \text{eV} \quad\Longrightarrow\quad \frac{v}{c}=\sqrt{\frac{2\times13.6}{511000}}=\frac{1}{137}\tag{6.6.1}$$

The 511000 eV in the denominator is the electron rest energy $\mu c^2$. And this ratio is none other than the fine-structure constant:

$$\alpha\equiv\frac{e^2}{4\pi\varepsilon_0\hbar c}\approx\frac{1}{137.036}\tag{6.6.2}$$

In plain words: $\alpha$ measures the strength of the electromagnetic interaction, and “ground-state electron speed = $\alpha c$” is its most tangible incarnation in the hydrogen atom. In fact the entire Bohr spectrum can be rewritten in terms of it:

$$E_n=-\frac{\alpha^2}{2n^2}\,\mu c^2\tag{6.6.3}$$

— the binding energy is $\alpha^2/2n^2$ of the rest energy.

How big is the relativistic correction? The next order in the kinetic-energy expansion is smaller than the leading one by $(v/c)^2$:

$$\frac{\Delta E_\text{rel}}{E_n}\sim\left(\frac{v}{c}\right)^2=\alpha^2\approx5.3\times10^{-5}\tag{6.6.4}$$

That is of order $10^{-4}$ eV — matching the doublet’s relative splitting $0.016\ \text{nm}/656\ \text{nm}\approx2.4\times10^{-5}$ nicely. The ruler fits: fine structure = Bohr levels × $\alpha^2$. And that is where the name “fine-structure constant” comes from — it was born for precisely these narrow gaps.

The three corrections

First: relativistic kinetic energy. Expand $E=\sqrt{p^2c^2+\mu^2c^4}-\mu c^2$:

$$E=\frac{p^2}{2\mu}-\frac{p^4}{8\mu^3c^2}+\cdots\tag{6.6.5}$$

The second term is the correction Hamiltonian $\hat H_\text{rel}=-\hat p^4/8\mu^3c^2$. It always pushes energies down, and pushes hardest on the low-$l$ states that hug the nucleus and move fast.

Second: spin–orbit coupling. The protagonist of section 5.7. Jump into the electron’s instantaneous rest frame: the proton circles the electron, and that loop of positive charge produces a magnetic field at the centre; the electron carries its own magnetic moment (spin), and a moment in a field has an orientation energy.

How strong is the internal fieldadvanced~6 minexpand

Order of magnitude for a current loop’s field: $B\sim\frac{\mu_0}{4\pi}\frac{ev}{r^2}$. Insert typical 2p numbers, $v\approx\alpha c/2\approx1.1\times10^6$ m/s and $r\approx4a_0\approx2.1\times10^{-10}$ m:

$$B\sim10^{-7}\times\frac{(1.6\times10^{-19})(1.1\times10^6)}{(2.1\times10^{-10})^2}\approx0.4\ \text{T}\tag{6.6.6}$$

The electron builds itself a magnet of a few tenths of a tesla — a hundred times stronger than a fridge magnet, the same league as a hospital MRI. The orientation energy $\sim\mu_B B\approx5.8\times10^{-5}\times0.4\approx2\times10^{-5}$ eV — again landing in the $\alpha^2$ bracket.

The proper form (with the Coulomb field made explicit):

$$\hat H_\text{SO}=\frac{e^2}{8\pi\varepsilon_0\,\mu^2c^2r^3}\,\hat{\vec S}\cdot\hat{\vec L}\tag{6.6.7}$$

There is a famous trap here: the naive “change of frame” derivation comes out too large by a factor of two. The electron’s rest frame is not inertial (it is accelerating), and the frame itself precesses — Thomas precession — contributing exactly the factor $1/2$. The formula above already includes it.

The eigenvalues of $\hat{\vec S}\cdot\hat{\vec L}$ follow at once from Chapter 5’s angular-momentum addition: with total angular momentum $\hat{\vec J}=\hat{\vec L}+\hat{\vec S}$, squaring gives

$$\hat{\vec S}\cdot\hat{\vec L}=\frac{\hbar^2}{2}\left[j(j+1)-l(l+1)-s(s+1)\right]\tag{6.6.8}$$

For $l\ge1$ there are two values $j=l\pm\frac12$, making $\hat{\vec S}\cdot\hat{\vec L}$ once positive and once negative — every level with $l\ge1$ is snapped in two. The mechanism of the doublets, found.

Third: the Darwin term. It acts only on $l=0$ states. Relativistic quantum theory (the Dirac equation of Chapter 12) shows that the electron’s position has an intrinsic “jitter” on the scale of the Compton wavelength $\hbar/\mu c=386\ \text{fm}=\alpha\,a_0$: what the electron actually feels is the Coulomb potential averaged over that little ball. For a $1/r$ potential, such smearing matters only at the origin where the potential is sharpest — so the correction is proportional to $\lvert\psi(0)\rvert^2$, and as the last section noted, only s states are non-zero on the nucleus. The magnitude is again $\alpha^2$.

The splitting diagram: n = 2 as the example

Notation: $n\mathrm L_j$, so $2\mathrm p_{3/2}$ means $n=2$, $l=1$, $j=3/2$.

Level	$j$	Shift relative to the unperturbed level	Degeneracy $2j+1$
$2\mathrm p_{3/2}$	$3/2$	$-1.13\times10^{-5}$ eV	4
$2\mathrm s_{1/2}$ and $2\mathrm p_{1/2}$	$1/2$	$-5.66\times10^{-5}$ eV	2 + 2

Three sentences to read the table:

1. Every level is pushed down (the overall $\alpha^2$ correction is negative);
2. larger $j$ sinks less: $2\mathrm p_{3/2}$ and $2\mathrm p_{1/2}$ are pulled apart by $4.5\times10^{-5}$ eV (about 10.9 GHz) — the main source of the $\mathrm H\alpha$ doublet, exactly what Michelson saw in 1887;
3. $2\mathrm s_{1/2}$ and $2\mathrm p_{1/2}$ remain degenerate (same $j$) — the surviving remnant of the Coulomb potential’s accidental degeneracy.

∎Key formulas

• Fine-structure constant

  $$\alpha=\frac{e^2}{4\pi\varepsilon_0\hbar c}\approx\frac{1}{137.036}$$

  Ground-state electron v/c = α; the dimensionless measure of electromagnetic strength

• The three corrections

  $$\hat H'=-\frac{\hat p^4}{8\mu^3c^2}+\frac{e^2}{8\pi\varepsilon_0\mu^2c^2r^3}\hat{\vec S}\cdot\hat{\vec L}+\hat H_{\text{Darwin}}$$

  Relativistic kinetic + spin–orbit (with the Thomas factor 1/2) + Darwin (s states only)

• Fine-structure levels

  $$E_{nj}=-\frac{13.6\ \text{eV}}{n^2}\left[1+\frac{\alpha^2}{n^2}\left(\frac{n}{j+1/2}-\frac34\right)\right]$$

  Depends only on n and j; 2s_{1/2} and 2p_{1/2} still degenerate (later broken by the Lamb shift)

• Eigenvalues of S·L

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

  j = l ± 1/2, one positive and one negative: every l ≥ 1 level splits in two

• The 21 cm line

  $$\Delta E_{\text{hf}}(1\mathrm s)=5.9\times10^{-6}\ \text{eV}\;\Leftrightarrow\;1420\ \text{MHz}$$

  Hyperfine structure; radio astronomy's premier line for mapping cosmic hydrogen

?Self-check4 questions

1. 1.

   The fine-structure corrections have relative size about α² ≈ 5×10⁻⁵. The most direct physical reason is:

   • In hydrogen the electron's v/c ≈ α, and relativistic effects enter as (v/c)²
   • Planck's constant is very small
   • The proton is much heavier than the electron
   • The speed of light is infinite

   Submit
2. 2.

   Which statements about spin–orbit coupling are correct? (Select all that apply.)

   Select all that apply

   • In the electron's rest frame the proton's circulation produces a field of order 0.1–1 T acting on the electron's spin moment
   • Thomas precession contributes a factor of 1/2; the naive frame-change derivation is too large by a factor of two
   • It splits every l ≥ 1 level in two, by j = l ± 1/2
   • s states (l = 0) have the largest spin–orbit splitting

   Submit
3. 3.

   The fine-structure formula says 2s_{1/2} and 2p_{1/2} are exactly degenerate, yet experiment finds a 1057 MHz interval. Its source is:

   • Vacuum fluctuations of the electromagnetic field perturbing the electron (the Lamb shift), a QED effect
   • The coupling of the proton's spin to the electron's spin
   • Experimental error
   • An applied magnetic field

   Submit
4. 4.

   Use the fine-structure formula to compute the energy interval between 2p_{3/2} and 2p_{1/2}, in units of 10⁻⁵ eV. (Hint: ΔE = (13.6/4)·(α²/4)·[2/1 − 2/2] eV, α = 1/137.04)

   ×10⁻⁵ eV30% relative tolerance

   Submit

What comes next

Fine structure is the splitting the atom carries built-in. In the next section we intervene from outside: apply a magnetic field, or an electric field, and watch the levels crack open on command. This is more than two extra effects — the field strength is a knob, the splitting pattern deforms continuously as you turn it, and it is the best training ground for the perturbative idea of “the bigger term calls the tune” — the final stepping stone to Chapter 7.