University Quantum Mechanics II · The Fine Structure of Hydrogen and the Lamb Shift · 8.1
The hierarchy of corrections set by α
Before this week computes a single fine-structure matrix element, we size them all. Hydrogen contains one pure number, α ≈ 1/137, and every correction is mc² times a power of it. Read off the power and you can rank relativity, radiative corrections and nuclear magnetism at a glance, and see where that ranking stops being trustworthy.
Build the model
Connect the measurement to the mechanism.
Strip hydrogen to its ingredients and only m, e²/(4πε₀), ħ and c remain, with the proton's mass in reserve. They make exactly one pure number, α = e²/(4πε₀ħc) ≈ 1/137, so every energy of the atom is mc² times a function of α, and the power of α tells you how big a correction is before you know what it does. Binding is the first rung: Eₙ = −α²mc²/(2n²), and the virial theorem puts the electron's speed at αc/n.
Relativity therefore corrects the Bohr energies at relative order (v/c)² ~ α²: the relativistic kinetic term, spin–orbit coupling and the Darwin term all land at α⁴mc² = 1.45 meV, divided by an n³ that records how they concentrate near the nucleus. Letting the electron emit and reabsorb photons of the quantised field costs one more α and a logarithm of the range of photon energies involved, which puts the Lamb shift at α⁵mc² ln(1/α). The proton's magnetism brings a second, independent small number, mₑ/mₚ = 1/1836, and puts hyperfine structure at (mₑ/mₚ)α⁴mc².
At n = 2 the rungs are 3.40 eV, 45 μeV, 4.4 μeV and 0.73 μeV. The ladder is dimensional analysis, and it stops where dimensional analysis stops: it drops coefficients that range from 1/4 to 7, so rungs closer than a decade cannot be ordered by it; it says nothing about signs or about which states move; and it fails outright once Zα or mₑ/M is no longer small, in the innermost shell of uranium or in positronium.
- Simple definition
- The fine-structure hierarchy is the ordering of hydrogen's energy corrections by their powers of the dimensionless coupling α = e²/(4πε₀ħc) ≈ 1/137 and of the mass ratio mₑ/mₚ, each rung being mc² times a higher power and therefore a smaller effect.
- Example
- Hydrogen, n = 2: binding 3.40 eV (α²mc²/8), 2P fine-structure doublet 45.4 μeV (order α⁴mc²), Lamb shift 4.37 μeV (order α⁵mc² ln(1/α)), 2s hyperfine splitting 0.734 μeV (order (mₑ/mₚ)α⁴mc²). Each rung is 6 to 75 000 times below the one above it.
The only pure number in hydrogen's Hamiltonian, so every energy of the atom is mc² times a function of α, and of mₑ/mₚ once the nucleus carries a moment.
e in C, ε₀ in F m⁻¹, ħ in J s, c in m s⁻¹; α has no unit. Quoted from CODATA 2022, measured to about 1 part in 10¹⁰
The first rung: binding is (Zα)²/(2n²) of the rest energy, 6.7 × 10⁻⁶ of it at n = 2, which is why a non-relativistic equation works at all.
Eₙ in eV, mc² = 510 998.95 eV. Electron mass: E₁ = −13.606 eV, E₂ = −3.401 eV; the reduced mass moves hydrogen's E₁ to −13.598 eV
Relativistic corrections are a series in (v/c)² = (Zα/n)², so the next rung sits a factor of order (Zα)² below the binding.
v in m s⁻¹, from ⟨T⟩ = −Eₙ. Hydrogen 1s: 2.188 × 10⁶ m s⁻¹, so v/c = α
Kinetic, spin–orbit and Darwin terms all land here, at (Zα)²/n of the binding. Power counting cannot rank them, which is why each is computed this week.
Hydrogen: α⁴mc² = 1.449 meV. The n = 2 doublet this unit derives is α⁴mc²/32 = 45.3 μeV, 10.95 GHz
One more α for emitting and reabsorbing a photon. Photons of every energy between about α²mc² and mc² contribute, giving ln(1/α²) = 9.84; the power count writes ln(1/α) = 4.92 and leaves the 2 in the coefficient.
Mainly s states. Hydrogen: α⁵mc² ln(1/α) = 52.0 μeV, so 6.5 μeV at n = 2; measured 4.37 μeV (1057.8 MHz)
A second small number, independent of α: the nuclear magneton is mₑ/mₚ of the Bohr magneton. With the quoted (4/3)gₚ = 7.45, hydrogen 2s gives 0.735 μeV, against the measured 0.734 μeV.
mₑ/mₚ = 1/1836.15, nuclear moment taken as one nuclear magneton. Hydrogen: (mₑ/mₚ)α⁴mc² = 0.789 μeV
Hydrogen has one pure number, and it is α
List what hydrogen's energy can depend on: the electron mass m, the Coulomb strength e²/(4πε₀), ħ, and c once relativity is admitted. One combination, mc², carries the energy; whatever multiplies it must be dimensionless, and there is exactly one candidate. e²/(4πε₀) is 2.307 × 10⁻²⁸ J m and ħc is 3.162 × 10⁻²⁶ J m, so α = e²/(4πε₀ħc) = 7.297 × 10⁻³, about 1/137. Every energy of the atom is therefore mc² times a function of α, and the proton adds the ratio mₑ/mₚ, plus order-one numbers such as its g-factor. The value is quoted, not derived here: it is measured to about 1 part in 10¹⁰ by atom-recoil interferometry and from the electron's magnetic moment, independently of any hydrogen line. Its meaning is a comparison. At any separation r the Coulomb energy e²/(4πε₀r) is α times ħc/r, the energy of a photon whose reduced wavelength is r, so electromagnetism is weak and each extra exchange with the field costs about 1/137. The same number orders the lengths: the Bohr radius a = 52.9 pm, ħ/mc = 386 fm = αa, and the classical electron radius 2.82 fm = α²a.
Binding sits α² below the rest energy
Quantum Mechanics I and Unit 2 derived the Bohr energies, quoted here in one line: Eₙ = −(Zα)²mc²/(2n²). With mc² = 511.0 keV that is a binding of 13.61 eV for hydrogen's n = 1 and 3.401 eV for n = 2, α²/8 = 6.7 × 10⁻⁶ of the rest energy at n = 2. The virial theorem for a 1/r potential, ⟨T⟩ = −Eₙ, turns this into a speed: ½m⟨v²⟩ = (Zα)²mc²/(2n²), so vᵣₘₛ = Zαc/n, 2.19 × 10⁶ m s⁻¹ in hydrogen's ground state. That small v/c licenses the Schrödinger equation, and it also says how relativity returns. Expand the kinetic energy: √(p²c² + m²c⁴) − mc² = p²/2m − p⁴/(8m³c²) + …, and the second term is ¼(v/c)² of the first. So the first relativistic correction is order α² times the binding, α² × α²mc² = α⁴mc² = 1.449 meV for hydrogen. Limiting case: α²mc² = m(e²/4πε₀ħ)² contains no c, while α⁴mc² falls as 1/c² and vanishes in the non-relativistic limit, as a relativistic correction must.
Three different effects share one rung
Power counting sizes the other relativistic effects too, and finds them on the same rung. In its rest frame the electron sees the moving proton's magnetic field B = −v × E/c². At r = a with v = αc, the spin moment μB = eħ/2m meets an energy μB B ≈ (α²/2)ħ²/(ma²) = α⁴mc²/2, 0.72 meV. The Darwin term, quoted from relativistic theory, smears the Coulomb potential over ħ/mc = αa, and that too gives order α⁴mc². So the kinetic, spin–orbit and Darwin corrections are the same size: power counting cannot rank them or say which state each moves, which is why the next five topics compute each one. What it does fix is the dependence on n. The shifts are set by ⟨1/r²⟩, ⟨1/r³⟩ and |ψ(0)|², which all fall as 1/n³ because they sample the region near the nucleus, where the electron moves fastest. The fine-structure scale is (Zα)⁴mc²/n³, of order (Zα)²/n of the binding. At n = 2 the doublet this unit derives is α⁴mc²/32 = 45.3 μeV, or 10.95 GHz; the measured value is 10.969 GHz.
One more α from the quantised field
So far the field has been a fixed Coulomb potential. Quantise it and the electron can emit and reabsorb photons. Each such loop brings the charge in twice, once at emission and once at absorption, so it costs one factor of α. Welton's picture, worked out later this week, supplies the mechanism: zero-point fluctuations jiggle the electron over a mean-square distance of order (α/π)(ħ/mc)² times a logarithm, where the Darwin term smeared it over (ħ/mc)². One more α, then: the Lamb shift is of order α⁵mc² = 10.6 μeV. The logarithm counts the photons that contribute, whose energies run from atomic values near α²mc² up to mc², so it is ln(1/α²) = 2 ln(1/α) = 9.84 up to factors of order one. The power count keeps α⁵mc² ln(1/α) = 52 μeV, falling as 1/n³ and acting mainly on s states. For n = 2 that is 6.5 μeV. The measured 2S₁/₂–2P₁/₂ splitting is 4.37 μeV, or 1057.8 MHz, and its accurate value is quoted from quantum electrodynamics. The logarithm is much of why the Lamb shift trails the n = 2 doublet by a factor of 10 rather than 137: it cuts 137 to 28, and the coefficients, 1/4 for the doublet and 0.67 for the Lamb shift, do the rest.
A second small number: the mass ratio
Hyperfine structure couples the electron to the proton's magnetic moment, and a new ratio enters. A magneton is charge × ħ over twice the mass, so the proton's natural unit, the nuclear magneton eħ/2mₚ, is mₑ/mₚ = 1/1836 of the Bohr magneton. Replace the orbital field that drove spin–orbit coupling with the proton's dipole field and the energy drops by that ratio: the hyperfine scale is (mₑ/mₚ)α⁴mc² = 0.789 μeV, or (mₑ/mₚ)α(Zα)³mc²/n³ for an ion. The mass ratio is fixed by the strong interaction, not by α, and numerically it is about α1.5, so hyperfine structure behaves like α5.5 mc² and lands below the Lamb shift. Its coefficient is quoted from a later unit: (4/3)gₚ = 7.45 for s states, with the measured proton g-factor gₚ = 5.586. That gives 5.88 μeV for 1s, against the measured 5.874 μeV (1420.4 MHz, the 21 cm line), and 0.735 μeV for 2s, against 0.734 μeV (177.6 MHz). Limiting cases: as mₚ → ∞ the nuclear moment and the splitting vanish, and in positronium, where M = m, the splitting rises to 0.84 meV, level with fine structure.
What the ladder does not tell you
Set the power counts beside hydrogen's measured n = 2 values: binding 3.40 eV, doublet 45.4 μeV, Lamb shift 4.37 μeV, 2s hyperfine 0.734 μeV. The bare counts at n = 2, 6.80 eV, 181 μeV, 6.50 μeV and 0.0986 μeV, miss them by coefficients of 0.50, 0.25, 0.67 and 7.4. A spread of one and a half decades is harmless where rungs are four or five decades apart and decisive where they are not: power counting puts the Lamb shift 66 times above 2s hyperfine structure, and the atom puts it 6 times above. The ladder is blind to direction and to states. Fine structure lowers every n = 2 level yet leaves 2S₁/₂ and 2P₁/₂ together, because it depends on n and j alone; the Lamb shift then lifts the s level; hyperfine structure needs a nuclear spin, and ⁴He⁺ has none. It also fails once its small numbers are not small. For U⁹¹⁺, Zα = 0.67, and the 1s binding quoted from the Dirac equation, 132.3 keV, exceeds the Bohr value by 17.1 keV, 13% of the total, so the series in (Zα)² converges slowly and every order matters.
Change one variable at a time
Make the relationship visible.
At Z = 1, n = 2 the gaps are 75 000, 38 000, 28 and 1836. Take Z to 92 and n to 1: Bohr crowds mc² and fine structure closes to within a factor 2.2 of Bohr, yet Lamb never catches fine structure, since its extra α is one Z cannot supply. The gap even widens from 28 to 344, but only because ln(1/Zα) shrinks to 0.40, where a leading logarithm no longer means anything. Changing n moves the three corrections together.
mc² ÷ BOHR BAR75115.5
BOHR ÷ FINE BAR37557.7
FINE ÷ LAMB BAR27.9
FINE ÷ HYPERFINE BAR1836
Live interpretationmc² ÷ BOHR BAR: 75115.5. BOHR ÷ FINE BAR: 37557.7. FINE ÷ LAMB BAR: 27.9. FINE ÷ HYPERFINE BAR: 1836
Catch the common trap
Explain before calculating.
In hydrogen the n = 2 fine-structure doublet is 1.3 × 10⁻⁵ of the n = 2 binding energy. Estimate the same ratio for the hydrogen-like ion Ne⁹⁺, with Z = 10.
Choose an answer to test the model.
Practice & worked examples
Reason from the model, then test the result.
EasyUsing e = 1.602177 × 10⁻¹⁹ C, ε₀ = 8.854188 × 10⁻¹² F m⁻¹, ħ = 1.054572 × 10⁻³⁴ J s and c = 2.997925 × 10⁸ m s⁻¹, evaluate α and confirm that it has no unit. Then find the rms speed of the electron in hydrogen's n = 3 level and use (v/c)² to estimate the energy scale of its relativistic corrections.
- Coulomb strength: e²/(4πε₀) = (1.602177 × 10⁻¹⁹ C)² ÷ (1.112650 × 10⁻¹⁰ F m⁻¹) = 2.307079 × 10⁻²⁸ J m, since C² per F m⁻¹ is C V m, which is J m.
- Quantum and relativistic scale: ħc = 1.054572 × 10⁻³⁴ J s × 2.997925 × 10⁸ m s⁻¹ = 3.161528 × 10⁻²⁶ J m. A ratio of two quantities in J m has no unit: α = 2.307079 × 10⁻²⁸ ÷ 3.161528 × 10⁻²⁶ = 7.29735 × 10⁻³ = 1/137.036.
- Speed, from the virial theorem: v₃ = αc/3 = 7.29735 × 10⁻³ × 2.997925 × 10⁸ m s⁻¹ ÷ 3 = 7.292 × 10⁵ m s⁻¹, a third of the ground-state speed.
- Then (v/c)² = (α/3)² = 5.917 × 10⁻⁶. The binding is |E₃| = 13.606 eV ÷ 9 = 1.5118 eV, so the relativistic scale is (v/c)²|E₃| = 5.917 × 10⁻⁶ × 1.5118 eV = 8.94 μeV.
- Check: this is α⁴mc²/(2n⁴) = 1449.0 μeV ÷ 162, order α⁴ as it should be. The n = 3 fine-structure spread derived later this week is 17.9 μeV, so the estimate has the right power of α and lands within a factor of 2, which is all power counting promises.
Answerα = 7.297 × 10⁻³ ≈ 1/137.04, a pure number; v = 7.29 × 10⁵ m s⁻¹; (v/c)² = 5.9 × 10⁻⁶, so the n = 3 relativistic corrections are of order 9 μeV (the derived spread is 17.9 μeV).
MediumHydrogen's n = 2 level has a measured 2P doublet of 10.969 GHz and a Lamb shift of 1057.8 MHz. Scale each rung to the ion ⁴He⁺ (Z = 2): predict its n = 2 binding, its doublet and its Lamb shift, the last with the logarithm ln(1/Zα) as well as the powers. Compare with the measured He⁺ Lamb shift of 14.04 GHz, and say what hyperfine structure ⁴He⁺ shows.
- Binding goes as (Zα)²: 3.401 eV × 4 = 13.61 eV.
- Fine structure goes as (Zα)⁴: 10.969 GHz × 16 = 175.5 GHz, or 0.726 meV. Relative to the binding that is 5.33 × 10⁻⁵, four times hydrogen's 1.33 × 10⁻⁵, the factor (Zα)² that the ladder predicts.
- The Lamb shift goes as α(Zα)⁴ ln(1/Zα). The powers give Z⁴ = 16, and the logarithm falls from ln 137.04 = 4.920 to ln 68.52 = 4.227, a factor 0.8591. So the estimate is 1057.8 MHz × 16 × 0.8591 = 14.54 GHz.
- Against the measured 14.04 GHz that is 3.6% high, while bare Z⁴ scaling, 16.93 GHz, would be 21% high, so the logarithm is not decoration. The doublet is 175.5 ÷ 14.04 = 12.5 times the Lamb shift, close to hydrogen's 10.4: the extra α keeps the Lamb shift a rung lower at every Z.
- Hyperfine: the ⁴He nucleus has spin 0 and no magnetic moment, so ⁴He⁺ has no hyperfine structure at all. The ladder gives the size such a term would have, (mₑ/mₚ)α(Zα)³mc²/n³, not whether its operator exists; ³He⁺, whose nucleus has spin ½, does split, by a measured 8.67 GHz in its ground state.
AnswerBinding 13.61 eV; 2P doublet about 175.5 GHz (0.726 meV); Lamb shift about 14.5 GHz against the measured 14.04 GHz; no hyperfine structure, because the ⁴He nucleus has no spin.
HardThe measured ground-state hyperfine splittings of hydrogen, muonium (μ⁺e⁻) and positronium (e⁺e⁻) are 1420.41 MHz, 4463.30 MHz and 203.39 GHz, and their positive particles are 1836.15, 206.77 and 1 electron masses. For each, compare the splitting with the fine-structure scale α⁴μc², where μ is the reduced mass, and test the power-count prediction that the ratio is of order mₑ/M.
- Reduced masses: μ/m = 1/(1 + m/M) is 0.99946, 0.99519 and 0.5, so α⁴μc² = 1449.04 μeV × μ/m = 1448.25, 1442.07 and 724.52 μeV.
- Splittings as energies, E = hν with h = 4.135668 × 10⁻¹⁵ eV s: 1420.41 MHz gives 5.874 μeV, 4463.30 MHz gives 18.459 μeV, and 203.39 GHz gives 841.15 μeV.
- Ratios to the fine-structure scale: 5.874 ÷ 1448.25 = 4.06 × 10⁻³; 18.459 ÷ 1442.07 = 1.28 × 10⁻²; 841.15 ÷ 724.52 = 1.16.
- Divide by mₑ/M = 5.446 × 10⁻⁴, 4.836 × 10⁻³ and 1: the leftover coefficients are 7.45, 2.65 and 1.16. The ratios span more than three decades and mₑ/M tracks them; what remains is of order one, (4/3)g with the quoted gₚ = 5.586 giving 7.45 and gμ = 2.002 giving 2.67.
- In positronium the mass ratio is 1, so nothing separates hyperfine from fine structure: the 0.84 meV splitting exceeds α⁴μc² itself. Its value quoted from quantum electrodynamics, (7/12)α⁴mc² = 845 μeV at leading order, includes a virtual-annihilation term with no hydrogen analogue. Here the two effects must be computed together.
AnswerSplitting ÷ α⁴μc² = 4.06 × 10⁻³ for hydrogen, 1.28 × 10⁻² for muonium and 1.16 for positronium; dividing by mₑ/M leaves 7.45, 2.65 and 1.16, all of order one. The mass ratio sets the rung, and in positronium hyperfine and fine structure are the same size.