University Quantum Field Theory I · The Lamb Shift Completed at Order α (Z α)⁴ · 15.8
The 2S(1/2)–2P(1/2) splitting assembled
Every piece of hydrogen's n = 2 Lamb shift is now on the table, derived in this course, derived in Advanced QM or quoted. Here we add them with their signs and pedigrees, find that 2P(1/2) moves too, and set the total beside the measured 1057.8 MHz and beside Bethe's 1947 estimate, whose closeness turns out to rest on errors that cancel.
Build the model
Connect the measurement to the mechanism.
Dirac's equation leaves 2S(1/2) and 2P(1/2) degenerate to all orders in Zα, so at order α(Zα)⁴m the whole splitting is radiative, and by now every radiative piece is in hand. The model is one electron bound to a static point nucleus, corrected once by its own virtual photon and once by a vacuum-polarisation loop, each piece a pure number times U = 4α(Zα)⁴m/(3πn³), 135.6 MHz at n = 2. On 2s the self-energy, joined across a photon energy K so that K cancels, gives ln(m/(2K₀)) + 11/24; the anomalous moment's contact term adds 3/8 and the Uehling term removes 1/5, a bracket of 7.662 and a rise of 1039.0 MHz.
On 2P(1/2) every contact term vanishes with ψ(0), leaving the anomalous moment's spin-orbit share, −1/8, and a small Bethe logarithm, +0.030: the level falls 12.9 MHz. The splitting, 1051.9 MHz, sits 5.9 MHz (0.56 per cent) below the measured 1057.8 MHz, about what a term one power of Zα higher should leave. The same ledger shows why Bethe's cutoff at m landed close: it overstates 2s by 8.1 MHz while ignoring a 12.9 MHz fall of 2P(1/2).
The model stops at relative order Zα and at an infinitely heavy point nucleus; recoil, the proton's size, α(Zα)⁵ and two loops come next, and for heavy ions, where the window (Zα)²m ≪ K ≪ m closes, adding pieces fails altogether.
- Simple definition
- At order α(Zα)⁴, hydrogen's Lamb shift is the radiative shift of 2S(1/2) minus that of 2P(1/2), each a sum of self-energy, anomalous-moment and vacuum-polarisation shares in units of 4α(Zα)⁴m/(3πn³).
- Example
- 2s rises 7.662 × 135.6 MHz = 1039.0 MHz and 2P(1/2) falls 0.095 × 135.6 MHz = 12.9 MHz, so the levels Dirac left degenerate split by 1051.9 MHz, 5.9 MHz below the measured 1057.8 MHz.
Joined self-energy, anomalous moment and Uehling term in one bracket: 7.662 U = 1039.0 MHz, with no cutoff K and no photon mass left in it.
U = 4α(Zα)⁴m/(3πn³), an energy, 135.6 MHz × h for n = 2; m = 0.511 MeV; K₀ = 226.4 eV for 2s, quoted
−0.125 + 0.030 = −0.095 U = −12.9 MHz: no contact term and no ln K reach l = 1, and the magnetic share outweighs the Bethe logarithm.
ln(K₀(2p)/(Z²Ry)) = −0.0300, quoted, for every Z; Ry = 13.606 eV; −1/8 is the g − 2 share of spin-orbit, with ⟨L⋅S⟩ = −1
9.8405 − 2.8118 = 7.0287 for hydrogen: the decades from the binding scale up to m, less Bethe's average over the spectrum. It also makes the Z dependence explicit.
2Z²Ry = (Zα)²m; ln(K₀/(Z²Ry)) = 2.8118 for 2s, quoted, and the same for every Z in Bethe's dipole theory
7.757 U = 1051.9 MHz for hydrogen against the measured 1057.8 MHz: 5.9 MHz short, the size of a term one power of Zα higher.
ln k₀ = ln(K₀/(Z²Ry)): 2.8118 for 2s and −0.0300 for 2p, quoted; 91/120 = 19/30 + 1/8
Bethe's 2s rise is 8.1 MHz too high, and the 12.9 MHz fall of 2P(1/2), which he took as zero, leaves the splitting 4.8 MHz above his number.
U ln(m/K₀) = 1047.1 MHz with the modern K₀ (UAQM-13.07); ln 2 − 19/30 = 0.0598
List the 2s pieces with their pedigrees
Natural units ħ = c = 1 and the metric (+, −, −, −) hold throughout; an energy E is read as a frequency E/h with h = 4.136 × 10⁻¹⁵ eV s. Every piece is a pure number times U = 4α(Zα)⁴m/(3πn³), 135.6 MHz at n = 2 (topic 15.01; 135.64 MHz to five figures, which would raise the total by 0.3 MHz). Four pieces act on 2s. Above a photon energy K the free vertex gives ln(m/(2K)) + 11/24, derived in this course (topics 15.02, 15.03 and 15.07). Below K Bethe's dipole sum gives ln(K/K₀), derived in Advanced QM (UAQM-13.04 to 13.07) and quoted, with ln(K₀/Ry) = 2.8118, a sum over the whole spectrum, also quoted. F₂'s contact term adds 3/8: a = α/(2π) is derived in this course (topic 14.03) and its reduction to a level shift quoted from UAQM-06.04. The Uehling term removes 1/5, derived in topic 13.08. Nothing else enters at this order.
Add the 2s pieces: K leaves and 7.662 remains
The self-energy parts add to ln(m/(2K)) + ln(K/K₀) + 11/24 = ln(m/(2K₀)) + 11/24, and K is gone, as it must be from a measured number; the terms dropped are of relative order K/m and (Zα)²m/K, small anywhere inside the window 27 eV ≪ K ≪ 511 keV. Because 2Ry = α²m, the logarithm is ln(1/α²) − ln(K₀/Ry) = 9.8405 − 2.8118 = 7.0287: the decades between the atomic binding scale and m, less Bethe's average excitation energy K₀ = 16.6 Ry = 226.4 eV. The constants give 11/24 + 3/8 − 1/5 = 19/30 = 0.6333. So 2s rises by 7.662 U: self-energy 7.487 U = 1015.2 MHz, anomalous moment +50.9 MHz, vacuum polarisation −27.1 MHz, total 1039.0 MHz, computed. The logarithm carries 92 per cent of it; the three constants together move 2s by 85.9 MHz, more than fourteen times what this order leaves unexplained.
2P(1/2): what survives when ψ(0) = 0
A p state vanishes at the origin, |ψ₂ₚ(0)|² = 0, so all three contact terms, F₁'s slope, F₂'s Darwin-like share and the Uehling term, give nothing at this order. Bethe's sum rule removes ln K as well (topic 15.06), so 2P(1/2) needs no matching. Two finite pieces remain. The anomalous moment multiplies the spin-orbit coefficient by 1 + 2a, and with ⟨1/r³⟩ = (Zαm)³/24 and ⟨L⋅S⟩ = −1 for 2P(1/2) the extra energy is −a(Zα)⁴m/24 = −U/8 = −16.95 MHz, UAQM-13.09's −17 MHz, checked here in one line. The low-energy self-energy leaves the Bethe logarithm itself, −ln(K₀(2p)/Ry) = +0.0300, quoted, worth +4.07 MHz. Net: −0.0950 U = −12.9 MHz. The magnetic piece sets the sign, because a stronger spin-orbit coupling pushes a j = 1/2 level down; the same factor lifts 2P(3/2), with ⟨L⋅S⟩ = +1/2, by U/16.
The ledger, and the measured number
Subtracting, ΔE(2S(1/2)) − ΔE(2P(1/2)) = (7.662 + 0.095)U = 7.757 U = 1051.9 MHz, computed. In megahertz the ledger reads: joined self-energy on 2s +1015.2, derived in this course above K and quoted from Advanced QM below it; anomalous moment +50.9 on 2s and −17.0 on 2P(1/2), with a = α/(2π) derived here and the reduction quoted; vacuum polarisation −27.1 on 2s, derived here; Bethe logarithm +4.1 on 2P(1/2), quoted. A 2P(1/2) entry enters the splitting with its sign reversed. The measured splitting, quoted, is 1057.8 MHz, so the assembly is 5.9 MHz, 0.56 per cent, short. That is the expected size: the first omitted terms are of relative order Zα = 0.0073, and the α(Zα)⁵ term, quoted ahead from topic 15.09, is about +7 MHz. No number in the ledger was adjusted to fit: the photon mass, the scale μ and the cutoff K have all gone.
Where Bethe's 1047 MHz sits
Bethe cut his non-relativistic integral at K = m, took 2P(1/2) as fixed, and compared U ln(m/K₀) with the splitting; with the modern K₀ that is 1047.1 MHz (UAQM-13.07). Since ln(m/K₀) = ln(m/(2K₀)) + ln 2, his estimate is the assembled 2s bracket with ln 2 = 0.6931 in place of 19/30 = 0.6333: 0.0598 U = 8.1 MHz too high. Taken apart, his constant beats the self-energy's 11/24 by 0.2348 U = 31.8 MHz, while the anomalous moment and vacuum polarisation he left out return a net 23.7 MHz. The 12.9 MHz fall of 2P(1/2) then adds to the splitting, so the assembled 1051.9 MHz lies 4.8 MHz above his number. Bethe had the logarithm right, and the logarithm is 92 per cent of the answer; the near-agreement of his constant was luck that only the relativistic pieces could confirm. His own 1947 figure, 1040 MHz, quoted, rested on a rougher K₀.
Checks, and where adding pieces stops
Dimensions: U is α(Zα)⁴ times a mass, an energy of 5.61 × 10⁻⁷ eV, and every bracket is a pure number, logarithms of energy ratios plus rationals. Limits: at α → 0 with Zα fixed every radiative term vanishes and Dirac's degeneracy returns; every share scales as 1/n³ through |ψ(0)|² or ⟨1/r³⟩, so n = 3 moves by about 8/27 as much; and the splitting grows as Z⁴ times a logarithm that shrinks with Z. The assembly rests on three approximations: the electron in the loop is free above K, Bethe's sum is dipole and non-relativistic below K, and the nucleus is an infinitely heavy point. For hydrogen each costs under a per cent. As Zα grows the window (Zα)²m ≪ K ≪ m narrows: it opens at 27 eV in hydrogen and at 109 eV in He⁺, and it has closed by uranium's Zα = 0.67, where the levels must be computed to all orders in Zα rather than piece by piece.
Change one variable at a time
Make the relationship visible.
Drag K from 316 eV to 100 keV: the two self-energy bars trade about 780 MHz and their sum never moves. Then move Bethe's cutoff: near 0.94 m his line meets the foot of the VP bar, the 2s level, and near 1.04 m the top of the 2P bar; K = m falls between.
FREE VERTEX ABOVE K501.6 MHz
BETHE BELOW K513.7 MHz
BETHE CUT AT KB1047.1 MHz
BETHE − ASSEMBLED SPLITTING-4.8 MHz
Live interpretationFREE VERTEX ABOVE K: 501.6 MHz. BETHE BELOW K: 513.7 MHz. BETHE CUT AT KB: 1047.1 MHz. BETHE − ASSEMBLED SPLITTING: −4.8 MHz
Catch the common trap
Explain before calculating.
With the modern K₀, Bethe's non-relativistic formula cut at K = m gives 1047.1 MHz for the 2s shift, while the assembled order-α(Zα)⁴ splitting is 1051.9 MHz. Which account of the 4.8 MHz between them is right?
Choose an answer to test the model.
Practice & worked examples
Reason from the model, then test the result.
EasyThe assembly does not care which n it is applied to. Using ln(K₀/Ry) = 2.7677 for 3s and −0.0382 for 3p, both quoted, find hydrogen's 3S(1/2)−3P(1/2) splitting at order α(Zα)⁴ and compare it with 8/27 of the n = 2 result.
- The unit carries the whole 1/n³: U₃ = 135.6 MHz × 8/27 = 40.18 MHz. The rational shares 11/24, 3/8, −1/5 and −1/8 do not depend on n, because each is a contact or spin-orbit coefficient times |ψ(0)|² or ⟨1/r³⟩, and those supply the 1/n³.
- 3s: ln(1/α²) − 2.7677 + 19/30 = 9.8405 − 2.7677 + 0.6333 = 7.7061, a rise of 7.7061 × 40.18 MHz = 309.6 MHz.
- 3P(1/2): −1/8 + 0.0382 = −0.0868, a fall of 0.0868 × 40.18 MHz = 3.49 MHz.
- Splitting: (7.7061 + 0.0868) × 40.18 MHz = 7.7929 × 40.18 MHz = 313.1 MHz. Scaling the n = 2 result instead gives 1051.9 × 8/27 = 311.7 MHz.
- The 1.4 MHz difference is the Bethe logarithms: K₀ is 15.9 Ry for 3s against 16.6 Ry for 2s, so the 3s logarithm is 0.0441 longer; the 3p logarithm, 0.0082 larger in size, gives a little back, and the bracket grows from 7.757 to 7.793, 0.46 per cent.
Answer313.1 MHz at order α(Zα)⁴, 0.46% above 8/27 of the n = 2 splitting (311.7 MHz), because the 3s Bethe logarithm is 0.044 smaller than the 2s one, slightly offset by the larger 3p logarithm.
MediumBethe stopped his photon integral at K = m as the natural place where his non-relativistic treatment fails. Find the cutoff Keff at which his formula U ln(K/K₀) alone would reproduce (a) the assembled 2s shift and (b) the assembled splitting, in keV, and say how sensitive his estimate is to that choice.
- Set Bethe's U ln(Keff/K₀) equal to U[ln(m/(2K₀)) + 19/30]. K₀ cancels: ln Keff = ln(m/2) + 19/30, so Keff = (m/2)e¹⁹⁄³⁰.
- (a) e¹⁹⁄³⁰ = e(0.6333) = 1.8839, so Keff = 0.9419 m = 481.3 keV.
- (b) The splitting adds 2P(1/2)'s 0.0950 to the bracket: Keff = (m/2)e(0.6333 + 0.0950) = (m/2)e(0.7283) = 1.0358 m = 529.3 keV.
- Sensitivity: the estimate changes by U Δ(ln K), so a 1 per cent change in the cutoff moves it by 135.6 × ln 1.01 = 1.35 MHz. K = m is 6.2 per cent above (a), 8.1 MHz too high for 2s, and 3.5 per cent below (b), 4.8 MHz too low for the splitting.
- Checks: Keff is m times a pure number, as a cutoff must be, and K₀ and Z drop out, because the logarithm Bethe got right is the same in both calculations; only the constant differs, and only the relativistic pieces fix it.
AnswerKeff = 0.942 m = 481 keV reproduces the 2s shift and 1.036 m = 529 keV the splitting; each per cent of cutoff is worth about 1.35 MHz, so K = m is 8.1 MHz high for 2s and 4.8 MHz low for the splitting.
HardHydrogen-like helium, He⁺, has Z = 2. Assemble its 2S(1/2)−2P(1/2) splitting at order α(Zα)⁴, taking the Bethe logarithms ln(K₀/(Z²Ry)) to be hydrogen's, compare it with the measured value of about 14 041 MHz, and compare the shortfall, in units of U, with hydrogen's.
- U scales as Z⁴: U(Z = 2, n = 2) = 16 × 135.6 MHz = 2169.6 MHz. Bethe's dipole sums scale exactly with Z²Ry, so ln k₀(2s) = 2.8118 and ln k₀(2p) = −0.0300 carry over unchanged.
- The logarithm shortens: ln(1/(Zα)²) = 9.8405 − ln 4 = 8.4542. The window now opens at (Zα)²m = 108.8 eV, still more than three decades below m.
- 2s: 8.4542 − 2.8118 + 0.6333 = 6.2757, a rise of 13 616 MHz. 2P(1/2): −0.0950 U, a fall of 206.1 MHz. Splitting: 6.3707 × 2169.6 MHz = 13 822 MHz.
- Against about 14 041 MHz, measured and quoted, the assembly is 219 MHz, or 1.6 per cent, short, against 0.56 per cent for hydrogen.
- In units of U the shortfalls are 5.9/135.6 = 0.044 for hydrogen and 219/2169.6 = 0.101 for He⁺, a ratio of 2.3. A term one power of Zα beyond this order is Z times larger in units of U, so it should roughly double: scaling topic 15.09's quoted α(Zα)⁵ term, about 0.052 U in hydrogen, gives about 0.104 U, some 226 MHz, for He⁺.
AnswerAbout 13 822 MHz (13.82 GHz), 219 MHz or 1.6% below the measured 14 041 MHz; the shortfall per unit U grows from 0.044 to 0.101, roughly doubling, as a missing relative order Zα predicts.