Covariant form of the Dirac equation A μ = (A, iφ), x μ = (r, ict) and p μ = -iħ∂/∂x μ = (-iħ▼, -(ħ/c) ∂ /∂t) = (p, iE/c) and γ μ = (-iβα, β) Definition of the Dirac Matrices Then H Ψ = i ħ∂Ψ/∂t Equation (1) where H = βmc 2 + α∙ (cp + e A) By substitution: [ (p μ + e A μ /c) ∙ γ μ - imc] Ψ = 0 let π μ = p μ +(e/c) A μ Definition of “Dirac Momentum” then [π μ γ μ - imc] Ψ = 0 Equation (2) This is covariant! It can be reduced to the Klein-Gordon form by multiplying by [π μ γ μ + imc]

Reduction to Klein-Gordon form (1) Thus [π μ γ μ + imc] [π μ γ μ - imc] Ψ = 0 multiplying out => [ Σ π 2 μ + m 2 c 2 + Σ´ π μ γ μ π ν γ ν ] Ψ = 0 But π μ γ μ π ν γ ν + π ν γ ν π μ γ μ = γ μ γ ν π μ π ν + γ ν γ μ π ν π μ = γ μ γ ν ( π μ π ν - π ν π μ ) Equation (3) But π μ π ν = (p μ + e A μ /c) (p ν + e A ν /c) = p μ p ν +(e 2 /c 2 ) A μ A ν +(e/c)[p μ A ν + A μ p ν ]

Reduction to Klein-Gordon form (2) Substituting in the last term => (e/c) [ …] = (e/c)[ (-iħ) (∂ /∂μ )(A ν ) + A μ (-iħ) ∂ /∂ ν ] = (-iħe/c) ∙ A ν ∙ ∂ /∂μ + (-iħe/c) ∙ A μ ∙ ∂ /∂ν + (-iħe/c) ∙ ∂ (A ν )/∂μ Only the last term survives the commutator {Eq (3)} And we are left with the electromagnetic field tensor: F μν = ∂A ν /∂x μ - ∂A μ /∂x ν Thus [ Σ (π μ 2 + m 2 c 2 ) + ħe/ic) Σ γ μ γ ν F μν ] Ψ = 0 Represents a charged particle in an electromagnetic field

Reduction to an exact non- covariant equation (a)First term π μ 2 = {(p μ + (e/c) A μ } 2 = p μ 2 + (e 2 /c 2 )A μ 2 + (e/c) (A μ p μ + p μ A μ ) =p 2 – (E 2 /c 2 ) + e 2 A 2 /c 2 + (e 2 /c 2 )(-φ 2 ) + (2e/c) A∙ p + (2e/c)(iE/c)(i φ) Substituting operators into the first term and dividing by -2m, & including m 2 term, & collecting terms, gives => (E-mc 2 ) + eφ + (ħ 2 /2m)∙▼ 2 + 1/(2mc 2 ) (E-mc 2 + e φ )2 +ieħ/(mc) ∙ A∙▼ - e 2 A 2 /(2mc 2 )

The 4-component Pauli terms are σ i = (τ i 0 ) and α k = (0 τ k ) ( 0 τ i ) (τ k 0 ) (b) 2 nd term Σ γ μ γ ν F μν ∙{ ħe/(2ic)} Spacelike terms:γ j γ j = iσ i Space-time terms:γ k γ 4 = iα k Note: the electromagnetic tensor terms are: spacelike H i = F jk ; and the time-like F i4 = (1/i) E i Substituting, and dividing by (-2m) => (-eħ/2mc) σ∙ H + ieħ/(2mc) ∙α∙ E Noting that the Bohr magneton is μ 0 = eħ/(2mc), and W= (E-mc 2 ) the whole equation becomes: [ W + eφ + {ħ 2 /(2m)}▼ 2 + {1/(2mc 2 )}(W + eφ) 2 +{ieħ/(mc)}∙A∙▼ -e 2 A 2 /(2mc 2 ) -μ 0 ∙σ∙ H + iμ 0 ∙α∙ E ] Ψ = 0

Pauli Approximation 2 coupled Equations for U A & U B U A & U B are each 2 component wavefunctions τ·(cp +eA) U B + (E 0 – eφ) U A =EU A τ·(cp +eA) U A - (E 0 + eφ) U B = EU B The Pauli approximation leads to a similar non-covariant equation, but with only reduced 2-component matrices… Assume that W= E – mc 2 is << mc 2 define E 0 = mc 2 or | E – E 0 | ≤ E 0 then <<mc Letting α = ( 0 τ ) and β = ( 1 0 ) and ( U A ) = ( ψ 1 ) ( τ 0 ) ( 0 1 ) ( U B ) ( ψ 2 ) ( ψ 3 ) ( ψ 4 ) We can write the Dirac equation [α∙(cp + eA) + βE 0 - eφ] Ψ = EΨ which becomes the 4-component equation: [ ( 0 τ ) ∙ (cp + eA) + ( 1 0 ) E 0 -e φ ] ( U A ) = E ( U A ) [ ( τ 0 ) ( 0 -1) ] ( U B ) = ( U B ) giving

This is exactly the Schrodinger equation for hydrogen – Hence Ψ 1 = Ψ 2, (same equation), no cross terms) and U A is the hydrogen wavefunction. Then do an iterative process…. calculate a new U B and substitute again for a new U B equation, solve etc… BUT, there is an easier way… Go back to the exact non-covariant solutions – the only term which couples the first 2 components (U A ) with the second 2 components is the last term (ieħ/2mc) ( α E ) Ψ In the second equation, we can make another approximation: E 0 + E >> eφ Hence, E 0 + E + e φ ≈ 2E 0 = 2mc 2 Then U B ≈ {τ /(2mc)} ∙ ( p +eA/c) U A Now substitute in the first equation to give: [ E-E 0 + e φ -(1/2m) (p + eA/c) 2 ] U A = 0

Ψ Use the approximation for U B (αU) A = τU B ≈ {1/(2mc)} [p + i(p x τ)] U A most easily justified by looking at component by component. Continuing to the Pauli approximation Now the last term can be rewritten without any cross linkage between U A and U B : [ ieħ/2mc αE ] Ψ => [ {iμ 0 /(2mc)} E p + {μ 0 /(2mc)} τ ( E x p ) ] Ψ which is just an equation for ( ψ 1 ) = U A ( ψ 2 ) The E p term has no classical analog, and is just the Darwin term The E x p term is the Lorentz force on the moving electron

Now solve for a central potential

Calculating the small terms… 3/4n)

Adding the small terms together

Angular momenta & normalization

Exact solution for the Coulomb potential case

The Dirac energy Expanding this equation in powers of (αZ) 2 yields the Pauli energy as the first 2 terms….

On the aZ expansion for the Dirac energy of a one-electron atom L J Curtis, Department of Physics, University of Lund, S Lund, Sweden J. Phys. B10, L641 (1977) Abstract. A procedure for directly prescribing a term of arbitrary order in an CLZ-expansion of the Dirac energy of a one-electron atom is presented, and utilised to obtain higher-order corrections to the Dirac fine-structure formula. These can then be combined with terms not included in the Dirac formalism and applied, for example, to semi-empirical charge- screening parametrisations of multi-electron atoms.

Expanding the 4 terms using the binomial coefficients The table gives C PQ for each P and Q; The row Norm is the common denominator for each column. P=0 gives C PQ =1, the rest energy mc 2. P=1 is the Balmer energy.

The n=2 levels in hydrogen

Radiative corrections: the Lamb shift Two terms – both calculated to infinite order 1.the photon self-energy term 2.The vacuum polarization term Removes the degeneracy between states of the same J, but different L angular momenta.

QED - theory - short history See your QM field theory text for more details…. First calculation – Bethe - 1 photon emission & absorption within a few percent of experiment. Precision… Feynman, Schwinger, Tomonaga – 1950s – developed calculational techniques for the higher-order diagrams – often by numerical integration Mohr - developed analytical theories to include all order diagrams of the 1- photon exchange and the pair production, plus parts of multiphoton exchange and multiple pair production… Sapirstein – continuing calculations in muonic systems, and higher-order terms – an active program. ΔE(Lamb) ~ Z 4 α 5 mc 2. F(Zα)/π where F(Zα) = A 40 + A 41 ln(Zα) -2 + A 50 + A 60 (Zα) 2 +…{higher powers of Zα}

QED - Experimental work – short history Late 1930’s – spectroscopy: Several measurements of Balmer-α suggested that the 2s 1/2 and 2p 1/2 levels had a different energies (10-30% precision) 1947 – microwaves – Lamb & Retherford – measured the 2s-2p difference directly to a few parts in 10,000 – ΔE=1058 MHz s –gradual improvement in H(2s-2p), other measurements in higher Z 1-electron ions – e.g. in hydrogen: Lundeen & students (at Harvard & Notre Dame) - in hi-Z ions –use of accelerators – up to chlorine (Z=17) 1970s onwards – lasers – measurements of Lyman α – 1s-2p, later 1s-2s

The Munich group’s precision measurements The Hydrogen Spectrometer Hydrogen atoms are excited by longitudinal Doppler-free two photon excitation at 243 nm from a frequency doubled ultrastable dye laser at 486 nm. The UV radiation is then resonantly enhanced in a linear cavity inside a vacuum chamber. A small electric field that mixes the metastable 2S state (lifetime 1/7 sec) with the fast decaying 2P state is applied. We set an upper limit on the second order Doppler-shift below 1 kHz. dye laser vacuum chamber Setup for exciting the 1s-2s transition Results: 1.Test of QED 2. proton radius 3.Variation of fundamental constants Phys. Rev. Lett. 92, (2004)