Relativsitic Quantum Mechanics. 3.1 Dirac Equation Summary and notation 3.1. DIRAC EQUATION SUMMARY AND NOTATION. April 22, 2015 Lecture XXXIII
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- Θεόδωρος Ταρσούλη
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1 3.1. DIRAC EQUATION SUMMARY AND NOTATION April, 015 Lctur XXXIII Rlativsitic Quantum Mchanics 3.1 Dirac Equation Summary and notation W found that th two componnt spinors transform according to A = ± σ ξ/ whr ± rfrs to th two indpndnt transformations that ar rlatd by parity, th dirction of th vctor χ is paralll to th vlocity, and ξ is th rapidity, tanh ξ = v. Two distinct transformations produc two distinct spinors that ar quivalnt in th rspt fram. To transform from on to th othr, transform to th rst fram with A + and thn back to th moving fram with A. χ + = σ ξ/ ζ χ = σ ξ/ ζ σ ξ χ + = χ (E σ p = mχ (E + σ p = mχ + χ ± ar ignkts of hlicity with ignvalus λ = ± /. In th ultra-rlativistic limit χ ± ar dcoupld and nvr mix. Hlicity is consrvd. And in th low nrgy limit, χ + = χ. Th coordinat stat rprsntation is (i t + iσ φ +(r, t = mφ (r, t (i t iσ φ (r, t = mφ + (r, t whr φ ± (r, t = d 3 p iet ip r χ ± (p Dfin A = Thn th Dirac quation is writtn φ ( φ+ ( A+ 0 = 0 A ψ 1 ψ ψ 3 ψ 4 ( σ ξ/ 0 0 σ ξ/ ( Σ i σi 0 = 0 σ i ( t + Σi x i + imγ0 ψ = 0 (3.1 ( or i t + iσ m ( φ+ m i t iσ = 0 φ 1
2 3.1. DIRAC EQUATION SUMMARY AND NOTATION Dfin γ 0 = ( 0 I I 0 Thn γ 0 = I and γ 0 Σ i = γ i = ( 0 σi σ i 0 Thn multiply Equation 3.1 from th lft by γ 0 and w hav γ 0 ( t + Σi x i + imγ0 ψ = 0 (γ 0 t + γi x i + imψ = 0 (γ γ i i + imψ = 0 (γ µ µ + imψ = 0 (iγ µ µ mψ = 0 Just as σ transforms as a thr vctor, γ µ = (γ 0, γ transforms lik a four vctor.
3 3.. CURRENT DENSITY 3. Currnt dnsity Go back to coordinat rprsntation Th complx conjugats ar i t φ + = iσ φ + + mφ (3. i t φ = iσ φ + + mφ + (3.3 (3.4 i t φ + = i (φ +σ + mφ (3.5 i t φ = i (φ σ + mφ + (3.6 Now multiply Equations 3. and 3.3 from th right by φ + and φ rspctivly and Equations 3.5 and 3.6 from th right by φ + and φ and add and w hav i t (φ +φ + = i (φ +σφ + + m(φ φ + φ φ + i t (φ φ = i (φ σφ + m(φ + φ φ +φ (3.7 (W hav usd Th continuity quation suggsts and (MΘ α = (M αβ Θ β = M αβ Θ β = Θ βm βα = (Θ M α (σφ α = (φ σ α ρ t + j = 0 ρ = φ +φ + + φ φ j = φ +σφ + φ σφ Not that thr is no mixing of lft and right stats. Mor notation. Also j µ = (ρ, j, µ j µ = 0 ρ = ψ ψ = ψ γ 0 γ 0 ψ = ψγ 0 ψ whr ψ = ψ γ 0 is th Pauli adjoint. Thn ( j = ψ σ 0 ψ = ψ γ 0 γ i ψ = 0 σ ψγψ and Coupling trm is ψγ µ ψa µ j µ = ψγ µ ψ 3
4 3.3. FERMION MAGNETIC MOMENT 3.3 Frmion magntic momnt First w introduc th EM fild by th usual stratgy, p p c A or in coordinat spac i µ i µ A µ. Thn th Dirac quation bcoms In trms of th lft and right handd spinors iγ µ ( µ A µ ψ = mψ [(i 0 V + σ ( i A]φ + = mφ [(i 0 V σ ( i A]φ = mφ + or mor compactly (P 0 σ Pφ + = mφ (P 0 + σ Pφ = mφ + Tak th sum and diffrnc and dfin Ψ = 1 (φ + + φ Φ = 1 (φ + φ Thn P 0 Ψ σ P Φ = m Ψ (3.8 P 0 Φ σ P Ψ = m Φ (3.9 Now dfin Substitution into th abov givs Φ = imt Φ, Ψ = imt Ψ Now that last quation can b rwrittn P 0 Ψ σ PΦ = 0 (3.10 P 0 Φ σ PΨ = Φ (3.11 σ PΨ = ( i 0 + V Φ Φ (3.1 in th non-rlativistic limit. Thn substitution into th nxt to last givs P 0 Ψ 1 σ Pσ PΨ = (P 0 1 (σ P Ψ = 0 4
5 3.3. FERMION MAGNETIC MOMENT which lads us to 0 = (i 0 V 1 (P iσ (P PΨ = (i 0 V 1 (P iσ k ɛ ijk P i P j Ψ = (i 0 V 1 (P i 1 σ kɛ ijk [P i, P j ]Ψ = (i 0 V 1 (P + 1 σ kɛ ijk [ i A j j A i ]Ψ = (i 0 V 1 (P + σ BΨ ( 1 ( i A + Th frmion magntic momnt µ = σ = g s g =. σ B + V Ψ = i 0Ψ 5
6 3.4. FINE STRUCTURE HAMILTONIAN 3.4 Fin Structur Hamiltonian Lt s writ an approximation of th Dirac quation to ordr (v/c 4. W bgin with a pair of coupld quations for th two spinors. In th non-rlativistic limit w solv for Φ in trms of Ψ and thn writ an quation with only Ψ which is th solution to th Schrodingr quation whn v = 0. W ar trying to driv th fin structur hamiltonian in th Schrodingr limit, sinc w will in th nd still rly on prturbation thory and that dpnds on knowing th unprturbd nrgis for H 0 = p + V. Rfring back to quations 3.10 and 3.11, in th non-rlativistic limit, Equation 3.11 bcom Substitution back into 3.11 givs Φ to nxt highr ordr Φ = ( P 0 and thn substituting into 3.10 P 0 Ψ = σ P ( P 0 Φ σ P Ψ (3.13 σ P + σ P Ψ σ P + σ P ( = σ P P 0 σ P 4m + σ P Ψ (3.14 Ψ (3.15 Our goal hr is to driv th Schrodingr quation to ordr (v/c 6. But Ψ is not th sam as ψ. Aftr all ψ d 3 r = d 3 r( Ψ + Φ d 3 r( Ψ + Ψ σ P σ P Ψ Thrfor, in ordr that ψ b proprly normalizd ψ = (1 + 1 ( σ P Ψ and Ψ = (1 1 ( σ P ψ and substitution into 3.15 givs an quation for th Schrodingr wav function P 0 (1 1 ( ( σ P ψ = σ P P 0 σ P 4m + σ P (1 1 ( σ P ψ Lt s xpand and rarrang that last ( P 0 ψ = σ P P 0 σ P 4m + σ P = = (σ P (σ P (1 1 ( σ P ψ + P 0 1 ( σ P ψ (σ P4 16m m ([P 0, (σ P ] + (σ P P 0 σ P 4m ([P 0, σ P] + σ PP 0 ψ (σ P4 16m m ([P 0, (σ P ] σ P 4m ([P 0, σ P] 6 (σ P 8m P 0 ψ
7 3.5. FINE STRUCTURE HAMILTONIAN (SAKURAI S TREATMENT V/C As w ar intrstd in a hydrogn atom, w know that P 0 = i t V and as w ar in an nrgy ignstat i t ψ = Eψ. Also σ P = σ p. (σ p (σ p4 (E V ψ = ( 16m m ([V, (σ p ] σ p (σ p ([V, σ p] 4m 8m (E V ψ = ( (p (p4 16m 3 + 8m ( V σ p (σ p (σ [V, p] 4m 8m (E V ψ = ( (p (p4 16m 3 + 8m ( V + (σ p (p [p, V ] iσ (p [p, V ] 4m 8m (E V ψ = ( (p (p4 16m 3 + 8m ( V + (σ 4m ( p V iσ ( ip V 8m (E V ψ = ( (p (p4 16m 3 + 8m ( V + 4m ( V iσ ( ip 1 dv (σ p r r dr 8m ( p ψ = ( (p (p4 16m 3 8m δ3 (r + 4m (σ L1 dv r dr ( p4 16m 3 ψ = ( (p (p4 8m 3 8m δ3 (r + 4m (σ L1 dv r dr ψ Th scond trm is th rlativistic corrction. Th third, th Darwin trm, and th last, th spin orbit coupling. Not that th factor of two that in th nonrlativistic approach coms from th Thomas prcssion is alrady thr. 3.5 Fin Structur Hamiltonian (Sakurai s tratmnt v/c To rcovr th fin structur hamiltonian w nd to kp trms to nxt ordr in v/c lik w startd to do in Equation?? and w nd to pay attntion to th normalization. Th solution to th Schrodingr must b normalizd and Ψ and Φ ar not, but rathr w should hav that ψ d 3 r = d 3 r( Ψ + Φ = d 3 r( Ψ (1 + p 4m c +... whr w us?? to lowst nonzro ordr. Now dfin so that ψ = ΩΨ = (1 + ψ = Ψ (1 + p 8m c ψ p 4m c Thn multiply?? from th lft by Ω 1 and for simplicity assum that A = 0. To ordr (v/c w hav [ p + V { p 8m c, ( p + V Ω 1 ( cp σ (c (1 + V E nr c cp σ + V Ω 1 ψ = E nr Ω ψ } σ p ( ] Enr V c σ p ψ = E nr (1 p 4m c ψ 7
8 3.6. DARWIN With som manipulation, using V = E and E = 0 w gt [ ] p + V p4 σ (E p 8m 3 c 4m c 8m c E ψ = E nr ψ Using w gt that th fourth trm is E = 1 r dv dr x 1 4m c r dv σ (x p = dr Th last is th Darwin trm. For hydrogn 3.6 Darwin 1 4m c r dv dr σ L = 8m c E = 8m c δ3 (x 1 c r dv dr S L W associat th Darwin trm with th fact that for a rlativistic lctron w cannot localiz it bttr than th compton wavlngth λ = 1/m. Thrfor, th intraction with th Coulomb fild is smard out and bcoms a bit wakr. W can stimat th siz of th ffct in ths trms by first considring th avrag of th Coulomb potntial ovr a small rgion of spac. Finally approximat δr 1/m and V (r = V (r 0 + V r i + 1 V δr i δr j +... r i r i,j i r j V (r 1 6 V (δr = 1 6 δ 3 (r(δr H D = 1 6 m δ3 (r which is prtty clos to what w gt from th Dirac quation. Not that it will only shift th nrgy of l = 0 stats, and it turns out by th sam amount as th contribution from L S whn l = 0 and spin orbit rally cannot b contributing at all. 8
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