8 Foldy-Wouthuysen Transformation

Transcription

1 8 Foldy-Wouthuysn Transformation W now hav th Dirac quation with intractions. For a givn problm w can solv for th spctrum and wavfunctions ignoring th ngativ nrgy solutions for a momnt, for instanc, th hydrogn atom, W can compar th solutions to thos of th schrödingr quation and find out th rlativistic corrctions to th spctrum and th wavfunctions. In fact, th problm of hydrogn atom can b solvd xactly. Howvr, th xact solutions ar problm-spcific and involv unfamiliar spcial functions, hnc thy not vry illuminating. You can find th xact solutions in many txtboos and also in Shultn s nots. Instad, in this sction w will dvlop a systmatic approximation mthod to solv a systm in th non-rlativistic rgim E m m. It corrsponds to ta th approximation w discussd in th prvious sction to highr ordrs in a systmatic way. This allows a physical intrprtation for ach trm in th approximation and tlls us th rlativ importanc of various ffcts. Such a mthod has mor gnral applications for diffrnt problms. In Foldy-Wouthuysn transformation, w loo for a unitary transformation U F rmovs oprators which coupl th larg to th small componnts. Odd oprators off-diagonal in Pauli-Dirac basis: α i, γ i, γ 5, Evn oprators diagonal in Pauli-Dirac basis:, β, Σ, ψ = U F ψ = is ψ, S = hrmitian 7 First considr th cas of a fr particl, H = α p + βm not tim-dpndnt. i ψ t = is Hψ = is H is ψ = H ψ 7 W want to find S such that H contains no odd oprators. W can try is = βα ˆpθ = cos θ + βα ˆp sin θ, whr ˆp = p/ p. 73 H = cos θ + βα ˆp sin θ α p + βm cos θ βα ˆp sin θ = α p + βm cos θ βα ˆp sin θ = α p + βm xp βα ˆpθ = α p cos θ m sin θ + β m cos θ + p sin θ. 74 p To liminat α p trm w choos tan θ = p /m, thn H = β m + p. 75 This is th sam as th first hamilton w trid xcpt for th β factor which also givs ris to ngativ nrgy solutions. In practic, w nd to xpand th hamilton for p m. 6

2 Gnral cas: H = α p A + βm + Φ = βm + O + E, 76 O = α p A, E = Φ, βo = Oβ, βe = Eβ 77 H tim-dpndnt S tim-dpndnt W can only construct S with a non-rlativistic xpansion of th transformd hamilton H in a powr sris in /m. W ll xpand to p4 m 3 and p E, B m. Hψ = i is ψ = is i i ψ t t + t is i [ ψ t = is H i t ψ is ] ψ = H ψ 78 S is xpandd in powrs of /m and is small in th non-rlativistic limit. is H is = H + i[s, H] + i in [S, [S, H]] + + [S, [S, [S, H]]]. 79! n! S = O to th dsird ordr of accuracy m H = H + i[s, H] [S, [S, H]] i [S, [S, [S, H]]] + [S, [S, [S, [S, βm]]]] 6 4 Ṡ i [S, Ṡ] + [S, [S, Ṡ]] 8 6 W will liminat th odd oprators ordr by ordr in /m and rpat until th dsird ordr is rachd. First ordr [O]: To cancl O, w choos S = iβo m, H = βm + E + O + i[s, β]m. 8 i[s, H] = O + β m [O, E] = m βo 8 i i 3 O3 [S, [S, [S, H]]] = 3! 6m 6m 3 βo4 84 [S, [S, H]] = βo m [O, [O, E]] 8m m O3 83 i 4 βo4 [S, [S, [S, [S, H]]]] = 85 4! 4m 3 Ṡ = iβȯ m 86 i i [S, Ṡ] = [O, Ȯ] 87 8m 7

3 Collcting vrything, H = β m + O m O4 8m 3 + E [O, [O, E]] i 8m [O, Ȯ] 88 8m + β O3 [O, E] m 3m + iβȯ m = βm + E + O 89 Now O is O, w can transform H by S to cancl O, m S = iβ m O = iβ β O3 [O, E] m m 3m + iβȯ m 9 Aftr transformation with S, H = is H i is = βm + E + β t m [O, E ] + iβȯ 9 m = βm + E + O, 9 whr O is O, which can b canclld by a third transformation, S = iβo m m H = is H i is = βm + E 93 t = β m + O m O4 + E [O, [O, E]] i [O, Ȯ] 94 8m 3 8m 8m Evaluating th oprator products to th dsird ordr of accuracy, O m [O, E] + iȯ 8m [ ] O, i 8m α E α p A p A = = m m m Σ B 95 = 8m iα Φ iα A = i 8m α E 96 = i [α p, α E] 8m = i α i α j i Ej 8m x i = i,j E + i 8m So, th ffctiv hamiltonial to th dsird ordr is H p A = β m + p4 m 8m 3 i Σ E 8m 8 Σ E + 8m + 4m Σ E p 97 + Φ m βσ B 4m Σ E p 4m Σ E p E 98 8m

4 Th individual trms hav a dirct physical intrprtation. Th first trm in th parnthss is th xpansion of p A + m 99 and p 4 /8m 3 is th lading rlativistic corrctions to th intic nrgy. Th two trms i Σ E 8m 4m Σ E p togthr ar th spin-orbit nrgy. In a sphrically symmtric static potntial, thy ta a vry familar form. In this cas E =, and this trm rducs to Σ E p = r Φ r Σ r p = r Φ Σ L, r H spin orbit = Φ Σ L. 4m r r Th last trm is nown as th Darwin trm. In a coulomb potntial of a nuclus with charg Z, it tas th form E = 8m 8m Z δ3 r = Z 8m δ3 r = Zαπ m δ3 r, 3 so it can only affct th S l = stats whos wavfunctions ar nonzro at th origin. For a Hydrogn-li singl lctron atom, Φ = Z, A =. 4 4πr Th shifts in nrgis of various stats du to ths corrction trms can b computd by taing th xpctation valus of ths trms with th corrsponding wavfunctions. Darwin trm only for S l = stats: Zαπ ψ ns m δ3 r ψ ns = Zαπ m ψ ns = Z4 α 4 m. 5 n 3 Spin-orbit trm nonzro only for l : Zα 4m r σ r p = Z4 α 4 m [jj + ll + ss + ] 3 4n 3 ll + l

5 Rlativistic corrctions: p4 8m 3 = Z4 α 4 m n n l +. 7 W find El = = Z4 α 4 m n n 8 = El =, j =, 9 so S / and P / rmain dgnrat at this lvl. Thy ar split by Lamb shift S / > P / which can b calculatd aftr you larn radiativ corrctions in QED. Th P / and P 3/ ar split by th spin-orbit intraction fin structur which you should hav sn bfor. El =, j = 3 El =, j = = Z4 α 4 m 4n 3 9 Klin Paradox and th Hol Thory So far w hav ignord th ngativ solutions. Howvr, th ngativ nrgy solutions ar rquird togthr with th positiv nrgy solutions to form a complt st. If w try to localiz an lctron by forming a wav pact, th wavfunction will b composd of som ngativ nrgy componnts. Thr will b mor ngativ nrgy componnts if th lctron is mor localizd by th uncrtainty rlation x p. Th ngativ nrgy componnts can not b ignord if th lctron is localizd to distancs comparabl to its compton wavlngth /mc, and w will ncountr many paradoxs and dilmmas. An xampl is th Klin paradox dscribd blow. In ordr to localiz lctrons, w must introduc strong xtrnal forcs confining thm to th dsird rgion. Lt s considr a simplifid situation that w want to confin a fr lctron of nrgy E to th rgion z < by a on-dimnsional stpfunction potntial of hight V as shown in Fig.. Now in th z < half spac thr is an incidnt positiv nrgy plan wav of momntum > along th z axis, ψ inc z = iz E+m, spin-up. 3

6 Figur : Elctrostatic potntial idalizd with a sharp boundary, with an incidnt fr lctron wav moving to th right in rsion I. Th rflctd wav in z < rgion has th form ψ rf z = a iz E+m + b iz E+m, and th transmittd wav in th z > rgion in th prsnc of th constant potntial V has a similar form ψ trans z = c iqz with an fftiv momntum q of Th total wavfunction is q E V +m + d iqz q E V +m, 3 q = E V m. 4 ψz = θ z[ψ inc z + ψ rf z] + θzψ trans z. 5 Rquiring th continuity of ψz at z =, ψ inc + ψ rf = ψ trans, w obtain + a = c 6 b = d 7 a E + m = c q E V + m 8 b E + m = d q E V + m 9 3

7 From ths quations w can s b = d = no spin-flip + a = c a = rc whr r = q E + m E V + m c = + r, a = r + r. 3 As long as E V < m, q is imaginary and th transmittd wav dcays xponntially. Howvr, whn V E + m th transmittd wav bcoms oscillatory again. Th probablity currnts j = ψ αψ = ψ α 3 ψẑ, for th incidnt, transmittd, and rflctd wavs ar w find j trans j inc = E + M, j trans = c q E V + m, j rf = a E + m. 4 = c 4r r = < for V E + m, j inc + r j rf r = a = > for V E + m. 5 j inc + r Although th consrvation of th probabilitis loos satifid: j inc = j trans + j rf, but w gt th paradox that th rflctd flux is largr than th incidnt on! Thr is also a problm of causality violation of th singl particl thory which you can rad in Prof. Gunion s nots, p.4 p.5. Hol Thory In spit of th succss of th Dirac quation, w must fac th difficultis from th ngativ nrgy solutions. By thir vry xistnc thy rquir a massiv rintrprtation of th Dirac thory in ordr to prvnt atomic lctrons from maing radiativ transitions into ngativ-nrgy stats. Th transition rat for an lctron in th ground stat of a hydrogn atom to fall into a ngativ-nrgy stat may b calculatd by applying smiclassical radiation thory. Th rat for th lctron to ma a transition into th nrgy intrval mc to mc is α6 π mc 8 sc 6 and it blows up if all th ngativ-nrgy stats ar includd, which clarly mas no sns. 3

8 A solution was proposd by Dirac as arly as 93 in trms of a many-particl thory. This shall not b th final standpoint as it dos not apply to scalar particl, for instanc. H assumd that all ngativ nrgy lvls ar filld up in th vacuum stat. According to th Pauli xclusion principl, this prvnts any lctron from falling into ths ngativ nrgy stats, and thrby insurs th stability of positiv nrgy physical stats. In turn, an lctron of th ngativ nrgy sa may b xcitd to a positiv nrgy stat. It thn lavs a hol in th sa. This hol in th ngativ nrgy, ngativly chargd stats appars as a positiv nrgy positivly chargd particl th positron. Bsids th proprtis of th positron, its charg = > and its rst mass m, this thory also prdicts nw obsrvabl phnomna: Th annihilation of an lctron-positron pair. A positiv nrgy lctron falls into a hol in th ngativ nrgy sa with th mission of radiation. From nrgy momntum consrvarion at last two photons ar mittd, unlss a nuclus is prsnt to absorb nrgy and momntum. Convrsly, an lctron-positron pair may b cratd from th vacuum by an incidnt photon bam in th prsnc of a targt to balanc nrgy and momntum. This is th procss mntiond abov: a hol is cratd whil th xcitd lctron acquirs a positiv nrgy. Thus th thory prdicts th xisstnc of positrons which wr in fact obsrvd in 93. Sinc positrons and lctrons may annihilat, w must abandon th intrprtation of th Dirac quation as a wav quation. Also, th rason for discarding th Klin-Gordon quation no longr hold and it actually dscribs spin- particls, such as pions. Howvr, th hol intrprtation is not satisfactory for bosons, sinc thr is no Pauli xclusion principl for bosons. Evn for frmions, th concpt of an infinitly chagd unobsrvabl sa loos rathr qur. W hav instad to construct a tru many-body thory to accommodat particls and antiparticls in a consistnt way. This is achivd in th quantum thory of filds which will b th subjct of th rst of this cours. 33