Resumo de Mecnica Quntica Relativista 1. Preliminares a. Tetravectores e Noes Relativistas E2= p2c2+ m2c4 g=g=100-100000000-100-1 x=ct, x, y, z=(ct, x) p=Ec, px, py, pz=Ec, p pp=E2c2-p.p=m2c2 A= A0, A b. Operadores = x= (ct), p=ix=i= i(ct), -

= = 2(ct)2- 2x2- 2y2- 2z2= 1c22t2- 2 c. Mecnica Quntica Clssica 1= 0110 ; 2= 0-ii0 ; 3= 100-1 .A.B= A.BI+i.AB ddtAt= iH, At+ Att

2. Equao de Klein-Gordon a. Formulao pp =m2c2 + m2c22=0

b. Solues Livres = e-ipx= e-iEt - p.x E= cm2c2+ p2 c. Corrente e Equao da Continuidade j= i2m*- * j'= ie2m*- *

j=0 d. Limite No-Relativista r, t= r, t e-imc2t T mc2 it T mc2 it= -222m e. Partculas de Spin-0 =Ae-iEpt - p.x Ep= cm2c2+ p2 = eEpmc2* A= mc2L3Ep f. Interaco Electromagntica p p- ecA gix- ecAix- ecA= m2c2

j'= ie2m*- *- e2mcA* g. Invarincia de Gauge A'x= Ax+ xx '= eiec h. Forma de Schrdinger = + ; it= mc2- it= -22m2+ +mc2it= -22m2+ - mc2 = it= Hf ; Hf= 3+i2p22m+ 3mc2

3. Equao de Dirac a. Formulao it= ci1x1+ 2x2+ 3x3+ mc2= Hf it= Hf= cipi+mc2 = 1234 b. Propriedades das Matrizes e i, j=2ijI i, =0 i2= 2= I i= i ; = ; = 1 Tri=Tr=0 i= 0ii0 ; = I00-I i'=UiU-1 ; '=UU-1; U= U-1 c. Corrente e Equao da Continuidade = ; jk=ck t+ .j=0 d. Solues Estacionrias x, t= xe-it x= Hfx = 1234 = =cipi+mc2=cipi - mc2 = 00e-ip.x - mc2I0- cipi0=0-cipi0+ + mc2I0=0 = Ep= cp2+m2c2 0= cipi+ mc20 ; 0=U= U1U2; UU= U1*U1+ U2*U2= 1 p, x, t=NUcipimc2+ EpUeip.x- Ept

N= mc2+ Ep2Ep pp, =pp, e. Spin e Helicidade S= 2= 200 Hf, .p=p, .p= 0 S= S.pp p=0, 0, p S= Sz= 2z00z= 2100-100000000100-1 p, , +12=N10czpmc2+ Ep10eipz- Ept p, , -12=N01czpmc2+ Ep01eipz- Ept f. Operadores de Sinal e de Projeco; Operadores Pares e mpares = HfHf2= c.p+mc2cp2+m2c2= c.p+mc2Ep = = -1 ; p, , Sz= p, , Sz = 12 A= A+ A2 ; A= A- A2 i=cpicp2+m2c2 ; =mc2cp2+m2c2 g. Operador Velocidade e Movimento de Pacotes de Ondas dxdt= 1ix, Hf=c=v ddt= 1i, Hf= 2icp- 2iHf t= 0- cpHfe-2iHft+ cpHf xt= x0+ c2pHft+ 0- cpHfic2Hfe-2iHft h. Electro em Repouso it= mc2 (1)= 1000e- imc2t(2)= 0100e- imc2t(3)= 0010eimc2t(3)= 0001eimc2t

r= r0e- irmc2t ; r= +1 , r=1, 2-1 , r=3, 4 i. Limite No-Relativista

p p- ecA it= c.+eA0+ mc2= Hf+H' ; H'= -ecv.A+eA0 = e-imc2t it= c.+eA0it= c.+eA0-2mc2 it mc2 ; eA0 mc2 = .2mc it= 22m-e2mc.B+eA0= p22m- e2mcL+2S.B+eA0

4. Covarincia a. Transformaes de Lorentz (x')=a x a a = deta = 1 b. Equao de Dirac na Forma Simtrica; Matrizes Gama i0x0+ 1x1+ 2x2+ 3x3-mc=0 0= ; i= i , =2gI ; = 00 i= 0i-i0 ; 0= I00-I p-mc=i-mc=0 '=UU ; U= U-1 c. Transformaes e Covarincia 'x'= 'ax=Sax=Saa-1x'x= S-1a'x'= S-1a'ax= Sa-1'ax S-1a= Sa-1 iSaS-1aa x'-mc'x'=0 SaS-1a= a d. Transformaes Infinitesimais a = + ; = - S = I- i4 S-1 = I+ i4 = i2, S = I+ 18, = In e. Transformaes Prprias In Ix x'= limNI+ NIxN x

x'0x'1x'2x'3= cosh-sinh-sinhcosh000000001001x0x1x2x3 ; tanh= ; cosh= = 11- 2 'x'= limNI- i4NInNx= e-i4In x SRij= e-i4ijij= e-i2.s ; SR= SR-1 SL= e-i201= e21 ; SL-1= 0SL0 f. Corrente e Spinor Adjunto j'(x')=a j(x) = 0 ; 'x'= xS-1 g. Reflexo Espacial a =g P= ei0 ; P-1= e-i0 ; P4= I h. Solues Gerais da Equao de Dirac rpx= e-i201 r0= cosh210010-tanh2-tanh200-tanh2tanh201001r0 -tanh2= pxcE+mc2 ; cosh2= E+mc22mc2 rpx= E+mc22mc210010pxcE+mc2pxcE+mc200pxcE+mc2pxcE+mc201001r0 S-v= S-pE=e-2.vv= E+mc22mc21001pzcE+mc2p-cE+mc2p+cE+mc2pzcE+mc2pzcE+mc2p-cE+mc2p+cE+mc2-pzcE+mc21001 p=px ipy rp= S-pEr0 rx= rpe- irpx p- rmcrp=0rpp- rmc=0 i. Electres Polarizados

sR.S.=0, s ; s=a sR.S. pR.S.=mc, 0, 0, 0 ; p=a pR.S.

ss= -1 ; ps=0 s=ez up,uz= 1pup,-uz= 2pvp,uz= 4pvp,-uz= 3p uzR.S.=0, uz=(0, 0, 0, 1)

5. Covariantes Bilineares a. 16 Matrizes Gama S= I ; V= ; T= ; P=i0123= 5 ; A= 5 n2= I nS m : n, m=0 TrnS=0 a, b, ab nS : a b=fabnn , fabn C 5, = 5, = 5, Sa=0 b. Formas Bilineares 'x'S'x'= (x)Sx 'x'5'x'=det(a)(x)5x 'x''x'=a (x)x 'x'5'x'=det(a)a (x)5x 'x''x'= a a (x)x

6. Operadores de Projeco a. Propriedades do Projector Prp= P(p, uz, r) Prpr'p= rr'r'pPrpPr'p= rr'Prp b. Projeco da Energia p= p+mc2mc +2= +-2= -+-=0++ -=1 c. Projeco de Spin uz= I+ 5uz2 ; uz0 uzR.S. uz010= 10uz020=0uz030=0uz040= 40-uz010=0uz020= 20-uz030= 30-uz040=0 s= I+ 5s2 d. Projeco Simultnea P1p= +puzP2p= +p-uzP3p= -p-uzP4p= -puz

7. Equao de Weyl a. Formulao it=c.p b. Partculas Direitas (+)t=- c. + += 12E23e-ip.xu+(p) .p u+p=p0 u+p p0= p p=0, 0, p u+= 10 .ppu+= zu+= u+ c. Partculas Canhotas (-)t= c. - -= 12E23e-ip.xu-(p) .p u-p=- p0 u-p p0= p p=0, 0, p u-= 01 .ppu-= zu-= - ud. Corrente e Equao de Continuidade =I, -=0 j= -j=0 e. Representao de Weyl i= i00-i ; = 0-I-I0 ; 5= I00-I i, j=2ijI ; i, =0 = (+)(-) i(+)t= c. +-mc2(-)i(-)t= -c. --mc2(+) P= I52 (-)=P- = 0(-)(+)= P+ = (+)0

8. Equao de Dirac num Potencial Central a. Formulao HD= c.p+ mc2+Vr jm= jlmx, tjl'mx, t P= eiP0 jlm= m', msl 12 j | m'ms m Ylm'12 ms 12 12= 10 ; 12-12= 01 P0 jlm= -1l jlm jlm=ig(r)jlmrrjl'm= -f(r)jl'mrr ; l'=2j-l= 2l+ 12-l=l+1 , j=l+ 122l- 12-l=l1 , j= l- 12

9. Representao de Foldy-Wouthuysen a. Transformao de Foldy-Wouthuysen U= Hf+ Ep2Epmc2+ Ep =U ; A= UAU H=Ep = = - cp.pEpEp+ mc2+ cpEp = 12 p, , +12=mc2+ 210cpmc2+ 0eipz232 p, , -12=mc2+ 2010-cpmc2+ eipz232 p 1 12= 1000eipz232 ; p 1 -12= 0100eipz232 p -1 12= 0010eipz232 ; p -1 -12= 0001eipz232 b. Campos Externos

10.Teoria de Lacunas a. Electro - Positro =Eelec pos en-Eelec neg en =cp2+m2c2- -cp2+m2c2=Eelec+Epos Epositron=cp2+m2c2 k= pelec pos en- p'elec neg en=pelec+ppos ppos= - pelec b. Conjugao de Carga C=i20= -C-1= -C= -CT c= C0*= CT Qc= c Q c= 2Q*2 * c. Estados Prprios psx= p+mc2mcI+ 5s2(x) psc= -p+mc2mcI+ 5s2c 11.Electromagnetismo e Invarincia de Gauge a. Formalismo Lagrangiano S= t1t2Lx, xdt S=0 Lx= ddtLx L= mv22-qx, t+qAx, t.v SI= t1t2-q+ qA.v dt=qA dx A A+ SI'= SI+ B- (A)