1 Gio trnh L THUYT HM GREEN Trong Vt L TS. V Quang Tuyn B mn Vt l l thuyt, Khoa Vt li hc Khoa hc t nhin tp. HCM 20092 HmGreen(HG)ckhaisinhbinhtonhcAnhGeorgeGreennm 1828. Sau HG cng dng bi ton in t v tr thnh mt cng c ton hcrtclcvphbingiinhngphngtrnhviphnkhngngnht trong ton cng nh trong vt l. Trong vt l lng t, nht l trong vt l cht rn, HG v ang l mt mt cng c rt quan trng nghin cu nhng h tng tc phc tp. Hc phn l thuyt HG ny s trnh by HG nh l mt phng php (ton hc) nghin cu vt l. Hai chng s trnh by phng php HG cho bi ton phng trnhviphntuyntnhkhngngnht-llpphngtrnhviphnthnggp trong vt l. Hu ht cc v d, bi ton c a vo trong nhng chng ny u c ni kt vi cc bi ton vt l.Tchng3chngtistptrunggiithiuphngphpHGtrongvtl lng t, c bit trong trng thi rn. S gii hn HG cn bng nhit 0. Trong thctccthnghimcthchinnhitkhckhng.Tuynhinmtcch gn ng ta c th s dng HG cho nhng th nghim nhit thp hoc nhng th nghim c cc i lng ta quan tm khng thay i nhanh theo nhit Thc t cho thy HG nhit 0 c s dng tnh ton v gii thch thnh cng nhiu qu trnh vt l nhit hu hn.Nu cn xt n nh hng ca nhit , ngi ta cng c th s dng HG nhit hu hn Matsubara. i vi h khng cn bng th cn n HG khng cn bng. ChngtischtrnhbymtstngchnhcaccloiHGnysinhvin (bc c nhn) nm c hng pht trin/m rng ca HG trong vt l lng t.Hcphnnynhmgipsinhvinnmci)phngphpcnbncaHG trong bi ton phng trnh vi phn khng ng nht, ii) phng php tm HG bng hm ring v tr ring, iii) nh ngha v v ngha ca HG trong vt l lng t, iv) phng php trin khai nhiu lon ca HG v mt s cch tnh gn ng ca HG v) phng php gin Feynman cho HG. 3 Ti liu tham kho * Hm Green v phng trnh vi phn tuyn tnh : Duffy G. D. (2001): Greens Function with Applications (Chapman & Hall/CRC) Tang K. T. (2006): Mathematical Methods for Engineers and Scientists (Springer, Berlin) Michio Masujima (2005) : Applied Mathematical Methods in Theoretical Physics (Wiley) * Hm Green v ng dng vo vt l lng t: AbrikosovA.A.,GorkovL.P.DzyaloshinskiI.E.(1975):MethodsofQuantum Field Theory in Statistical Physics (Dover Publ., New York) DoniachS.,SondheimerE.H.(1998) :GreensFunctionsforSolidStatePhysicists (Imperial College Press, London)EconomouE. N. (2006): Greens Functions in Quantum Physics (Springer-Verlag Berlin) EnzC.P.(1992) :ACourseonMany-BodyTheoryAppliedtpSolid-StatePhysics (World Scientific, Singapore) FetterA.L.,WaleckaJ.D.(1971):QuantumTheoryofMany-ParticleSystems (McGraw-Hill, New York)Kadanoff L. P., Baym G. (1962): Quantum Statistical Mechanics (Benjamin, Berlin) Mahan G. D. (1981) : Many-particle Physics (Plenum, New York) Nguyn Vn Lin (2003): Hm Green trong Vt L Cht Rn (i Hc QG H Ni) * L thuyt in t c in: Jackson J. D. (1999) : Classical Electrodynamics (Wiley, New York) 4 Chng 1 1Dn nhp vo HG SinhvinvtlthnghaythitisaocnhcHG,ulldovtlcho vic hc ny, ? thy c vai tr cng nh ngha ca HG, trong chng ny s im li vai tr ca HG trong mt s bi ton ph bin trong vt l (c in); ng thi cng cho thy trc mi tng quan trc tip ca HG vi nhng i lng vt l c th o c trong th nghim, qua gip cho sinh vin, ngay t u, cm nhn c vai tr v ng dng thit thc ca HG trong vt l (lng t). 1.1HG trong vt l c in (HG c in) : 1.1.1Th Coulomb : HG c s dng ph bin trong bi ton th Coulomb to bi in tch im. intchenhvtiimcvectorvtr' r

storathin( ) r o

tiimr

c tnh nh sau : 0( )4 | ' |err rotc=

.(1.1.1) Ta c th m rng cho trng hp nhiu in tch im hoc trng hp phn b lin tc vi hm phn b in tch( ) r p

.Trong trng hp sau, th nngl30( ')( ) '4 | ' |rr drr rpotc=

. (1.1.2) Th ny tha mn phong trnh Poisson20( ) ( ) / r r o p c V = . (1.1.3) Ta a vo HG c nh ngha nh sau : 01( )4GRR tc=vi| ' | R r r = .(1.1.4) 5 HG ny tha phng trnh Poisson i vi in tch im vth Coulomb c tnh qua HG nh sau : ( ) ' ( ) ( ') r drG R r o p =

.(1.1.5) 1.1.2Phng trnh in t trng (Maxwell) : Xt phng trnh Maxwell cho th in t : 222 21( , ) ( , ) A A rt j rtc tuc V = c

220 2 21( , ) ( , ) / rt rtc to o p cc V = c .(1.1.6) Ngi ta c th gii hai phng trnh trn bng cch dng HG( , ) GR T

tha phng trnh( ', ') ( ') ( ') ( ) ( ) G r r t t r r t t R T o o o o =

(1.1.7) v nghim ca phng trnh (1.6) c xc nh qua HG di dng01( ) ' ' ( , ') ( ', ') r dr dt GR t t r t o pc=

(1.1.8) Xt mt trng hp c bit :0GGTc= =cti T=0. Khi ta thu c HG c dng { }( )( , ) ( ) ( )4c TGR T R cT R cTRuo ot= + (1.1.9) Bi v R v T dng, s hng cui bng 0 v c( , ) ( )4cGR T R cTRot= .(1.1.10) Th in tr thnh 0( ) ' ( ', / ) /4cr dr r t Rc R o ptc=

(1.1.11) v y chnh l dng quen thuc ca th tr trong in ng lc hc. 6 1.2H lng t : y ch tm tt mt s mi lin hgia HG v nhng i lng vt l c th o c trc tip hay gin tip trong thc nghim.- ng nng, th nng v nng lng ton phn c tnh trc tip qua HG - HG cho thng tin trc tip v mt ht ( ) ( ) ( ) n x x x v v = , mt dng, mt spin, - HG cho thng tin trc tip v cc hm tng quan- Cc thng tin v tnh cht ph (nng lng), mt trng thi, tit din tn x c bao gm trong HG tr. - Ngoi ra HG cho mt l thuyt nhiu lon rt h thng. 7 Chng 2 2Hm Green trong phng trnh vi phn tuyn tnh Phng trnh vi phn tuyn tnh khng ng nht l mt bi ton rt thng gp trongtonl.PhngtrnhthCoulomb,phngtrnhsng,phngtrnhtruyn nhit,... thuc v dng ton trn. V th gii nghim ca phng trnh ny l mt vic rtquantrngtrongvyltonvHGchngtlmtcngcrthiuqu. Chng ny s trnh by phng php HG trong vic gii phng trnh vi phn tuyn tnh.2.1Phng trnh vi phn tuyn tnh : Khostccbitonvtlthngdnnphngtrnhviphncdngthcnh sau : ( ) Lu f x = .(2.1.1) Trong L l ton t vi phn, f(x) l hm bit v u l nghim cn tm. Vi ci nhn vt l ta c th xem hm f(x) bn v phi ca phng trnh biu din lc- tc ng ln h, v nghim u(x) ca phng trnh biu din s phn hi ca h. * Ta ch xt ton t tuyn tnh, tc l L phi tha mn( ) L f g Lf Lg o | o | + = + ,(2.1.2) y o v|l nhng s phc, f v g l hai hm trong khng gian hm.Vi v d ton t tuyn tnh L: i)nhn bi mt hng s v hng a:Lu au =ii)o hm bc n ca mt hm (u) : nndLu udx=hoc nndLdx=gi l ton t vi phn.8 * Ngoi tnh tuyn tnh ta cng ch xt n nhng ton t Hermite (cn gi l ton t self-adjoint), tc lL L+= . yL+c gi l ton t lin hp ca ton t L v c nh ngha: | | f Lg Lf g+( , = ( , (2.1.3) Ch ta s dng nh ngha tch trong: *| ( ) ( )baf g f x gx dx ( , vi f v g l nhng hm xc nh trn [a,b], f* l lin hip phc ca f .Thc t,V d : Tm ton t lin hip ca 2 2/ L d dx = :2 22 2| * *d d d df Lg f g f gdx f gdxdx dx dx dx ( , = = = Tch phn tng phn22** *d d dg df dg dg df df gdx f dx f g f gdxdx dx dx dx dx dx dx dx = = + 22dg df df g f gdx dx dx = + Vi iu kin , 0 f g khix th ton t lin hip ca 22dLdx=l 22dLdx+=tha mn nh ngha (2.1.3).Nhn xt: trong trng hp ton t vi phn, thng gp phi vn pht sinh do bi iu kin bin. Thc vy, nh thy trong v d trn, v phi ca phng trnh (2.1.3) c th c nhng s hng b sung lin quan n cc iu kin bin ca f v g. Nu f v g c cng iu kin bin th cc s hng bin s kh nhau. Ti sao quan tm ton t Hermit ? Ta ang xt bi ton vt l. Trong vt l lng t mt i lng kho st c biu din bi ton t O v gi tr k vngca i lng ny c xc nh bi| O v v ( , trong v l hm sng m t trng thi ca h ang xt. Nh vy| O v v ( , phi thc, tc l| * | O O v v v v ( , = ( ,Bi v| * | O O v v v v ( , = ( ,(tnh cht ca bra-ket : | | * o v v o ( , = ( , ) ta c : | | O O v v v v ( , = ( ,Nh th ton t biu din mt i lng quan st c phi l Hermite.9 Tngtnhtrongbitontrringcamatrn,tacngcbitontrring ca ton t vi phn : n n nLo o = ,(2.1.4) vi n l mt hng s. Vi mt chn trc, hm so tha mn (2.1.4) v iu kin bin c gi l hm ring tng ng vi;c gi l tr ring.i vimt tontHermite,cctrringlsthcvcchmringtothnhhtrcgiao (,|n m n mo o o ( , = ) v y ( | | 1k kko o ,( =).2.2Hm Green v phng trnh vi phn tuyn tnh : 2.2.1Phng php HG cho phng trnh vi phn tuyn tnh: Trong phn ny chng ta s trnh by cch dng HG tm nghim phng trnh vi phn tuyn tnh khng ng nht. Phng php HG c th p dng cho nhng lp phng trnh vi phn ph bin gm c phng trnh vi phn thng v phng trnh o hm ring phn (xem v d Duffy D. (2001)).Gi s chng ta mun gii phng trnh vi phn ( ) ( ) Lux f x = ,(2.2.1) trn min xc nhxeO, tha iu kin bin cho trc v L l ton t vi phn tuyn tnh Hermite. Nhn hm G(x,x) (hm cha bit, nhng y chnh l HG ta s tnh sau ny)vo hai v ca(2.2.1) v ly tch phntheo x trn minO(tc l ta ang thc hin tch trong ( | *( ) ( ) f g f xgx dx ( , =) ca c hai v phng trnh (2.2.1) vi hm G(x,x) theo bin x). Ta c : ( , ') | ( ) ( , ') | ( ) Gx x Lux Gx x f x ( , = ( , .(2.2.2) VinhnghatontlinhipL+caL ,nhnhnxtv(2.1.3),vtrica (2.2.2) c th c vit li nh sau: ( , ') | ( ) ( , ') | ( ) Gxx Lux L Gxx ux+( , = ( , + s hng bin,(2.2.3) hay( , ') | ( ) ( , ') | ( ) L Gxx ux Gxx f x+( , = ( , + s hng bin(2.2.4) By gi ta chn hm G(x,x) tha mn ( , ') ( ') L Gxx x x o+= , (2.2.5) vviiukinbincchnthchhpkhittcnhngthnhphncha c bit trong s hng bin.Hm G(x,x) c gi l HG.10 Thay (2.2.5) vo phng trnh (2.2.4) v lu n tnh cht ca hm delta-Dirac, ta thu c : ( ') ( , ') | ( ) ux Gx x f x = ( , + s hng bin bit.(2.2.6) Vi f(x) bit trc v s hng bin bit, mt khi tnh c G(x,x) ta s tm c nghim u. (2.2.6) c th c vit li nh sau : ( ) ( , ) ( ) ux G x f d c c cO=+ s hng bin bit.(2.2.7) Tmligiiphngtrnhviphntuyntnhkhngngnht( ) Lu f x =bngHG, trc ht ta gii phng trnh vi phn( , ') ( ') L Gxx x x o+= vi iu kin binchnlathchhptmnghimG(x,x).Sautasccnghimuca phng trnh vi phn xc nh qua (2.2.7).2.2.2 ngha vt l ca HG : Hm Green c th c gii thch nh sau. Nu phng trnh vi phntuyn tnh, v d (2.2.1), m t h vt l tuyn tnh, th nghim u(x) biu din s phn hi (tuyn tnh) ca h di s kch thch ca lc ngoi f(x). Khi HG G(x,x) biu th phn ng ca h vt l i vi lc ca mtngun im ti im x. Ta cng c th xem HG G(x,x) nh hm truyn kch thch hoc nh hm tng quan gia lc kch thch vs phn hi ca h. 2.2.3Tnh i xng ca HG : HG c tnh i xng, cn gi l tnh cht o, nu L l ton t Hermite. Tht vy, xt( , ') ( ') LGx x x x o = ,(2.2.8) ( , '') ( '') LGx x x x o = .(2.2.9) Ly tch trong ca (2.2.8) vi( , '') Gxxt bn tri v (2.2.9) vi( , ') Gx xt bn phi :( , ''), ( , ') ( , ''), ( ') Gx x LGx x Gx x x x o ( , = ( , , ( , ''), ( , ') ( ''), ( , ') LGx x Gx x x x Gx x o ( , = ( , . VtontLlHermite nn( , ''), ( , ') ( , ''), ( , ') Gx x LGx x LGx x Gx x ( , = ( , .Thai phng trnh trn c c( , ''), ( ') ( ''), ( , ') Gx x x x x x Gx x o o ( , = ( , , tc l *( ', '') ( '', ') G x x Gx x = .(2.2.10) y l tnh cht o ca HG.Nu( , ') Gx xl thc th HG c tnh i xng :11 ( ', '') ( '', ') Gx x Gx x = ,(2.2.11) 2.3Hm Green cho phng vi phn khng ph thuc thi gian: Trong phn ny ta s kho st phng trnh vi phn tuyn tnh khng ng nht khngphthucthigian.BitonIctnhminhhaphngphpHGtrnhby trong phn trn.Trong bi ton II, khi dng HG gii phng trinh vi phn,ta cng gii thiu phng phphm ring v tr ring tm HG.Cc ktquthuc cho php tm hiu nhng tnh cht ca HG cng nh ngha vt l ca n. 2.3.1HG khng ph thuc thi gian I: Ta hy kho st phng trnh Schrdinger 1 chiu vi th U(x) : 222( ) ( ) ( )dk x Ux xdxm m + = (2.3.1) v gi thit( ) 0 Ux khix . G s cc iu kin bin : (0) a m = , '(0) b m = , (2.3.2) v quan tm n nghim vi x>0.Vit li phng trnh (2.3.1) di dng( ) ( ) L x f x m = (2.3.3) vi 222, ( ) ( ) ( )dL k f x Ux xdxm = + = .(2.3.4) Hm Green : Nhn hai v ca (2.3.3) vi g(x,x) v ly tch phn theo x t0 n: 0 0( , ') ( ) ( , ') ( ) gx x L x gx x f x m = .(2.3.5) Tch phn tng phn (hai ln) trn v tri (2.3.5), thu c : 0 0 00( , ')( ( , ')) ( ) ( , ') '( ) ( ) ( , ') ( )xxxxdgxxLgxx x dx gxx x x gxx f x dxdxm m m== ==+ = (2.3.6) Trong cc s hng bin,(0) mv'(0) m bit. kh nhng s hng bin cha bit ( ( ) m v'( ) m ) i hi iu kin bin cho g(x,x): ( , ') 0 g x =v ( , ')0dg xdx= .(2.3.7) V chn g tha phng trnh12 ( , ') ( ') Lgx x x x o = .(2.3.8) T (2.3.6) ta tm c : 00(0, ')( ') (0, ') '(0) (0) ( , ') ( )(0, ')(0, ') ( , ') ( )dg xx g x gxx f x dxdxdg xbg x a gxx f x dxdxm m m= += +(2.3.9) Tm nghim HG g(x,x) : vit li phng trnh cho HG di dng: 222( , ') ( , ')dk gxx xxdxo + = (2.3.10) trn(0, ) x e vi' (0, ) x e v iu kin bin :( , ') 0 g x = (2.3.11) ( , ')0dg xdx= .(2.3.12) i vi' x x : ( , ') sin( ) cos( ) gx x C kx D kx = + .(2.3.14) S dng cc iu kin bin (2.3.11) v (2.3.12) cho :0 C D = = . Vy( , ') 0 gxx =vi' x x > .(2.3.15) Ly tich phn hai v ca (2.3.10) theoxt' x c n' x c +v ly gii hn0 c : ' ' 2220 0' 'lim ( , ') lim ( , ')x xx xdk gxx dx xx dxdxc cc cc co+ + + = . (2.3.16) Vi i hi tnh lin tc ca( , ') gx x khi qua' x ta c : ( ' , ') ( ' , ') gx x gx x c c + = (2.3.17) v ( ' , ') ( ' , ')1dgx x dgx xdx dxc c + = .(2.3.18) Ly 0 c+ta c h phng trnh xc nh h s A v B trong (2.3.13) : sin ' cos ' 0cos ' sin ' 1/ .A kx B kxA kx B kx k+ = + =(2.3.19) H ny cho kt qu : cos ' kxAk= v sin ' kxBk= (2.3.20) v HG cn tm l: 13 sin ( ')( , ')kx xgxxk=vi' x x .(2.3.21) Hm m(2.3.9) tr thnh : '0sin ' sin ( ' )( ') cos ' ( )xkx kx xx a kx b f x dxk km= + +.(2.3.22) Ta c th vit li hmmcn tm vi( ) ( ) ( ) f x Ux x m =nh sau : 0sin sin ( )( ) cos ( ) ( )xkx kxx a kx b U dk kcm c m c c= + +.(2.3.23) Nhn xt: phng trnh (2.3.23) l phng trnh tch phn Volterra loi hai. 2.3.2HG khng ph thuc thi gian II: Gi s phng trnh vi phn tuyn tnh khng ng nht (2.2.1) c dng | |( ) ( , ) ( ) r r f r m =

L (2.3.24) trong( ) f r

lhmchov lbinphc,nghimcntm( , ) r m

thamn iu kin bin trn S thuc minOchar

v' r

.( ) r

L l ton t vi phn tuyn tnh Hermite v khng ph thuc thi gian t.gii(2.3.24)tanhngha( , ', ) G rr

lnghimcaphngtrnhviphn khng ng nht | |( ) ( , ', ) ( ') r G rr r r o = L ,(2.3.25) cng tha mn iu kin bin nh (2.3.24). Ta c th vit phng trnh (2.3.25) di dng Dirac | |( ) 1 G = L .(2.3.26) tm nghim ca (2.3.26), ta xt bi ton hm ring v tr ring ca ton tL . Gi|no ,v nE l hm ring v tr ring caL: | |n n nE o o , = , L .(2.3.27) Cc hm ring ny s to h y v trc giao v tr ring nEl thc.i)Gis khngthucphtrringcaL ,khicctrringca | | L khc 0, t (2.3.26) ta tnh c HG nh sau: | | 1 1( ) | |n nn nn nnGEo o o o ,(= = ,( = L L.(2.3.28) Nu ph ca ton tL gm c gin an v lin tc th HG (2.3.28) c vit di dng tng qut14 | | | |( )n n E EnnG dEE Eo o o o ,( ,(= + (2.3.29) hoc trongr

-biu din: * *( ) ( ') ( ) ( ')( , ', )n n E Ennr r r rG rr dEE Eo o o o = +

.(2.3.30) Nghim( , ) r m

ca phng trnh vi phn(2.3.24) c tnh qua HG( , ', ) G rr

v c dng : ( , ) ( , ', ) ( ') ' r G rr f r dr m = (2.3.31) ii)Xttrnghp thucphtrringcaL .Tinhngim nE = hm Green( , ', ) G rr

khngxcnh.Khiphngtrnh(2.3.24)khngcligii ngaitrtrnghphm( ) f r

trcgiaovittccchmring|no , tngng vitrring nE .Thtvy,tavitliphngtrnh(2.3.24)vi nE = didng Dirac | || |nE fm , = , L ,(2.3.32) nhn|no ( vi hai v v ly tch phn theor

ta c | | | || | | |n n n n n nE E E f o m om o ( , = ( , = ( , L .(2.3.33) phng trnh c nghim ta cn|nf o ( ,phi bng 0 vi mi|no ( , tc l( ) f r

trc giao vi tt c cc hm ring.iviphlintc,( ) G khngxcnhtrnminE = ,nhnglungii tch ti ln cn trnE i c = +v ln cn diE i c = vi mi0 c > . Ta a vo hm gii hn c nh ngha 0( , ', ) lim ( , ', ) G rr E G rr E icc++= + ,(2.3.34) 0( , ', ) lim ( , ', ) G rr E G rr E icc+= . (2.3.35) Nhng gii hn trn lun tn ti. Khitrng vi tr ring E trong min ph lin tc, nghim ca phng trnh vi phn khng thun nht c xc nh qua hm gii hn G nh sau : 0( , ) ( , ) ( , ', ) ( ') ' r r G rr f r dr m m = +

,(2.3.36) trong 0m l nghim ca phng trnh vi phn thun nht | |0( ) ( , ) 0 r r m = L .2.3.3Tnh cht v ngha ca HG: -T (2.3.30) ta tm li tnh cht o ca HG: 15 * *( , ', ) ( ', , ) G rr Gr r =

(2.3.37) Vnghavtl,iunychothyphnhitir

ivingunimti ' r

bng vi phn hi ti' r

i vi ngun im tir

. -Tr ring ca ton t HermiteLl thc HG( , ', ) G rr

gii tch khp trn mtphngphc ngaitrnhngim(min)kdtrntrcthcctrngvitrringEn(E)caL .Tinhngim trntrcthckhng trngvitrringcaL ,HG( , ', ) G rr

lHermitevt(2.3.37)tac *( , ', ) ( ', , ) G rr Gr r =

; v khi ( , , ) G rr

l thc. -Cc cc ca HG xc nh ph tr ring gin an ca ton tL .-Thng d ca HG ti cc cc n cho thng tin v hm ring caL: { }*( , ', ) | ( ) ( ')nE n nResG rr r r o o =

,(2.3.38) { }2( , , ) | | ( ) |nE nResG rr r o = .(2.3.39) -Trong trnghp ph lin lc caL :G+ khcG v nh vy tn timt ct nhnh dc theo trc thc m t ph lin tc ng vi trng thi lan truyn. -Hm gii hn c mi lin h sau : *( , ', ) ( ', , ) G rr E G r r E +=

,(2.3.40) tc l Re ( , ', ) Re ( ', , ) G rr E G r rE +=

,(2.3.41) Im ( , ', ) Im ( ', , ) G rr E G r rE +=

,(2.3.42) Ngi ta cng quan tm n hm ~G nh ngha nh sau : ~( , ', ) ( , ', ) ( , ', ) G rr E G rr E G rr E+ =

(2.3.43) khi : ~( , ', ) 2 Im ( , ', ) 2 Im ( , ', ) G rr E i G rr E i G rr E+ = =

.(2.3.44) -GiN(E)lmttrngthi(strngthitrongmtnvnnglng, ( ) ( )nnNE E E o = )v( , ) rE p

lmttrngthitrnmtnvth tch( ( , ) ( ) r Edr NE p = ).Nhsthydiy,cchmGv ~Gchota thng tin v N(E) v( , ) rE p

.S dng ng nht thc Dirac 16 00 01 1( )0P i x xx x i x xto+| |= | \ .) (2.3.45) choG (ch n (2.3.28)) ta c : **( ) ( ')( , ', ) ( ) ( ) ( ')n nn n nn nnr rG rr E P i E E r rE Eo ot o o o=

) (2.3.46) v ~*( , ', ) 2 ( ) ( ) ( ')n n nnGrr E i E E r r t o o o =

.(2.3.47) Ly vt (trace, k kiu Tr) ca cc yu t cho( , , ) G r r E ca (2.3.46) cho : 1( ) | ( ) | ( )n n nn n nnTrG E G E P i E EE Eo o t o = ( , = ) (2.3.48) hay 1( ) Im Tr ( ) NE G Et= ) .(2.3.49) Hm mt trng thi c th vit di dng : *( , ) ( ) ( ) ( )n n nnrE E E r r p o o o =

(2.3.50) Khi ta c 1( , ) Im ( , , ) r E G rr E pt= ) (2.3.51) Hoc ~1( , ) ( , , )2r E Grr Eipt= ) .(2.3.52) -HG G c th c biu din qua ~G hoc( , ) rE p

: * *( ) ( ') ( ) ( ')( , ', ) ( )n n n nnn nnr r r rG rr dE E EE Eo o o o o = =

~( , ', )2i G rr EdEE t =

(2.3.53) v ( , )( , , )r EGrr dEEp=

(2.3.54) 17 2.3.4V d: Xt ton t 2= V L trn V l 3R . iu kin bin l cc hm ring caL phi hu hn v cng. Ta s tm HG cho phng trnh vi phn (2.3.25) vi ton t trn. Phng trnh cho HG : 2( , ', ) ( ') G rr r r o +V =

.(2.3.55) Phng trnh Schrdinger ca ht t do 2 2( / 2 ) E m v v V = h (vi nng lng E2/ 2 p m = ) cho ta bit ton t 2= V Lc tr ring v hm ring 2 2 2/ E p k = h ,(2.3.56) 1( ) |ik rkr r k eVv= ( , =

.(2.3.57) HG cn tm c cho bi (2.3.30) c dng : *( ')2 3 2 3 2( ) ( ')| | ' 1( , ', )(2 ) (2 )ik r rk kkr rr k k r V eG rr dk dkk k kv v t t ( ,( ,= = =

(2.3.58) Ta s dng :... ...(2 )DDkLdkt

vi D l s chiu ca khng gian. y D=3.Vi' r r p =

v gi ul gc giap

vk

th 2cos2 222 202 21( , ', ) sin(2 )1(2 )14ikik ikikkdkG rr d ekkdk e ek ikkdkei kp up pp u ut t pt p ===

(2.3.59) Trong trng hpkhng thuc ph caL , viIm 0 > , tch phn nh nh l thng d cho | '|1 1( , ', )4 4 | ' |i i r rG rr e er rp tp t= =

.(2.3.60) Nuthuc ph caL ,E = , HG c xc nh qua cc hm gii hn (2.3.34) v (2.3.35), ta c : | '|( , ', ) ; 04 | ' |i E r reG rr E Er r t = >

.(2.3.61) Xt trng hp =0, phng trnh (2.3.55) chuyn v dng Laplace : 18 2( , ') ( ') Grr r r o V =

,(2.3.62) vi nghim l 1( , ')4 | ' |G rrr r t=

.(2.3.63) PhngtrnhviphnkhngthunnhttngngviHGnylphngtrnh Poisson cho th tnh in m: 2( ) 4 ( ) r r m tp V = .(2.3.64) Th tnh c xc nh qua HG v c dng : ' ( ')( ) 4 ' ( , ') ( ')| ' |dr rr drG rr rr rpm t p = = (2.3.65) 2.4Hm Green cho phng trnh vi phn ph thuc thi gian:2.4.1Phng trnh vi phn cha o hm bc I v Hm Green Xt phng trnh vi phn khng ng nht ph thuc thi gian c dng ( ) ( , ) ( , )ir rt f rtc tmc = c

L (2.4.1) v phng trnh ng nht tng ng( ) ( , ) 0ir rtc toc = c L ,(2.4.2) vi c l hng s dng vLl ton t vi phn tuyn tnh Hermite. Nghim( , ) rt m

ca (2.4.1) c th c tnh qua HG tng ng tha phng trnh( ) ( , ', , ') ( ') ( ')ir grr t t r r t tc to oc = c L .(2.4.3) y cc hm( , ) rt m

,( , ) rt o

v( , ', , ') grr t t

tha cng iu kin bin.Phngphpchungtnh( , ', , ') grr t t

lchuynphngtrnhphthucthi gian (2.4.3) sang dng khng ph thuc thi gian (2.3.25) bng php bin i Fourier nh sau : 1( ) ( )2ig d e gett e et=.(2.4.4) Ta t( ) ( ') ( , ') g g t t g t t t =( gn cc bin, ' rr

tm thi n)v phng trnh (2.4.3) cho thy g ph thuc hai bin thi gian di dng' t t . Thay( ) g tvo (2.4.3) ta c19 ( ) ( , ', ) ( ') r grr r rcee o = L . (2.4.5) y chnh l phng trnh cho hm Green G khng ph thuc thi gian c kho st mc trc. Nh vy( ) ( / ) g G c e e = ,(2.4.6) trong ( ) G (vi/ c e = ) l nghim ca phng trnh (2.3.25). Dng php bin iFourier(2.4.4)tastmc( , ', ) G rr t

;vttnhhm( , ) rt m

.Tuynhin cn ch rng, tng t hm( ) G , hm( ) g ecng khng gii tch mt s im ginonhayctnhnhtrntrcthc.Dokhngthlytchphn(2.4.4)trn trc thc, tc l khng th tnh( , ) rt m

trc tip t hm( ) g ttrn min thuc ph ca L - min c ngha vt l v cha nhng thng tin vt l.Nh vy ta cn n cc hm gii hn ca( ) g t(PT. (2.4.4)) trong tch phn c ly trn ngPe c gii hn n trc thc RP: 1( ) lim ( / )2 RP P iP P Pg d e G ce ee eett e et=.(2.4.7) Ta ch quan tm n hai cch chn c ngha vt l tng ng vi hai hm gii hn G c nh ngha (2.3.34) v (2.3.35) : 1( ) ( / )2ig d e G cett e et =,(2.4.8) Vi 0| |( / ) lim/n nnnG cc E ico oee c+,(= .(2.4.9) T mi lin h caG (2.3.40) ta cng c h thc gia hai hm gii hng: *( , ', ) ( ', , ) g rr g r r t t +=

.(2.4.10) V tng t hm ~G trong (2.3.43) ta cng a ra hm ~gc nh ngha : ~( ) ( ) ( ) g g g t t t+ = .(2.4.11) Hoc s dng tnh cht (2.3.47) ca hm ~G ta tnh c : 20 ~ ~1( ) ( / )21( 2 ) ( / ) | |2| |niin n nnicEn nng d e G cd e i c Eic eetettt e ete t o e o oto o== ,(= ,(.(2.4.12) Ta c th vit li (2.4.12) di dng Dirac nh sau: ~( ) | |( )ic icn nng ice iceicUt tt o ot = ,( = = L L(2.4.13) Trong ta nh ngha ( ')( ')ic t tUt t e =L(2.4.14) v chnh l ton t tin ha : | ( ) ( ') | ( ') t Ut t t o o , = , ,(2.4.15) vi| ( ) t o , lnghimcaphngtrnhngnht(2.4.2).Nhvynghimca phng trnh vi phn ng nht c th c biu din qua HG ~g , trongr

-biu din ta c: ~( , ) ( , ', , ') ( ', ') ' 'ir t grr t t r t dr dtco o =

(2.4.16) Ch rng gia ~gvg+ hocg c mi lin h trc tip. Tht vy, hy kho sttchphn(2.4.8).Thay(2.4.9)vo(2.4.8)vchuynsangtchphnchu tuyn ng l na hnh trnC(hocC+) bn knh v hn thuc na mt phng di (hoc trn) nu0 t >(hoc0 t < ) . Vi cchchn nh thtch phn trn chu tuyn v cc bng 0. Bng vic s dng nh l thng d v nh l Cauchy m rng nh gi tch phn chu tuyn v vi nh ngha (2.4.11) ta tm c ~( ) ( ) ( ) g g t u t t= ,(2.4.17) vi( ) 1 u t =nu0 t >v( ) 0 u t =nu0 t < .Tngt(2.3.36),nghimcaphngtrnhviphnkhngngnht(2.4.1)c tnh qua HGg+ hoc ~g nh sau : ( , ) ( , ) ' ' ( , ', ') ( ', ')tr t r t dr dt g rr t t f r t m o+= + (2.4.18) 21 2.4.2Phng trnh vi phn cha o hm bc II v Hm Green Phn ny trnh by tm tt phng php HG ca phng trnh vi phn cha o hm bc hai theo thi gian. Xt phng trnh khng ng nht 22 21( ) ( , ) ( , ) r rt f rtc tmc = c

L(2.4.19) v phng trnh ng nht 22 21( ) ( , ) 0 r rtc toc = c L .(2.4.20) HG tng ng tha phng trnh 22 21( ) ( , ', , ') ( ') ( ') r grr t t r r t tc to oc = c L (2.4.21) trong cc hm( , ', , ') grr t t

,( , ) rt m

v( , ) rt o

tha cng iu kin bin trn mt S baor

v' r

. tm( , ', , ') grr t t

ta dng phpbin i Fourier chuyn phng trnh(2.4.21) sang dng khng ph thuc thi gian. Ch n s ph thuc ca HG vo t-t , ta c: 1( ) ( )2ig d e gett e et=.(2.4.22) Thay vo (2.4.21) thu c : 22( ) ( , ', ) ( ') r grr r rcee o = L .(2.4.23) Nh vy 2 2( ) ( / ) g G c e e = (2.4.24) vi( ) G l nghim ca (2.3.26) vi 2 2/ c e = . Tng t( ) G , HG( ) g e cng khng gii tch ti nhng im trn trc thc c 2 2/ c e trngvitrringEcaL .Nu0 E < thccccca( ) g e lphc.Tuy nhin ta ch quan tm n trng hp cc im k d thuc trc thc.Ngi ta cng nh ngha cc hm gii hn tng t (2.4.7) v ty vo ng ly tch phn chn ra nhng hm c ngha vt l nh sau : * HG nhn qu : 2 201( ) lim ( / )2c ig d e G c ietct e e ct = +(2.4.25) * HG tr: 22 2 201( ) lim ( / Sign( ))2r ig d e G c ietct e e c et = +(2.4.26) * HG sm: 2 201( ) lim ( / Sign( ))2a ig d e G c ietct e e c et = (2.4.27) Trong 1Sign( )1e+ = ,, 00ee >< Ngoi ra ta cng nh ngha hm ~g ~( ) ( ) ( )r ag g g t t t = (2.4.28) v tm c : ~sin( )( ) | |nn nnnc Eg cEtt o o = ,(,(2.4.29) hoc di dng ton t : ~sin( )( )cg ctt = LL vi' t t t = .(2.4.30) Nghim ca phng trnh vi phn khng ng nht c biu din qua HG nh sau : ~( , ) ( , ) ' ' ( , ', ') ( ', ')( , ) ' ' ( , ', ') ( ', ')trtr t rt dr dt g rr t t f r trt dr dt grr t t f r tm oo= + = + ,(2.4.31) vi( , ) rt o

l nghim ca phng trnh ng nht tng ng. Bi tp Bng phng php HG hy tm dch chuyn( ) uxca si dy (hai u c nh) di tc ng ca lc phn b( ) f xcho trc.Phng trnh xc nh( ) uxcho bi22( )( )duxf xdx=, (0,1) xevi iu kin bin : (0) 0 u =v(1) 0 u =23 Chng 3 3Hm Green trong Vt l lng t HG trnh by chng trc ch thun li cho bi ton mt ht trong trng (lc) ngoi, trong ta c th xc nh HG trc tip qua tr ring v hm ring (ca ton t L -v d l ton t Hamilton H). Trong h nhiu ht thc t cn phi xt n tng tc gia cc ht v khi v nguyn tc khng xc nh c tr v hmring ca H. Do cn mt hnh thc lun HG m rng hn, d nhin thng l gn ng, khng da vo thng tin trc tip ca hm v tr ring.Chng ny s dn ra HG c xy dng trn ton t trng trong lng t ha ln II. chun b vic dn xut cho HG nh th, trc ht chng ta s trnh by li mt s kin thc v cc biu din v S-ma trn. 3.1Cc biu din : mtsphthucthigiancacchvtl,trongclngtngita thng dng mt trong ba biu din, cng thng gi l bc tranh, tng ng sau:biudinSchrdinger,biudinHeisenberg,biudintngtc(cngilbiu din Dirac).3.1.1Biu din Schrdinger: Trong bc tranh Schrdinger (mt s ti liu cn thm tn gi khc l bc tranh trng thi)ton t c lp vi thi gian : Hhm sng hay vector trng thiph thuc vo thi gian :| ( )St v ,T c lng t ta c phng trnh chuyn ng :24 -i vi nhng trng thi thun khit| ( )St v , trong h c lng t: | ( ) | ( )S Si t H tt v vc, = ,ch(3.1.1) -v i vi nhng trng thi pha trn ( )St p(dng cho h v m c m t mt cch thng k): | |( ) ( ),S Si t t Ht p pc=ch .(3.1.2) ( )St pthng c gi l ma trn mt v c nhng tnh cht sau: i)| |S m m mm p p v v = ,(,(3.1.3) vi mpl xc sut h trong trng thi|mv , ,ii) gi tr k vng ca i lng kho stA (,tng ng vi ton t A c xc nh( )SA Tr A p (, = (3.1.4) ( ( ) TrA : vtcaA. i khi ngi ta cng dng k hiu Sp thay cho Tr), viii) ( ) 1STr p = .(3.1.5) Phng trnh (3.1.1) cho nghim hnh thc : ( ')0| ( ) | ( )iHt tS St e t v v , = ,h(3.1.6) vi 0t l thi im c nh no v ngi ta thng chn 00 t = . 3.1.2Bc tranh Heisenberg: Bc tranh Heisenberg (i lc cn c tn gi bc tranh ton t) m t bi ton c lng t bng cch xthm sng (vector trng thi) khng i theo thi gian :|Hv , ,trong khi ton t ph thuc thi gian : O(t).S ph thuc vo thi gian ca ton t c xc nh nh sau 0 0( ) ( )0( ) ( )i iHt t Ht tH HA t e A t e =h h(3.1.7) hoc chn 00 t =: ( ) (0)i i i iHt Ht Ht HtH H SA t e A e e A e | |= = |\ .h h h h.(3.1.8) Ta cng c th vit li (3.1.8) di dng phng trnh chuyn ng : | |( ) ( ), i A t A t Htc=ch .(3.1.9) 25 StngnggiabctranhSchrdingervHeisenberg :hyxtphntma trn m t s chuyn di gia hai trng thi 1 v 2 :- trong bc tranh Schrdinger phn t ma trn ca ton t A l : 1 2 1 2( ) (0) ( ) (0) (0) (0)i iHt Htt A t e A e v v v v+ +( , = ( ,h h,(3.1.10) - trong bc tranh Heisenberg ta c : 1 2 1 2(0) ( ) (0) (0) (0) (0)i iHt HtA t e A e v v v v+ +( , = ( ,h h.(3.1.11) Nh vy hai bc tranh cho cng phn t ma trn- chng tng ng nhau.3.1.3Biu din tng tc (biu din Dirac): Biu din tng tc c xem nh lai trung gian ca hai biu din trn. Trong biu din ny ton t v vector trng thi u ph thuc thi gian.Xt ton t Hamilton H ca h. Ta chia ton t H thnh hai phn : 0H H W = + .(3.1.12) y ton t 0Hl phn khng nhiu lon v thng c chn sao cho c th gii chnh xc. W l th tng tc ph thuc thi gian. S ph thuc thi gian ca hm sng v ton t trong bc tranh tng tc c xc nh nh sau : 0 0( ) (0)i iH t H tIAt e A e=h h,(3.1.13) 0| ( ) | (0)i iH t HtI It e e v v, = ,h h.(3.1.14) i vi vector trng thi, ta vit li (3.1.14) di dng phng trnh chuyn ng : 00 0 00( )| ( )| ( ) ( ) | (0)| (0)Ii iHt HtI Ii i i iHt Ht Ht HtIVttt ie H Heti e Ve e evv vv ,c, = ,c= ,h hh h h h.. Nh vy | ( ) ( ) | ( )I Ii t Vt tt v vc, = ,ch .(3.1.15) Phngtrnhchuynng(3.1.15)chothythtngtcphthucthigianW(t) quy nh s thay i theo thi gian ca vector trng thi, trong khi s ph thuc thi gian ca ton t c quyt nh bi phn Hamilton khng nhiu lon 0H . 26 Ch rng vi tnh ph thuc thi gian (3.1.13) v (3.1.14), nu xt phn t ma trn chuyndigiahaitrngthi 1 2( ) ( ) ( ) t A t t v v+( , ,stmthystngnggia biu din tng tc v hai biu din trn. Ton t tin ha : Trong (3.1.14) t 0( )i iH t HtUt e e=h h,(3.1.16) ta c| ( ) ( ) | (0)It Ut v v , = , .Nhvy( ) Ut lmttonttinha thigian:di tc ng ca nvector trng thiti 0 tin sang trng thi tit. Ton t nyc c tnh :(0) 1 U =v tun theo phng trnh( ) ( ) ( ) i Ut Wt Uttc=ch .(3.1.17) Nghim ca (3.1.17) l1 1 10( ) (0) ( ) ( )tiUt U dt Wt Ut = h hoc 1 1 10( ) 1 ( ) ( )tiUt dt Wt Ut = h.(3.1.18) Bng php tnh lp cho (3.1.18) ta c th biu din( ) Utthnh chui v hn: 121 1 1 2 1 20 0 0( )1( ) 1 ( ) ( ) ( ) ...1 ( )t t tnniUt i dt W t dt dt W t WtU t=| |= + + |\ .= + h(3.1.19) trong 1 1( )1 2 1 210 0 0( ) ... ( ) ( )... ( )nnt t tnn nniU t dt dt dt Wt Wt Wt =| |= |\ . h(3.1.20) (1 2 0...nt t t t t > > > > ). Vi ton t thi trnh (ton t trnh t thi gian) T c nh ngha | |1 21 22 1( ) ( ),( ) ( )( ) ( ),Wt WtTWt WtWt Wt= 1 22 1t tt t>>(3.1.21) hoc | |1 2 1 2 1 2 2 1 2 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) TWt Wt t t Wt Wt t t Wt Wt u u = + (3.1.22) (ul hm bc nhy), ta c th vit li chui ca ton t tin ha nh sau : 27 1 110 01( ) 1 ... [ ( )... ( )]!nt tn nniUt dt dt T Wt Wtn=| |= + |\ . h.(3.1.23) Chui trn l khai trin ca hm e ly tha v nh vy ta c dng vit gn ca ton t tin ha : 1 10( ) exp ( )tiUt T dt Wt = h.(3.1.24) 3.2S-ma trn : Smatrncmilinhtrctipvitonttinhavakhosttrnvctngquanchtchvibinxcsutchuyndi,vithitdintnxcacc tng tc khc nhau. S ma trn c nh ngha nh l ton t thay i vector trng thi| ( ')It v ,thnh trng thi| ( )It v ,:| ( ) ( , ') | ( ')I It St t t v v , = , .(3.1.25) S ma trn c nhng tnh cht saui)( , ') ( ) ( ') St t Ut U t+= (3.1.26) (S dng nh ngha ton t tin ha| ( ) ( ) | (0)I It Ut v v , = ,cho| ( ')It v ,trong (3.1.25) ta c| ( ) ( , ') ( ') | (0)I It St t Ut v v , = , . T y ta c mi lin h gia S v U) ii) ( , ) 1 St t =iii)( , ') ( ', ) S t t St t+=iv)( , ') ( , '') ( '', ') St t St t St t =v) S tha phng trnh : ( , ') ( ) ( ') ( ) ( ) ( ') ( ) ( , ')i iSt t Ut U t Wt Ut U t Wt St tt t+ +c c= = = c c h h,(3.1.27) vi)V nh vy, tng t ton t tin ha, ta c th biu din S di dng nh sau : 1 1'( , ') exp ( )ttiS t t T dt Wt = h.(3.1.28) H thc Gell-Mann v Low. Trong phng trnh xc nh s bin i theo thi gian ca hm sng| ( )It v ,(3.1.14) cng nh trong cc dn xut ra ton t tin ha v S 28 ma trn chng ta a ra hm sng ti0 t = | (0)Iv , . Ch rng y cng chnh l hmsngkhngphthucthigian|Hv , trongbiudinHeisenbergvlhm sngti0 t = | (0)Sv , trongbiudinSchrdinger ;vvvytagichungl | (0) v , . Ta cn phi tm ra mi lin no gip ta xc nh| (0) v , v n nh l iu kin ban u t c th xc nh trng thi ti nhng thi im khc.Trongbiudintngtc,tontHamiltonctchralmhaiphnWv 0H . Trong phn khng nhiu lon 0Hc chn sao cho c th xc nh chnh xc h hm ring v tr ring ca n. Gi s trng thi c bn ( bit) ca 0Hl 0| o , . Khi mi lin h gia 0| o , v trng thi cha bit| (0) v , c Gell-Mann vLow kho st v ch ra : 0| (0) (0, ) | S v o , = , .(3.1.29) tmhiuhthcny,tahydnra| (0) v , tnhnghaSmatrn :tcng (0, ) S t lnphngtrnh| ( ) ( , 0) | (0) t S t v v , = , vsdngtnhchtiv)vii)caS ma trn, ta c : | (0) (0, ) | ( ) S t t v v , = , . Ly gii hnt cho| (0) (0, ) | ( ) S v v , = , .(3.1.30) So snh (3.1.30) vi (3.1.29) tac 0| ( ) | v o , = , . V ta din t h thc Gell-Mann v Low da trn gn ng on nhit nh sau: qu kh rt xa, lc khng c tng tchtrngthi 0| ( ) | v o , = , .Khithigiantngdnt,thtngtcW c bt m v c a dn dn vo h cho n khi0 t =th tng tcc a vo hon ton vi hm sng tng ng l| (0) v , . Ni cch khc ton t(0, ) S chuynmtcchonnhithmsng 0| ( ) | v o , = , nthiimt=0tith tngtcWhindinhonton,vnhvyhmsngtngng| (0) v , tit=0 chnh l hm ringca Hamilton ton phn H.Hy xt thm trng hp h tin v tng lai xa,t + : th tng tc s c tt i mt cch on nhit v ti+ khng cn tng tc, h s tr v trng thic bn | ( ) v , , (gi thit rng) c lin h vi trng thi c bn 0| o , bng mt tha s pha : 0| ( ) |ieov o , = , . Nh vy ta c : 0 0| | ( ) ( ,0) | (0) ( , ) | ( ) ( , ) |ie S S Soo v v v o , = , = , = , = ,(3.1.31) v0 0| ( , ) |ie Soo o = ( , (3.1.32) 29 3.3Hm Green : TrongphnnychngtasgiithiuHGtrongvtllngtvnhni, chng ta gii hn trng hp nhit 0. 3.3.1nh ngha Hm Green fermion: Chng ta bt u vi nh ngha HG electron : ( , ') | ( ) ( ') |iG t t Ta t a tvv v+= ( ,h .(3.1.33) yv lslngtvthngbaogmvectorsngk

vspino:( , ) k v o =

. |, ltrngthicahvnhit0trngthinyphiltrngthicbn;nu Hamilton ca h l H th|,l trng thi ring ca H. Cc ton t ph thuc thi gian ( ), ( ) a t a t+ trong biu din Heisenberg v tin ha theo qui lut : / // /( )( )iHt iHtiHt iHta t e a ea t e a ev vv v+ + ==h hh h(3.1.34) vi( 0), ( 0) a a t a a tv vv v+ + = = =m t trng thi ca 0H . T l ton t thi trnh: t ton t c bin thi gian sm hn v bn phi (hoc d nh : tr sang tri) v ch i vi ton t fermion cn thm du tr cho mi ln hon v; nh vy ta c th vit li nh ngha HG nh sau : ( , ') | ( ) ( ') |iG t t a t a tvv v+= ( ,h nu' t t > ,(3.1.35) ( , ') | ( ') ( ) |iG t t a t a tvv v+= + ( ,h nu' t t > ,(3.1.36) hoc( , ') ( ') | ( ) ( ') | ( ' ) | ( ') ( ) |i iG t t t t a t a t t t a t a tv vv v vu u+ += ( , + ( ,h h,(3.1.37) vi hm ul hm bc nhy. ngha ca HG : t nh ngha (3.1.35) ( ' t t > ) ta thy HG m t qua 1 trnh trong mthtctorati' t ritruynnt vbhy.Trongtrnghp' t t > , (3.1.36), HGmt qu trnh hymtht (tc l to 1 l trng) titsau ht li c to ra (l trng b hy) ti' t . Nh vy HG m t qu trnh truyn ht (l trng) v v th thng gi l hm truyn. 30 S dng (3.1.34) ta c ' '( ') ( ')( , ') | || |i i i iHt Ht Ht Hti iEt t Ht tiG t t e a e e aeie a e avvv vv + += ( ,= ( ,h h h hh hhh(3.1.38) tronggisEtrringngvitrngthicbn cahviHamiltonH: | | H E , = , .Biu thc (3.1.38) cho thy HG ph thuc vo' t t . Tng t ta cng c tnh cht ny i vi trng hp' t t > . Ngoi HG (nhn qu hay cn gi HG theo trt t thi gian) va nh ngha trn, ngi ta cng a ra cc HG tr, sm, nh v ln nh sau : ( , ') ( ') | [ ( ), ( ')] |( , ') ( ' ) | [ ( ), ( ')] |( , ') | ( ') ( ) |( , ') | ( ) ( ') |raiG t t t t a t a tiG t t t t a t a tiG t t a t a tiG t t a t a tvvvvv vv vv vv vuu++< +> += ( ,= ( ,= ( ,= ( ,hhhh(3.1.39) Thc ra khi xt h cn bng, v nguyn tc, ta ch cn mt HG, v dHG nhn qu(3.1.37), v tt c cc HG (3.1.39) c th biu din qua n. Tuy nhin ty trng hp m mi HG c u im ring v v th i khi ngi ta vn s dng chng : i) ,( , ')r aG t tvccutrcgiitchtt,chathngtinvtnhchtph,mttrng thi,. ii) ,( , ') G t tv< >cmilinhtrctipvinhngtnhchtnghcnhmt ht/hm phn b, dng.c bit, nu xt h khng cn bng th tt c cc HG trn u quan trng v khng th b qua. Ngi ta cng thng s dng nh ngha HG nhit 0 nh sau : 0 00 0| ( , ) ( ', ') |( , ; ' ')|T x t x tiG x t x tu vuvv v +(+ + ,= (+ + , h(3.1.40) Khng t sinh vin gp lng tng trong tham kho ti liu khi gp cc nh ngha khc nhau ca HG. nh ngha HG ny c gii thiu cng nhm gip sinh vin lm quen vi nhng tip cn khc nhau.31 trong, u v lslngtspin ;trongtrnghphkhngnhhngtv khng trong t trng thu v = . 0| + ,l trng thi c bn ca h bao gm tng tc.Taxthngnht,khiHGphthucchvohiuccbinx-x,t-t. , v v +l nhng ton t trng (trong biu din Heisenberg), chng c th trinkhai chuyn sang ton t sinh hy, a a+ : *( , ) ( ) ( )( , ) ( ) ( )k k kk k kx t x a tx t x a tuu uuu uv mv m+ +==

(3.1.41) vi ml hm sng (mt ht), trong gn ng n gin nht l sng phng: ( )ikxkexVuum o=

,(3.1.42) uol ch s spin. Khai trin (3.1.41) chnh l php bin i Fourier chuyn t khng gianx

sang khng giank

. Gi s trng thi c bn chun ha, tc l 0 0| 1 (+ + , = , khi HG trong khng gian Fourierk

( ')( ') ( ', ')ik x xkG t t dxe G x x t tuu =

(3.1.43) tng ng vi HG trong (3.1.33). Php bin i ngc li l ( ') ( ')31 1( ', ') ( ') ( ')(2 )ik x x ik x xk k kG x x t t e G t t dke G t tVuu ut =

(3.1.44) Ngi ta cng thng dng HG ph thuc tn s qua php bin i Fourier theo thi gian ( ')( ) ( ')i t tk kG dte G t teu ue= (3.1.45) v ngc li ( ')1( ') ( )2i t tk kG t t d e Geu ue et = .(3.1.46) Vi ton t thi trnh T ta c th vit li (3.1.40) nh sau 0 00 00 00 0| ( , ) ( ', ') |( , ; ' ') ( ')|| ( ', ') ( , ) |( ' )|x t x tiG x t x t t tx t x tit tu vuvv uv vuv vu++(+ + ,= (+ + ,(+ + ,+ (+ + ,hh(3.1.47) 32 HG trong biu din tng tc. HG trong biu din Heisenberg khng thch hp cho l thuyt nhiu lon. Ta s chuyn HG sang biu din tng tc. p dng php bin i theo thi gian (3.1.13) cho cc ton tav vav+ trong (3.1.34) ta c 0 00 0 ( ) ( ) ( ) ( )i i i iHt Ht Ht Hti i i iHt Ht Ht Hta t e e a t e ea t e e a t e ev vv v + +==h h h hh h h h(3.1.48) Vi a ltonttrongbiudintngtc.Sdngnhnghatonttinha (3.1.16) 0( )i iH t HtUt e e=h hvtnhchti)caSmatrn( , ') ( ) ( ') St t Ut U t+= chokt qu ( ) ( ) ( ) ( ) (0, ) ( ) ( , 0) ( ') ( ) ( ') ( ') (0, ') ( ') ( ', 0)a t U ta t Ut S ta t S ta t U ta t Ut S t a t S tv v vv v v++ + + += == =(3.1.49) Gi 0| o ,l trng thi c bn ca Hamilton khng nhiu lon 0H , khi trng thi c bn ca Hamilton ton phn Hcho bi (3.1.29)0| (0, ) | S o , = , . Thay kt qu ny v (3.1.49) vo (3.1.37) ta c : 0000 ( , ') ( ') | ( , 0) (0, ) ( ) ( , 0) (0, ') ( ') ( ', 0) (0, ) | ( ' ) | ( , 0) (0, ') ( ') ( ', 0) (0, ) ( ) ( , 0) (0, ) |iG t t t t S S t a t StS t a t St Sit t S S t a t StS t a t St Svvv vvu oou oo++= ( ,+ ( ,hh(3.1.50) V 0 0(0, ) | (0, ) |iS S eoo o , = ,v vi (3.1.32) ta c00 00 0| ( , 0)| ( , 0) | ( , 0)| ( , ) |iSS e SSooo oo o( ( = ( =( ,. S dng ng thc ny v cc tnh cht ca S ma trn, HG (3.1.50) c dng : 0 00 00 01( , ')| ( , ) | [ ( ') | ( , ) ( ) ( , ') ( ') ( ', ) | ( ' ) | ( , ') ( ') ( ', ) ( ) ( , ) | ]iG t tSt t S ta t St t a t Stt t S t a t St ta t Stvvvvvo ou o ou o o++= ( , ( , ( ,h(3.1.51) S dng ton t thi trnh T cho phn trong [], v d s hng u, ta c 0 00 0 ( ') | ( , ) ( ) ( , ') ( ') ( ', ) | ( ') | ( ) ( ') ( , ) |t t S ta t St t a t Stt t Ta ta t Svvvvu o ou o o++ ( ,= ( , 33 Tht vy, ton t( , ) S gm ba ton t tc ng trn ba khong thi gian( , ) t , ( , ') t tv ( ', ) t . Ton t thi trnh T s t ng sp xp chng theo ng trnh t cnthit :ittri(tr :t)sangphi(sm :t)th( , ) S t bntri( ) a t ,rin ( , ') St t , k l ( ) a t+ v sau cng l( ', ) S t . Nh vy ta c c dng n gin ca HG trong biu din tng tc : 0 00 0 | ( ) ( ') ( , ) |( ')| ( , ) |Ta t a t SiG t tSvvvo oo o+( , = ( , h .(3.1.52) Xt trng hp c bit khng c tng tc, W=0, v v th S=1. Khi HG c dng 0, 0 0 ( ') | ( ) ( ') |iG t t Ta t a tvv vo o+ = ( ,h.(3.1.53) HG nym tht khng tng tc v thng c gi l HG t do hayHG khng nhiu lon. N c vai tr quan trng v qua n ta c th tnh cc HG c tng tc nh s thy trong phn khai trin nhiu lon ca HG.3.3.2Phng trnh chuyn ng ca HG : By gi ta ly o hm HG theo thi gian t dn ra phng trnh chuyn ng ca HG. Chng ta chn HG c nh ngha trong khng gian x (3.1.47) v mun c ssosnhviphngtrnhHGtrongbitonphngtrnhviphnkhngng nht. Vi( ') / ( ') t t t t t u o c c = v( ' ) / ( ') t t t t t u o c c = v gi s trng thi c bn 0| + ,chun ha cng nh tm b qua cc ch s spin ta c 0 00 0( , ; ', ') ( ') | [ ( , ), ( ', )] |( , )| ( ', ') |i Gx t x t t t x t x tti x tTi x tto v vvv++c= (+ + ,cc (+ + ,chhh(3.1.54) Trong (3.1.54) ta dng ton t T gp cc s hng cho gn. S dng h thc phn giao hon[ ( , ), ( ', )] ( ') x t x t x x v v o+= cho ta 0 0( , )( , ; ', ') ( ') ( ') | ( ', ') | .i x ti Gx t x t t t x x Ti x tt tvo o v +c c= (+ + ,c ch hh(3.1.55) Ch vi s hng u bn v phi ta c c phng trnh HG tng ng vi phng trnhviphntuyntnhkhngngnhtkhosttrongchngtrc.Nhng 34 nhn chung s hng th hai trong v phi khc 0 v cng thng phc tp ; v vy ni chungHGtrongvtllngtkhngcnghatonhcnhHGtrongphng trnh vi phn tuyn tnh.Ta hy xt trng hp (c bit) ht t do. Phng trnh chuyn ng ca hm sng cho bi phng trnh Heisenberg : 2 2( , )[ ( , ), ] ( , )2xx ti x t H x tt mvv vV c= = chh .(3.1.56) Thay vo phng trnh (3.1.55) ta c phng trnh chuyn ng cho HG t do 0G: 2 20( , ; ', ') ( ') ( ')2i G x t x t t t x xt mo oc V+ = c hh.(3.1.57) Nh vy phng trnh vi phn cho HG ht t do l phng trnh Schrdinger khng ng nht, thuc lp phng trnh vi phn tuyn tnh khng ng nht ph thuc thi gian (qua o hm bc I) chng ta bn ti trong chng 2.3.3.3Mi lin h gia HG v mt s i lng vt l : Cu hi thng c t ra : nhng HG tru tng trn c vai tr nh th no i vi cc i lng c th o lng (trong thc nghim) ? Di y ta s ch ra mt s tng quan in hnh gia HG v cc i lng c th kho st trong thc nghim.

Mt ht c cho bi( ) ( ) ( ) nx x x v v+( , = ( , . Ta thy ngay i lng ny c lin h trc tip vi HG (xem (3.1.40)): ( ) ( , ; , ) nx i Gx t x t+( , = h (3.1.58) trong 0lim( ) t too++= + bo m theo ng trt t thi gian.Gi tr k vng (expectation value) ca ng nng T v nng lng ton phn E c xc nh nh sau [Fetter v Walecka (1971)] : 2 22 20 00 02 2' '| ( , ) ( , ) |22 |limlim ( , ; ', ')2x x t tx t x tmT dx dxmi dx Gx t x tmv v++ V(+ + ,V( , = =(+ + , V= hhhh (3.1.59) 2 2' 'limlim ( , ; ', ')2 2x x t tiE H dx i Gx t x tt m+ c V= ( , = c h hh;(3.1.60) chuyn sang khng gian Fourier vi (3.1.44) v (3.1.46)35 2 2( ') ( ')4' '2 2401limlim ( )(2 ) 2Vlim ( ) ,(2 ) 2i t t ik x xkx x t tikT i dx dkd e e Gmki dk d e Gmeecce ete et++ V( , = =

hh

hh (3.1.61) 2 2( ') ( ')4' '2 2401limlim ( )(2 ) 2Vlim ( ) .(2 ) 2i t t ik x xkx x t tikE i dx dkd i e e Gt mki dk d e Gmeecce ete e et++ c V= c = +

hh h

hh h(3.1.62) Ngoi ra ngi ta cng c th tnh mt spin, mt dng ht, qua HG. 36 Chng 4 4Khai trin nhiu lan ca hm Green HG trong biu din tng tc (3.1.52) ph thuc vo S-ma trn( , ) S ; ma trn ny l chui v hn ca tng tc W: 1 11 11( , ) exp ( )11 ... [ ( )... ( )]!nn nniS T dt Wtidt dt T Wt Wtn= = | |= + |\ .hh(3.1.63) Nh vy HG cng l mt chui v hn ca nhng s hng ph thuc vo th tng tc.4.1Khai trin nhiu lon v nh l Wick kho st HG, trc ht ta vit t s ca HG di dng chui ny, khi : 11 10 | ( ) ( ') ( , ) |( ')| ( , ) |1 1 ... | [ ( )... ( ) ( ) ( ')] || ( , ) | !nn nnTa ta t SiG t tSidt dt T Wt Wt a ta tS nvvvvv+++=( , = ( ,| |= ( , |( ,\ .hh(3.1.64) Nh vy ta c mt phc nhiu lon cho HG.Chng ta hy xt ring phn t s ca HG. Phn t cn quan tm l 1 | [ ( )... ( ) ( ) ( ')] |nT Wt Wt a t a tvv+( ,(3.1.65) c cu trc c xc nh qua th tng tc. Hy xt hai tng tc in hnh l tng tc e-e (qua th Coulomb) v tng tc e-phonon.Tng tc ht tch in qua th tng tc Coulomb c dng: 0 ', ,1( ) ( ) ( ) ( ) ( ) ( )2k q k q k kkk qWt V q a t a t a t a t+ + +'=, (3.1.66) vi 20 204( )eV qv qtc= . Tng tc e-phonon: , ,( ) ( ) ( )[ ( ) ( )] ( ) ( ) ( )q k q k q q q k q k qk q k qWt M a t a t b t b t M a t a t B t+ + ++ += + = (3.1.67) 37 Mi loi tng tc cha 3 hay 4 tan t trng, trong s ton t trng fermion l chn v s ton t sinh bng s ton t hy. V vy ta cn nh gi nhng s hng kiu nh 1 2 31 2 3| [ ( ) ( ) ( ) ( )] |k k k kT a t a t a t a t+ +( , ,(3.1.68) v cc s hng tng t bc cao hn. lm vic ny ngi ta dng nh l Wick: Biu thc (3.1.68) (trung bnh T-tch (tc ton t thi trnh)) l tng cc bt cp kh d.Ch rng: i)nhng bt cp ca tan t sinh vi sinh hoc hy vi hy khng c ng gp: 2 2| [ ( ) ( )] | | [ ( ) ( )] | 0 T a t a t T a t a tu v u v+ +( , = ( , = , dubng hoc khc v .ii)v do vy ch cn nhng cp gia 1 ton t sinh v mt tan t hy kiu nh 22| [ ( ) ( )] |k kT a t a t+( , ; chng s ch khc khng khi 2k k = , tc l: 222 2,| [ ( ) ( )] | | [ ( ) ( )] |k k k kk kT a ta t T a ta t o+ +( , = ( , . V nh vy nu c n ton t sinh v n tan t hy, ta c n! cp. iii)| ( ) | 0kA t ( , =v v th tt c cc T-tch c s l cc tan t u bng khng. Do , vi tng tc e-phonon, trong chui (3.1.64) ch nhng s hng vi n chn tn ti, nhng s hng vi n l trit tiu v s ton t trng ca phonon l l. Ngai ra, cn mt s nguyn tc sau: a)Mi ln hon v tan t fermion: nhn thm (-1) nu s ln hon v l l th thm du tr b)Khi xt T-tch ca cp gia tan t sinh v hy 22| [ ( ) ( )] |k kT a t a t+( ,-nu hai tan t cng thi gian th tan t sinh c t bn tri:2 2 2 20, , , ,| [ ( ) ( )] | [ ( , )]k k k kk k k k k k k kTa ta t iG t t n n o o o o+ +( , = = ( , =-nu khc thi gian th ton t sinh c t bn phi:22 202 2, ,| [ ( ) ( )] | ( )k k kk k k kTa ta t i G t t o o+( , = c)T-tch cha cc tan t khc loi (v d electron v phonon)v giao han nhau th c th tch thnh cc T-tch ca mi lai:2 1 1 22 1 3 3 1 2 3 3| [ ( ) ( ) ( ) ( )] | | ( ) ( ) | | ( ) ( ) |k k k k k k k kT a t B t a t B t Ta t a t TB t B t+ +( , = ( ,( , 38 Vi nh l Wick, ta hy kho st HG trong trng hp tng tc e-phonon. S hng vi n=1 bng khng, nh ni trn. T s ca HG c dng: 1 2 1 2 1 1 1 2 2 21 2 1 2301 21 2 1 1 2 2( ) ( )( )( ')2!| ( ) ( ) | | ( ) ( ) ( ) ( ) ( ) ( ') |...kq q q q k k q k k q k kq q k kphonon electroniG t t dt dtMM TB t B t Ta t a t a t a t a t a t+ + + ++ + + ( , ( ,+ . . (3.1.69) Phn T-tch ca ton t phonon cho: 1 2 1 2 101 2 1 2( ) | ( ) ( ) | ( )q q q q qphonon i TB t B t D t t o+= ( , = (3.1.70) Phn T-tch ca cc tan t electron cho 6 t hp ca cc T-tch ca cc cp: ( )( )1 1 1 2 2 21 1 1 2 2 22 2 2 1 1 11 1 2 21 1 2 22 2 1 1( ) | ( ) ( ) ( ) ( ) ( ) ( ') || ( ) ( ) | | ( ) ( ) | | ( ) ( ') || ( ) ( ) | | ( ) ( ) | | ( ) ( ') | 2|1k k q k k q k kk k q k k q k kk k q k k q k kkelectron Ta t a t a t a t a t a tTa t a t Ta t a t Ta t a tTa t a t Ta t a t Ta t a tTa+ + ++ ++ + ++ ++ + ++ += ( ,= ( , ( , ( ,+( , ( , ( ,+( ( )( )( )1 1 1 2 2 21 1 1 2 2 22 2 1 1 1 21 1 2 21 1 2 22 1 1 2( ) ( ) | | ( ) ( ') | | ( ) ( ) || ( ) ( ') | | ( ) ( ) | | ( ) ( ) || ( ) ( ) | | ( ) ( ) | | ( ) ( ') |3| ( )45k q k k k q kk k k q k k q kk k q k q k k kk kt a t Ta t a t a t a tTa t a t a t a t a t a tTa t a t a t a t Ta t a tTa t aTT TT+ + ++ ++ + ++ ++ + ++ +, ( , ( ,+( , ( , ( ,+( , ( , ( ,+( ( )1 2 2 2 1 11 2 2 1( ') | | ( ) ( ) | | ( ( 6 ) ) |k k q k k qt Ta t a t Ta t a t+ + ++ +, ( , ( ,(3.1.71) Hoc chuyn i sang HG t do v ton t s ht ta c: ( )( )( )( )( )1 1 2 12 1 1 11 1 21 2 1 21 2 123 0 0 01 1 2 23 0 0 02 2 1 12 0 00 1 100 02 0 00 2 23( ) ( ) ( ')( ) ( ) ( ')( ) ( ')( '1)( ) ( ')2345k k q k k k q kk k q k k k q kq k k k k kq q k k kq k k k k kki G t t G t t G t ti G t t G t t G t ti nG t t G t ti nnG t ti nG t t G t tiooo oo oo oo= + = = = += == == = + + + + ( )1 1 1 1 10 0 01 2 2 1( ') ( ) ( 6 )q k k k k qG t t G t t G t t= + (3.1.72) 4.2Gin Feynman Phng php gin trong khai trin chui ca HG c Feynman xut v t ra rt hu ch gip ta c mt ci nhn c th v trc quan v qu trnh tng tc. Phng php ny theo quy tc nh sau: 39 i.HG fermion 0( ')kG t t c biu din bng ng linnt vimi tn i t ' t ntii.HGphonon(bosonvhng) 0( ')qD t t cbiuthbngngcchnt ni' t vt . HG boson vect c biu th bng ng sng ni' t vt . iii.Th Coulomb c biu din bng ng sng hoc ziczac ni' t vt . iv.Mt n c biu th bng vng kn bt u v kt thc ti cng mt im v.Bo tan xung lng ti mi nh c th hin qua hmo 40 4.3Phng trnh Dyson 4.3.1Kin thc h tr: -Hamiltonian c bn:e ion e ion exH H H H H= + + + (3.1.73) 0e e e eH H H = + (3.1.74) 0 0 0ion ion ion ion ion ion ion phH H H H H H = + = + +(3.1.75) 0e ion e ion e phH H H = + (3.1.76) Phn ion ionH c tch thnh 2 thnh phn: 0ion ionHm t tng tc khi cc ion v tr cn bng; phH l phn hiu chnh tnh n dao ng mng (phonon). Cng th i vi e ionH -M hnh jellium: dnh cho kh electron. Trong m hnh ny ngi ta xt cc ion nh mt nn in tch dng khng i. Khi tt c cc tng tc lin quan n mng ion c biu th qua th mng ion LU : 0e L exH H W U H = + + + (3.1.77) -Hamilton kh electron 1 1 1 11 2 1 2 1 2 2 1( ) ( , ) ( , )( ( , )1( , ) ( , ) ( ) ( , ) ( , )2Ht dx x thx t x tdx dx x t x t Vx x x t x tv vv v v v++ +=+ (3.1.78) y 2 201 1 1 1 1 1( , ) ( ) ( , ) ( , ) ( , ) ( , )2e ex L ex Lhx t H x H x t U x t H x t U x tm V= + + = + +h(3.1.79) v 1 2( ) Vx x l th Coulomb. 4.3.2Phng trnh Dyson trng hp nhiu lon khng ph thuc thi gian: (tham kho, v d HG ca Nguyn Vn Lin trang 93-95, hoc Greens Functions in quantum Physics ca Economou, trang 55-56) Xt trng hp Hamiltonian 0H H W = +khng ph thuc thi gian, ta c c: 0 00 0G G GWGG GWG= += +(3.1.80) y c xem nh phng trnh Dyson cho HG.Trong r-biu din: 0 1 2 0 1 1 2 2( , ', ) ( , ', ) ( , , ) ( , ) ( , ', ) G r r Gr r drdr Gr r Wr r G r r = +(3.1.81) 41 v trong k-biu din: 0 0 1 1 2 2( , ', ) ( , ', ) ( , , ) ( , ) ( , ', ) G k k Gk k Gk k Wk k G k k = +(3.1.82) 4.3.3Phng trnh Dyson trng hp nhiu lon ph thuc thi gian: -Phng trnh chuyn ng ca HG (3.1.55): 0 0( , )( , ; ', ') ( ') ( ') | ( ', ') | .i x ti Gx t x t t t x x Ti x tt tvo o v +c c= (+ + ,c ch hh(3.1.83) vi ( , )[ ( , ), ( )]x ti x t Httvvc=ch .(3.1.84) S dng Hamiltonian (3.1.78) ta c: 2 2 2 2[ ( , ), ( )] ( , ) ( ) ( , ) ( , ) ( , ) x t Ht h x t dx Vx x x t x t x t v v v v v+= + .(3.1.85) Do vy ' ' ' ' ' '1 1 1 1 1 1 1 1 1 1 1 11' '2 1 2 0 2 1 2 1 1 1 1 1 0( , ; , ) ( ) ( ) ( , ; , )( ) | ( , ) ( , ) ( , ) ( , ) |i Gx t x t t t x x hGx t x ttidx Vx x T x t x t x t x to ov v v v+ + +c= +c (+ + ,hh(3.1.86) 1 1t t+= l bo m trt t trong (3.1.85). Ta c th vit gn li vi vic t 1 11 ( ) x t = : 2 110 0(1,1') (1 1') (1,1')2 (1 2) | (2 ) (2) (1) (1') |t ti G hGtidV Tov v v v+ + +=c= +c (+ + ,hh(3.1.87) y ta t 2 2t t+= v 1 2 1 2(1 2) ( ) ( ) V Vx x t t o = ; 2 22 d dx dt = . Hoc: 2 122110 0(1,1') (1 1') ( ) (1,1')22 (1 2) | (2 ) (2) (1) (1') |ex Lt ti G H U Gt midV Tov v v v+ + +=| | c + V = + + |c\ . (+ + ,hhh(3.1.88) Nu h electron c lp (khng tng tc) th ta c ngay phng trnh cho HG t do: 22 011(1,1') (1 1')2i Gt mo| | c + V = |c\ .hh (3.1.89) (HG o:42 20 1 211(1,1') (1 1')2G it mo| | c= + V |c\ .hh (3.1.90) vi 10 02 (12) (21') (1 1') d G G o= (3.1.91) ) -HG hai ht i vi h tng tc, HG(1,1') Gph thuc vo thnh phn gm 4 tan t trng. H 4 tan t ny tng ng HG hai ht 2G c nh ngha nh sau: ' '2 21(121' 2') | (1) (2) (2 ) (1) |( )G Tiv v v v+ += ( ,h(3.1.92) PT cho HG c th c vit li: 2 122112(1,1') (1 1') ( ) (1,1')22 (1 2) (121' 2 )ex Lt ti G H U Gt mi dV Go+=| | c + V = + + |c\ . hhh (3.1.93) Nh vy trong trng hp c tng tc, phng trnh cho HG mt ht khng phi l phng trnh ng! V v s tng tc, tm HG 1 ht, ta phi bit HG 2 ht v c th ta phi bit HG n ht. Mt s nt v HG 2GHG 2Gm t qu trnh truyn ca hai ht c thm (sinh) vo h ti 1' v2' v xut hin (hoc c di (hy) khi h) ti 1 v 2. Ni chung chuyn ng ca cc ht c tng quan vi nhau v cc ht tng tc vi nhau. Gin cho HG 2G : 2(121' 2') G = Ch : thnh thong ngi ta dng gin trn minh ha cho trng hp ' '1 2 1 2, t t t t = =; v gin cho 2Gtrong trng hp tng qut l HG 2G dng m t h hai ht tng tc vi nhau, v d exciton- ht c to thnh do s bt cp gia electron v l trng qua tng tc Coulomb. Gn ng cho HG 2Go gn ng bc nht ta c th b qua tng quan ca cc ht v xem rng cc ht truyn qua mi trng hon tan c lp vi nhau:1 2 1 2 1 2 1 2 43 2(121' 2') G= = = (11') (22') (12') (21') G G G G (3.1.94) (du + dnh cho boson, du tr dnh cho fermion) Gn ng ny chnh l gn ng Hartree-Fock (HF). S hng th nht tng ng gn ng Hartree, s hng th hai tng ng gn ng Fock. Trong (3.1.94) xt n tnh ng nht ca cc ht: v cc ht ng nht nn ta khng th phn bit cc qu trnh i) ht c thm 1 v xut hin 1 vi cc qu trnh ii) ht c thm 1 v xut hin 2. [Gn ng HF thuc v lai gn ng trng trung bnh (mean field)] oGn ng bc cao hn: Gn ng thang (ladder approximation; i khi cn gi l gn ng Bethe-Salpeter mc d gn ng BS ny tng qut hn gn ng thang) 2(121' 2') G= = + = 2(11') (22') (12') (21') 4 5 (14) (25) (45) (451' 2') G G G G i d d G G W G + h (3.1.95) Nu ta thc hin php lp gii phng trnh tch phn trn th s c chui v hn cc gin thang (ladder diagram) 2(121' 2') G =+++ + s hng trao i HG trong gn ng HF Vi gn ng HF (3.1.94) phng trnh chuyn ng ca HG (3.1.93) thnh 1 2 1 2 1 2 1 2 2 1 2 11 2 1 2 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 44 2211(2)(1,1') (1 1') ( ) (1,1')22 (1 2)[ (11') (22 ) (12') (21')](1 1') [ 2 (1 2) (22 )] (1,1') 2 (1 2) (12') (21')ex LHartree Fockex Lni G H U Gt mi dV G G G GH U i dV G G i dV G Goo++| | c + V = + + |c\ . = + + + hhh. .h h.(3.1.96) S hng Hartree v LU thng kh nhau. Ta c PT: 2211(1,1') (1 1') (1,1') 2 (1 2) (12') (21')2exi G HG i dV G Gt mo| | c + V = + + |c\ .hh h (3.1.97) - Phng trnh DysonH PT (3.1.93) khng kn; v vy ta tm mt h phng trnh ng mt cch hnh thc bng vic a vo ton t nng lng ringEc xc nh nh sau: 23 (1 3) (131' 3 )2 (12) (21') 3 (1 3) (11') (33 )i d V Gd G i d V G G++ = E hh(3.1.98) Vi nh ngha ny ta c phng trnh Dyson cho HG: 2211(1,1') (1 1') (1,1') 2 (12) (21')2exi G HG d Gt mo| | c + V = + + E |c\ .hh (3.1.99) (Ta xem s hng Hartree v LU kh nhau.) Dng thc c th ca nng lng ring s c tho lun sau. Vi HG o (3.1.90) c th chuyn PT Dyson sang dng PT tch phn 0 0 0(1,1') (1,1') 2 (1, 2) (2,1') 2 3 (1, 2) (23) (31')exG G d G HG d d G G = + + E

(3.1.100) Hoc dng gn hn: 0 0 '(1,1') (1,1') 2 3 (1, 2) (23) (31') G G d d G G = + E(3.1.101) vi(12) (12) (12) (1 2) (12)exHoo o = E + E = + E (3.1.102) Nng lng ringE: Ngi ta dn ra c rng: (12) 4 5 (51) (14) (425) i d d W G E = Ih (3.1.103) 45 Trong Ic gi l hm nh v c cho bi (12)(123) (1 2) (1 3) 4 5 6 7 (46) (673) (75)(45)d d d d G GGo ocEI = + Ic,(3.1.104) W l th tng tc chn(12) (1 2) 3 4 (2 3) (34) (41) W V ddV L W = + (3.1.105) vi L gi l hm phn cc v c dng (12) 3 4 (13) (342) (41 ) L i dd G G+= Ih (3.1.106) Cc PT (3.1.100) - (3.1.106) to thnh mt h PT kn. gii h ny ngi ta phi thc hin nhng gn ng khc nhau, ty cnh hung cho php. Gn ng HF chn (SHFA): Ch gi li s hng u ca hm nh: (123) (1 2) (1 3) o o I = (3.1.107) v gn ng ny gi l b qua hiu chnh nh. C khi cn gi l gn ng HF chn (SHF- screened HF). Vi gn ng ny ta c c (12) (21) (12) i W G E =h ,(3.1.108) (12) (1 2) 3 4 (2 3) (34) (41) W V ddV L W = + (3.1.109) vi(12) (12) (21 ) L i G G+= h (3.1.110) Gn ng ny cho hm phn cc L tng ng vi gn ng RPA (Random Phase Approx.). V th thnh thong thut ng RPA v SHFA c dng ch cng mt gn ng (trong khi RPA ch nhm n hm phn cc L, cn SHFA th dnh cho c nng lng ring v L).Vi h ng nht (ch ph thuc vo ta tng i' r r p =v ' t t t = ) cc PT trn c th c vit trong khng gian xung lng; v d PT (3.1.109) cho th chn tr thnh: ( , ) ( ) ( ) ( , ) ( , ) Wq Vq Vq L q Wq e e e = + (3.1.111) Hay ( )( , )1 ( ) ( , )VqWqVqL qee=(3.1.112) 46 Khi xem xt th chn, ngi ta thng a vo khi nim hm in mi (dielectric function)( , ) q c ec nh ngha nh sau: ( , ) ( ) / ( , ) Wk Vq q e c e = (3.1.113) T (3.1.112) ta c( , ) 1 ( ) ( , ) q Vq L q c e e = (3.1.114) i vi kh electron, vi nh ngha ca hm phn cc, c th kh d dng tm c: ( , )( )k q kkk q kn nL qiee o c c=+ + h(3.1.115) V ( , ) 1 ( )( )k q kkk q kn nq Vqic ee o c c= + + h(3.1.116) y l cng thc Lindhard cho hm in mi. Xt kh electron 3 chiu, ngi ta ta c th dn ra kt qu, khi ly0 e , 22( , 0) 1 qqkc = + (3.1.117) vikgi l s sng chn 204 e n tkc uc=c(3.1.118) Nh vy 23 2 204 1( ) ( , 0)seV q WqL qtc k =+(3.1.119) Kt qu ny cho thy th chn kh i s phn k ca th Coulomb ti0 q . Php bin i Fourier ca(3.1.119)cho 2 23 2 20 04 1( ) ( )iq r rs sq qe eV r V q e eL q rktc k c = = =+ (3.1.120) y l th Yukawa ! Gn ng HF Nu b qua phn chn trong th tng tc, tc l ch gi li s hng u trong (3.1.109) th nng lng ring c dng n gin l(12) (2 1) (12) i V G E = h ,(3.1.121) thay vo (3.1.99) th ta c li PT HG trong gn ng HF dn ra trong (3.1.97) . Khng xt tng tc vi trng ngai v gi s h ng nht, ta c th vit PT Dyson trong khng gian k nh sau: 47 2 2( , ) 1 ( , ) ( , )2kG k k G kme e e e| | = + E |\ .hh (3.1.122) Hay 01( , )( , )kG kkeec e= E h(3.1.123) (cht ch hn, mu s phi c thmiovi0 o .) So vi trng hp ht t do, PT (3.1.123)cho thy nng lng ht t do c thay th bng 0( , )kk c e +E do h qu ca tng tc.E r rng l mt hiu chnh nng lng v mang tnh t thn ca h tng tc.V vy ngi ta gi l nng lng ring.