-fgtlandi/courses/04---second-quantization.pdf · to 'y-t"l ll,rn eaf, e uitj- @.!.to...

31
".1ra^"t^d AXi "L ^ eDal^ -.*. Fo,-k\r A @ s4^" ,.^o^^^e t'^zc,ro'tdv pr*^^k?a-l-.-'o^^t' ,.{-ar-o{rr. {&. a- ar'fi,1.<:la-^-tl *"1 -d .r*iHn'r*f d.,cq..$^,n f{"e ,fv.bb "f 7r-*L^r^ -^/,1L ol,ta,.,.;c9, {-t^*I ,'t\ a&ol% ^^.rik 6{- {.o./' otraQ^,",tr *-fl^ o.- try ^ry.;-'*hg.r # *C*A-L" pox-!.'&; , I^n g*r.^^L* ,v.eahq,.nrcS LLre rr.\Lr q -&aA t u-rln^'cJrr Pax.fuU /.^ ",,j\ u,l^icli^ l.La[+*'. B.\ (+,r ,.nao",t.-6 i&,.-].i{cc^-t" fra:c"l. c.tl;^, }tnel '+n Foo ,*^le-od, & xa..cat 'd- g*xto"t't t-c' a o.f'+'o"^ tr a h^.r-r, *t^o.i- *ilry cLo''*gaa u)e "* V{"r} } .t;rLto'f qt-Llc,.-rt. ld.a.on-'t*'<a {a.od- N""*Y" Llrg.,,J rtqout"\'*a& U ^i 11"" e$o^'H <t't/ Pe ec*-t'L Llac- ,Ie*ferb 3e,ccl.-tL 2o[.or.r, !t.e \oa/vt^ "f .^-^*.,osf l^/./c Or"A,A gQ x.c-e, f .* tM otot'x,.olJ-'>*4* l\^^'e tt. \-f @[''"'* j) ^[-'lt p.-. ar-r. L.^n n-l g.cL(}-^l-^- cS , M **H[{^,t"^-^8 taelore a,e0-r''th'-i"t'u{A tt l,,.'a['d'' i4^^r"\,.. olr*,,{ We PrbLfu4^^ f,""{,"-a.u^c..-'5' ae^**-d gr^o"*Vtct-k'o'* l-o a' xL...:or*.,-ri" {* ?l/e' F*t !a"na n ol't"t.le'^6" \oi[ P.;-Vy. f eat->.Ld- "2,rcr1,l % lo *Lratt A.r &la.i-0- .uL.uu t ^U w^r.,.e,\ {^ro,xn, 8.,, t l't^o+ ,,-t;;ucr0,!--T ,{-.A lo. L "r" ^U"!nac'l- o*rd. "^-+ -"rfr P{:r,du,cd*ve' X Vn-xtfare- PrPf* ?'De' d"o {4^i, ,t'- L -tkfs' Ir.^ Wu r cJ.l-o-, ,r.riH,^ar L fi'ov otsS pvoofs' &tn qtx/-)- o'!9aru u5 to f^^? A' I"*r5,1,.f }r2 A f a**k'o'n''s- M% k &n t^t-l'^usAu'e'*ia-'< llljll "!xA * ^*1"{3 ttt" ' b'L ^^^@U \oe .ii u);tl le lre' Y* S-A^t' ^^oi Serc^^-t ?"ata - U^..t l\a F.u o-ho- L0"rrccxg il 6*i{^lx aa ^d^tcEo-A- ' t l- n I *-cr..hAg :cS a^".t- 6)o.4/\ },urO-"A-+U Jrtv\ N ,^^^ ^f o.^^{d rlne ro-tr't f' sbLe@$ Vp"o*rt 14, ".l.o"al- c.kd[-

Upload: others

Post on 12-Nov-2020

0 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

".1ra^"t^dAXi "L ^ eDal^ -.*. Fo,-k\r

A @s4^" ,.^o^^^e t'^zc,ro'tdv pr*^^k?a-l-.-'o^^t' ,.{-ar-o{rr. {&. a- ar'fi,1.<:la-^-tl *"1

-d .r*iHn'r*f d.,cq..$^,n f{"e ,fv.bb "f 7r-*L^r^ -^/,1L ol,ta,.,.;c9, {-t^*I ,'t\

a&ol% ^^.rik 6{- {.o./' otraQ^,",tr *-fl^ o.- try ^ry.;-'*hg.r # *C*A-L"pox-!.'&; , I^n g*r.^^L* ,v.eahq,.nrcS LLre rr.\Lr q -&aA t u-rln^'cJrr Pax.fuU/.^ ",,j\ u,l^icli^ l.La[+*'. B.\ (+,r ,.nao",t.-6 i&,.-].i{cc^-t" fra:c"l. c.tl;^, }tnel '+n

Foo ,*^le-od, & xa..cat 'd- g*xto"t't t-c' a o.f'+'o"^

tr

ah^.r-r,

*t^o.i- *ilry cLo''*gaa

u)e "*V{"r} } .t;rLto'f

qt-Llc,.-rt. ld.a.on-'t*'<a

{a.od- N""*Y"Llrg.,,J rtqout"\'*a&

U

^i 11"" e$o^'H <t't/

Pe ec*-t'L Llac- ,Ie*ferb3e,ccl.-tL 2o[.or.r, !t.e \oa/vt^ "f .^-^*.,osf l^/./c

Or"A,A

gQ x.c-e, f .* tM otot'x,.olJ-'>*4* l\^^'e tt. \-f @[''"'*

j) ^[-'lt p.-.ar-r. L.^n n-l g.cL(}-^l-^- cS , M **H[{^,t"^-^8

taelore a,e0-r''th'-i"t'u{A tt l,,.'a['d''

i4^^r"\,.. olr*,,{ We PrbLfu4^^

f,""{,"-a.u^c..-'5' ae^**-d gr^o"*Vtct-k'o'* l-o a' xL...:or*.,-ri" {* ?l/e'

F*t !a"na n ol't"t.le'^6" \oi[ P.;-Vy. f eat->.Ld- "2,rcr1,l % lo *LrattA.r

&la.i-0- .uL.uu t ^U w^r.,.e,\ {^ro,xn, 8.,, t l't^o+ ,,-t;;ucr0,!--T,{-.A lo.L

"r"^U"!nac'l- o*rd. "^-+ -"rfr P{:r,du,cd*ve' X Vn-xtfare- PrPf* ?'De' d"o

{4^i, ,t'- L -tkfs' Ir.^Wu

r cJ.l-o-, ,r.riH,^ar L fi'ov otsS pvoofs' &tn qtx/-)- o'!9aru u5

to f^^? A'I"*r5,1,.f }r2 A f a**k'o'n''s- M% k &n t^t-l'^usAu'e'*ia-'<

llljll "!xA * ^*1"{3 ttt" ' b'L ^^^@U\oe .ii u);tl le lre' Y* S-A^t'

^^oi Serc^^-t ?"ata -U^..t l\aF.u o-ho-

L0"rrccxg il 6*i{^lx aa ^d^tcEo-A- ' t l- nI *-cr..hAg :cS a^".t- 6)o.4/\

},urO-"A-+U Jrtv\ N

,^^^ ^f o.^^{d rlne ro-tr't f' sbLe@$ Vp"o*rt

14, ".l.o"al- c.kd[-

Page 2: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r. d.c,f-t+eu,*;"e-

".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs o-,-^Lr*r-l+o^r-.

d4i^ olt^]r- ^.-t "I ^-^-o!.,r. .^./\ cpt't-o,.,^.ol n * L3 .1,sr. d,-er.ia>lt b ^bP 'Aqgrl <.p-,.1J- ^^ol bt' pl^no.4iaecl- ^; **U a'e-U -

-1/ -

1*{-erlEr' *gL'Ba&e,re" u)eJ ^I**i

*uler&

*,.t,lr- pcrstk& a{ote-' d^r alae-evrn6

,r.[^otr^ o[ro:c.o,clr<*i ?eS o- )*t-8lL

T -o.^^[- hc e9N;'"{U &{^e- ,}l-t.tu tt a/

pa>tl-uJ+-l-,-r.^.,^ ^[ot' )* )

€t*

I *) -- 9o,ti'''n' a[ *Jc r (J"f{"| - b^:'d'^''K')

l!e> jwuo'n/v1Q/',ll-'^'* *l*k

\\" ,<f) = "/^/vlo'^nn a^^

A*a

\ J?- tl.li'Q'l* t^l'u[*- A^-t)z. u.*

lur o.,to\ AP'\ /r f z tl

"l--LaaJ\ lx) , A""ot LD<'

^rt^nerrd,{"e- Hi,uUe..rl ^f ae q

\ M' t, Mt, cr) = r{g cf"'<'ogun orEl*ol

l7-) ?o* l*'o.,Q,. J go-r'i{'og^a ":( -

C'o^^ti ;*U Orls '^'\grtA't"o'^/' C I no ) '

-, I4ilJ.t- -,.!*[t] oo'-"-f-t'ukq c.\2. euc.r:. ok:r*Z e & p8 PL P*k J"r"

o-u:;,&'dr- -o-

oLA).tl^^^t- H"' j afo-

ilrc,r"e,ovd(r de o,:'t'r'l,\l'e t-t# ttt* a^^ o.Wo ru-o,n^/1/\"O-t \aa'aYi^

S"? (t)("1 t>

Page 3: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

boso^na Ferant,to,nas3:*Nor.r.-l o.^Peo^. tdo-.,

t- d.ir;u-be- ua,,l.r a^61+u^

N i &^^1,.' prcVcltt. A ga*r,ib.U ba*^^

; Wt ?r-er*c)- ba'lia I

Ir.qrua

ldr, otp

"t -'- d,.l

)(.r ) o @ l*r) (L)

(g)

(+t

(ar

a1^M YLre- .r.'{-q-!c C f-l"e -f!<.l.,'^. ^^log bz uvTovtdta" oa

)+) 7] cC*r,--', *') I{.r"' "r)

pr cax[ crj'^^

po,-k'llr i',^',,'

c.oe-l6lie-.'^^k c (a''" - tn.l )'

.^^-^-od^^0-\^ [-^^^" c^6r'"alo-lo',

Fq. ,^lo r..o , il .reLrJe uttld- ta.cra/e

laol;e Fuo

({)

(s)

l+)-- Z c(rlar,b') lr\4'r['t-2)tkl , ,l*-Z

too.r-r.t :rl:*'aW i/n*l C ({.l '

, (.) = (olr"- olPl^+)

?ro\o ('., "', *') lc t<l , -'', d^')lz

A,P

kc{a Qct

!*rd"''n- V4'\

ic{r,r,\i-' e-l , W q.3 \a-V2J ^@-td'?"1 -d )4^'e- Po*

cl^o,.,\ge + gt1d BI9 ?o*{*-!ALw exi

?.o\, (---,",

Ft.AL c

, dj,"') = ?vnot ("', {r', "'' <t' "' )

mno{ &erarge bd a} a'r'ros I o- p hozul-

'it"^ /\^^eA S

C("' ot; "'"i '- ) tl c(..*s... ,,-")

Page 4: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

AS€.x-t<o,L, A.o e-zi+ = I . \^r\^r^^& rge "\r\,rt,Sl l,.e'ala i. o q'-

lf I alt"^fl't Alv\@Jrts

r;r-r4,r;r-,..-...,

(!,i^ @.t^"Atro/i^^J @^rl/ryl!^ut,ct ."^% d t{"c- f-+ ,'t^J f{^g

u)e

+=

D.,\/1

l^eru

{^ q.

(8)

Ft"e W#-olr"Pavdx

gax.\-.'{rx AJt.,- .r.^r*dltJ<^"€"rr.,a h6-lo&.

o\l fr,ut urld^^ lL"a^/la- r0e )JJIR- '/ + tr oa,.ol *rlno*,' /-l',&t)-f-'et <*'e tJ&' { - rr 7

f.,..n .t 1<r o Por.,-ti go b!,,rxL^e'r- a, eo^fa, ap-llrd- 4 ,Ilrre @)6 U"l***,.\T61-li^t*

BoS olrrn9 t "^^l-tE€-r Af i',.r'

f g.r^^^^-o\M.5 i t^old -^.l*IE{'- 4P"t'

cs .ll^-nu lu )>V.auLL)ed', ^-l.l,I.,,-,..6l "*

!\

^€4'l o ^eL

k.A *f t't.e Lo,.'c,'^|L gno'f, Ja^*[- ilnc' d\n'c,,'a- { 4+t\ or r-*

o.r% o ^ H^e i!g1n "{ L'r^e- laoot-t-' clz ':'^ lt'o"* I

v n^6*

^,r"+

lt q n)

d^r,n n a- ca^^Jr('rt-I4^^<, ? l,t^€, ,r.r,v1i<J-aJ\trtj- tf *H eW 4k^""l'l"r> ,J ^,21'c{pt- Lc^rur"'tl2 tt'e,,^""'4ot',.""a'ldovunE'

,^^r.\eeotr @e u)*19- fo;;ila' ;'lWa u>r;$ ^,r',.ol-

JDttvua-s o- bo,/.;.n P<>sk"Q.ak

c(--',dL,-- d )("- , v.r', 4t)c(.Jr-

Page 5: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

3oa.o^n-S

L"l {ocrlS c!,v\ BoSo'-,S (*" a-.*,rc,e,'^*. S"*fp.o4l. ,re e^c-;|c,el ar.r.rr

H,,ra (ac ^fr'^i'^ q Cst ,^^q'Y

r-r.l

^l-a[* a.r I = l, 2, ], 1, . - . We ecr.^/\ *lAua^d\ .z(cl

.\ucx,c_!c xzl d ^l-r-[.a. A pou;ba c ocorfi.r.e,ua^;-

R"U" /rsok- !i!'l'-t

c(J,J,!,int

z

L , L'L,

Me.

45, 9, J'

cQ, T ,3, r,r, a7s 3, zz, lJr Zt ) (s -)

u)e-

(r ( )

a.- ,L {-*k [.c,rr*^s,l-t\^

(t z)

,ry H^c

l*3 p^r.[^*fuo, W eor'e-t te q, @r',vf'&k "r."^a,rA. 6rl 't4- BoSoazr5

u)e a:ct o-g.if,atued- l"o e#et^@'^^8g H^g ardtsu ? l.-t^e- ^/'^oU:cl*b , b[-ur

f\Aned :cr[vor.lp ltne fr^A^^lt*^"" aaeu.,r'rb<rrg ^^^ Cl o) aa

fr'l 5

/il^n ;^.1lr..&rpe-n>= il^' t Plnt i&s- * "lJ'br^^"tt*-^E

t *ev \.

fu t ^.,^p to '"prto.^"a* y*"1" a ol"o^' "e,.^o;15

1.r.,.-t+rubc'r >c'e4>c't'['A*rl ofCO'- a\ o- ba^'i^ {-" ktc- f*

l^l)= f f(*,,n^z,'"')lMr'Me' ")\, szr

l}^

\o*^r . Tt/rc.l i/

(rs)

^nor n nno'Prnl<zerr t po-r-l-."-{u' i^'^ n'l"oL l'1>

w\,.1c, W €(*t, ML, " ) arc.r a- ^r'4rd '^*l- of oc*-ffi<'</"r'--,'-|S '

t5]

Page 6: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

',1

Lc,l. ^/\.1g claxj{-U .rlhori-' tl^c- FooV loqli^ r\t*@.rv1g. Fon *n" ,^1XA-f p, * U^c ,ru"-6rr - f ax-k& Flr a,lp e-l- +po,<r Lrer o[rra,tl,rn 'r"i"oa,r d.

dnaj- .ra r o( = J ,2, , ", &' d4^;l J\rne'sJ\^E ?1"*L H^"r ant' d* Po-'^ibfu

,rLo!..r o- po,o!- & ilv1el.' *=fu c'.-ol. th.e F-o.,L b"r i^ ox;'t'r*l's k\"'

o, p.r*k'",rr, .,,. Qd^ or\^e 5o t4'e F'oc,.- bod'u usatQd' 'boQ-

,rJ,^c- \ nn.l, M2, ..', v&).

q1^. *,.. r.r.r*\ogr. -f trtcL ^hak'"f-. A,,. ^ ^rpt [o^^l' ^{-*l.- ; HA-I

A,\ W\AqJvS p or"i*.c0r'r r}x De

? *=k e))t , u-l\^^'"I^

lo) = Io,o,

t t er.*rr,r",- l)^w err'-,. be

ar,J,l Wz vo<luJ/w'r-C, C$

t)JEuoj&'t^ /\/\Jo

(''1)

Oi^rr rrg ho.u e- 6t- *Lol't^ trl-t:$l'a *"4 o\ ^e p oa'-k- *0^

\o, l, o, ) \OrOr-'r 't>\ t, o, o, - -')

,f or or l, o. . - ' ) W *.-[ es..t,+,& ,

l-.:& -tlok l(: ts>'

kha r".o,r,^,.g. a-L Wc ^ {-oJer.,\.rh. r[."tle-

^ry P.*^l-,o[or"

.udL z Paxkcfua ' 'l"tr'e1^ oA

N ox[ uJg V^.cua & lo."^"eJ" "t

\2,o,o,\DI,L t- ,*ADk Wu- *l-1"" [*o p'r.. a-.^r"Is "f *8b - (o'*l-"a

Ivr . *z ) . B,Jl- +.l,'i/ r'i' ^* F '\o {*n+ 'io4' a^'ol- '*k,"'l'ld ^,.l.l

E d,-o l-k i ,\i^ F{^e of}ux- "'e-L t ""ok'l' i^'n ecud* P 'Dt(-

gl- ^i PoJ"'^J-

o[oA.,r, aiu€-

^*r"U ,,n4f^,f

i^^\<.r.r^kEt.

@

Page 7: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

A-l fh.* po-,*" [, &{ru,,r al-urcerg,+,0"azr xLlal*A"d- *{-r1;4"$ l'to tcu-,.a<'-^-^cl Pu{t--0r

,.Fo.[L t r> /vv\6'd ho,r..,,e Md palL.'c&,'. n^ i+ 3 o l,,'ue- *q# *xmtA

# ou,.,..d- fu,ko^ ct '^ * o.'^^/^'L hi -Q^o-h o'n^ 7-*t,cl-L^tdA+i/\^e etc..O

r^r\^ie[ ,rr"e q &^l^'y PoreJ*e{la ,; Alolc !o'> ' d""'i 'u

ffi l,-.,w-',"--)aJ l.-., Morr-") Jrt-Il'-", ^^v

* Jr( rs)

wL\r,\^ u + P ( d',.'"d'F)tl'^{ a{'o{tr)'

Br\ u\ar,^-r. u * t, t4a11 b*fto'ue a4"'

l.Ou^n^r4,"-Cc o)'d'Ll4-b-'

Wfl o,,t-c od ^^f PlbJ"d ;^r'cfuf)-'

? o0uro-r6-r. Frl o* \"o-[ r^'\ iL ' la q^r.o-.-&

Mrl, 1I ,'al-*'ts*

E

(r n)

obre{.

"-kly |rlt ri, \+c (or."

we- o-{.^.o *^lutt\ ol"{+^^e' H^e- a,c\u*tr5ar @

<p;*L1 t0"e n11;wrLer

(t+\

ha)

q6

t*A"l PtCo*, at l

^tl =

Iav ae(nnor

0

Page 8: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

&. qtx^h"" .io. t{,'.l>ta+{qre- **1.1. W^e n,atur^be-6 "€ parL|*;,c(A

j.^r. .\f,o.tL \ *). = ( u.Ja - o^^ { lo oor-r*[ V^e f. {a,.r( ,r,',tt-t r*\6 2r. .gpo*-k"/,r4 .l.^^ [Le- a"-l0l.L*r.^^r Lue @(r t{'.\^e *t'r.e- op"rrc-kv'

(rt1

a.t P"f"r4tx's

it \ -,, Mr, - ) L*" (vo)

Loa

N al

Page 9: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

F"ar.r,,^^or,^s

e{."r, trre. c,or-^i-;'^,re-

t^@'"^1 aiJ" Lt\^ e.t"a,t,gx,t Fc,r"nvraio.vvl s -

tal'a *y4..

6d^^f loo&---.&h t-t^e

^^^ H^e+or^r,^6r(oDAt- "{

t- q ( u) , @Q '^-"'dr-A,, \no'are

a (^ '^, di, . *j,...) =^ C(^"",dj, .-.,4;t."- ) (2, )

n ,rla f,*?p-* t5 c,Lnoo-r.f- o(i. -- or.,, t{,,i"^ ^nL Bc"t- X. -' -' :tL, ult{o-\e .*

lo4.-,-,,-[.t,o.ar. i,r .:rt-=o" ffu:g

C(o(r,d2,".., {,*r)=o ;f .^4d l-t^ro oli axl 9"e (zz)

4r.^i * Hne P*'lL J.-o Fe,a.n^^"Cj*S Q6\zr,nfJt

Wg @,c,c^^frB gt^e, .'\dr^^.'1e- ^[*keelr.-r.i<l,r.r^,^ra t"" L^AL fLC

/u1d @\J\^ "4 [-at,td" ]lc' Jo'oAeA o

FocV t<>61-A& , lor l- ^).@,1.o flt',e-

r J. fuut,

(0<>. 8oso..ru"5(tE)

k- Fer;;+io"ws

.lrr\r' --- O( J/ Plt".No ,,\^^/rSF /u'fl'L^) O^^{i '

We "t.,.na,$

We- +-{^o

C a^^2,-

o,J.Lr3,--'

J-t-|-.-.r.,"-e e-r.o,alt"etr'',r O^l&

a*. tsJ[ fo 'u"\buce +&-"J-

O.L

{

iO"{ 4 ! $,e

Q-u( r*"Pl

OO.r".*Or^"lLC

(z<)

Page 10: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

To !.r*d.J.)Ltdo,^^d- \rr\" d t{^ i ,^^^r4 te & , a^otc H"^t qf $*f-- F rr.lJl(/,"a..

zan (*i)"= o as(

l. l.'a- ar o.ar.,te*' l"r-lr,Jr;

I

qt-h ^.l. f#.a QouO; f^*^*L , J"4d fio "t*okn69t {{o^^" l-/,,.ct<+ria a^^€L P 6cl" }:e-tP,

.^^erd .rrroh olPqV o'-c-,tJ-lty trt)C, €t'\

"J I o) = tJ., )

(..{ )" lo> = 6f 1r*) -* o

}*-r. trrt,.rez,. 1* LrH to o^^t-;l"L0-k Fq.}'cr*

-t t -

W *2{t.L".- ? So-:'a"as c^ "tt Fot.r.rraloa^s' (,rue *U

o- r.r.,pl..4r:rot u,ad. ['o ots H^^t' ol{|l^;.i.e

<1.e, lcr'rnnC i/'

Cd^^drr.lr^ -trf,

ulr,rl.. t{zru^^ -t^^

(\""\^^ r-0e ^-*d

BoSo,naS

F gf"^^r^.an St I +t-t

(x)

(zt)a^^t fn.B1g - AB- L bA

*rilc (re) a^^ol' (?A) aA

(2a1t q"l +O\9 1L - $"

P

[a", q? ]t= o

Page 11: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

EtrM€_ ^r^, L,s^s Q-o-ttgtqe 4 ,*a-poL-----: =- ?,^a *l-' zaJiar^t{,na

J,rn ordatt Lo o).r.,,\r **r? *@, u)c r-^.u/Sf L. ay,.Qt [.9

Car^a"tk*^arfA , i.ne .,.,^-!<>4^/\4^^[, *^ "p-r-rIo- (; J D, f^rJ fo aa a-tr€- {-td-/tr,:r.-^ ^p{r-). rfhc- ,.l^-o^lnrr-['r.^"^ 7"Xa,-lar (6'r & ^--^S"" POJ6"J"L:€,g. 1^ p. ts"rl-

;Op ule tG^E N ,.^^a,^-r.L^5x-rrixUo.l"A pa-\-""2,4 , * ,ruw-{urr. a-r.o A.e/",t*t4- t'o b4k-

oloall H".e aaa-@*un,r-^,r['r^* "{ Vl,^f^ c,- t-t"o[ 8-8. whJ utL .<,tee-ot' b c>t'

r^ to Q,<,<,q- ,f [&g fo [-of M D,rur"\rr\/r [-'"n'-

?= 0." { f,.l

ottox-e.tA v<Na ^rr\Hs3l Cd^^ AD e' .r {-te (6J.,.tt q3c "F Z-lf^ f,-or^h f..l-.'a-,.J

W€Al , de ta-u\dr^/ t/,no'| t{^e ^,^,..,o.rr,r.1+rl't;am *-l*"6du go"<-k:c'll "^'S'* l-*kA or(.t

l\a> {tnrr5 a-f, a,- )')' {,l'\a ''^l"nn'tbe6 o{erx;oJ"oc o*'""}+*ry We I-o l"{

ru,rr.u.rt'en I p"Y-k"&A1.. (L,n"

a"l

L,.a.uA.tl rr^ Fc>

&,{ft.^e- ke',e tolo.L A^/t'o^^/'r-^^tu'n'*

?IZ lr- a-- o-r^-

w

I." ^e,L efr.rh ,vwv^&'r crrrc'ro.(* Otf att -

l.rt r [4^ ^^tr- iL ^'-^o'-iLr+ l't*'!**

( z?)

T/cD ,.r^,vfq ^L^J

*^^ oDA^ " l'\'

k-p,.,o,r.r.r t"l^i^ r.\^-^ '{ >r'u6ov'or/9'

Page 12: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

5r""":r:*, U + _E'd,^r

patt.ch, # 2/* +

dL"^.ok- Flne t^r..ehL %L trrT.-j

l{"c- [o hD4 u^"^,\e]& ,^^{rtt$ , d.l^n ,rre *g

,ulZtuv

(ao2

-rjt{tW Luu.az

rL-1' .B

A^^ {-"}, vx. eo\^ er\)er.), ar.x^d a- ?^u@! fr<rr-,tl*:lo.- Lz-( 4 ba %,

^^ ^*4^-g*ukrt- goe:.-o,klr a^.^or ff?o* * ,.,.g^^al*'44 + -><zpot^

Al*) = \* l*) (.3r)

d-- *s Ly'* ?-**;zeA-,^.-{rcdo\^^- '*^aJ 6<- ..r*l-l* 4A

/' f \' aio" (sL)

(zz1

al

6t,!u,a} i^ vekf,<-r

Eo.A 3

c q

,l>r-..Rt* dV errryo'"*er baa& '

.4", lAlor)

1,r;^ i^ h:t:c {t % f*"It"

Page 13: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

No4+ } uro*^l so -orciL a^^ Trxa.l"* u.^ a bar" <rl^ot id ^% -..\[te €u.qrgr/.,\ot. % db H^af , tt!* tu *x;,l xe)^ er.r\;-b<.n

l" &ryS" bat*e & g/.O^,rt-Lr/^^ ,"rrL*o\,l^ic s- L-{- t"r) cr^r.{- t t >

\..-^-ro or*{z.o !'v\<.rr/'\4 \> ali^ /tAL , fl"'u" ,

\-{^e ol-'t/u;L I u)e *^f% .,.-,:.-ilt

cl^ o^,.6 c- OParc"* a*@

l"t > Z I i >< i l.{)

{'-

Lr@"^,

1..

t--

( 31)

/3 s)

)i)

r-

To -,r'rl,ra-\,.g {'{^e 'ei^'*U!

xqA^a^^^t et^ ['ata]

,t1l..,i4, tgJt) be/cxs;uar^

"*\,Ob*co.mt4"<. ad,6o.,-^"'[, ffl

tl.o)+ Z <i)"/>ai

:ETEJ"tr t^rar^" u-E- T-aUlu.6r

r.,-ri|la A&e*)ld- f ^g^"+-2a-k:o"r I t^)? ** ,'t<"eo( Lo

*J to)C lo>

d/g

uJe

ry Z (,2)i) a;

(@5, e.^^n^^ L-;la-'kcr'* "ftt*!,;l'g J"^o"J4dt'|'-^ {ik' ttu,-<

bd7r"

Page 14: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

f.Jcu.l It tbo&- Fo (3e) a.^,\oL *^^^e,r.l (3s), C36):

A E (.erAr") [ r <, '*>

ai ] tt ('ri, o, J

t ol "r; I < i I'r ) ('u lA\ oz) ("'li )ra.,j

4rrr. ^,ur\'\ o'fi;r^ ol l^

f l*>4*l A\ *) {*\ = za(,

X( loe)(o.l - A

axrA\ /\

&

fl,urs (rc- ea*pe**at {'L'.Dl

A

d^ i. ,r-\aorr)s V\ra*'r) k o-l<J< cf^@''^^ *^^d f--."}*q xt-a,o*& 9,^a^ ktsa+da^^ ' Noka \'@t"* ^a'^e-xea

*oln",*"b^^^"^l'5(irA[)r'o+-htx-c'oln-t*'^^o^'da*a^i\o"s-

<{... "p.r-t-.. &( (gz ) i^ a- y-,*.- *.h"b *w.b{, *rrrr., t{rUrL

(34

A ^),\ {'{"e €cqg{'te

^^,1gr^ -pax'[a"1"

t^j{o.-tda^/' GP4,tca--

utlr*-da,-j-.[J J@^'d +,.x..,]'iors ll.rlo - b od"t1'E- ??&rs,d;ufrl-^bc- l&^e i*

I,t^rr ^ "; o- ^'.o-"o(

brl..,u.,lt'^"tJ-'r,lrL

. wL w'-U- d-i*trt'ttl

I (ittl1') ala;ij/-

Page 15: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

Coo.&-^^o,k lo'ko.^"

A^^ ;^^r*{ o^^f ,:^ ,^-g/-r- - gaxk"-l^ Vel ia {-01e po->i..io,an t.a-l lx).11^/r\r ree..is o-0.A,o *o* q- ba-t^;, t{rJ -l ,i o.- zi{&A aUkw'l<*,r,),/cL tt% u% ecr^,.k-.r^.t oLrS 2e , d1^"^ -,,rtsplrnng H^rr./

{*lB) -- f (x-A)

[^* It)(zcl = J

5rr.<t l'a-) ,i C),- .u"^18e- - 7ox-ldoL Ve+, tNQ' cct^^

Ceq)(x-0.\2o,^,^6[i/r€) o.r..r. n^r,-lai AoJ. o^^ 3*,t^-t" Lx' '

ru-o.:*s^;r '^,.o o.^./\e- .*rc,t[rl if l;w PtJ - % *qhe r@€-\<<a^^ n L{^** , in raox,t? lv/'^Ar t [l''';^ *H) b

-&-b^.^o#anv 7 evQ^^ {-4^€.rC'J^ "L n "tl- 4 or"L

$\i'*e'B-l [*,ilc. i(l,"un, u*

kil^ al

D..^ib.r-tio{

a.A P Oo)

A* B,a{ g^l'"e

-tl^"^"'^b

o.-

(3?)

etfA- tU Or.rr.re *

{ lbr\o)D t

t&*[

<1,,,". ad,vvl *^ul-'k^^ "*-0-^-ka^^5t=r^^k.^."^, O<tS-Q7

;t f-/! O* BiI @4Detr at {^t^l . 5' ""(A

f to) * Gra =' o.zr'rvia't t^'f .t-'dtA o+ ?ep"^d-d or.nzr 1't-

G1)

W * wl a)u a- e;HtL d^{r#/.e^^L

C.I^ * Grl,J. a,,rt ."l. € (zs),t>l-t-r<*'>L "lA'><*<'L

\4^^d ,<2P&

(< o)f+tr>, f+cr)]r

I I c',o1, + t- )J:

[(* ->c')

o

6

Page 16: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

-t^e alxr z^ ^^^^^tlr- t-" i(er-.\,.!4r,\,rter^ ; z-ralo,o. K,r@uaeclr-e,r f ! hD.,ro<, 6i. A!.^o, xufLe,.<L .,r"Lr"^.".^s % ,-^*kg-o.{, , Fo'r )r,^o[a^,ta,tl,,e of ralJ-@,/ *+C*> * @) -t-q---okr l&. d!^,* oP ',,,,.k'/-l"^ --{< --Poli,{4al^^ x- . qlnt f. L"a .^,.r-.,.r.^r.1a e,.r -I p ax-[.-'"!r^r j,^ H/tb^o,

J* Q* c*l.l cro IN

i-'L,.a c,oor/ olr^^o.k *ldo.^r.. tl, ,ug.nD

H.t 7

(q<)

.L-- !** <,'lAl>c-') 'l+cnl*(n') Clzl

I ' l- t@. [no,a o^o]o*n "lu'**^-'kFo" ^ ^^!arrrcr, We -r.u.Oarvtlptl1.l*v' aPY<D-W l,,zto<'a' A 'a

; g f,Cr.-ro') c13)Cx.'

g!..u^ !'l^c. tol-a{ "^^^,a^r\^I^^ hrr Tp (o- b ea^^ -t t

J9 d4-drtu' ; A d(x-xt) I *\*t * c'"'ta zL'

Ia"^-'; { (Y--x" ) I +* 1r'-t *cr')i i-LalL.

o.r

dr" r,\ ob^'o a'koX ;&^^{n-'os l"o t&e tr.lat y o@^//fl)tL "P#'l-ok Arv\

vobtu <tt ? t+> i"^ t',o^'@

+ - 6I^,^ tr'*^b'

e I*" ++t*t I oL -r I *r*to

"lr

4t-^t<)trat YaYg, W-a+ I l-|rc'L r,)-O/Ll)

Page 17: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

wc @d-^"' tb', do t'L-re @j/\e- +1^,.-^.3t f.o. l-h. Ffo,^.,.^.l. k,^.,,-o^r'vr.<f" r[o"""'^' 'L?oaut a^^ t o* .*-EL p *x-kcr:.-_ * o.ra aacr<;r-,,ro-.( p-LQr^k ^4U (X) ^^

D(2'p,a

rl * U C>C)

*..+x-edo.zrzr u)rJJ l-t.tt -. bc

(es)

(<a)

TIo - "rr; IL- t". a*4

TL J**+ l+ (lc) ^?U>c

Zr" * tnt+ U(x)

L, b'

- LCV-b')L

crcl-o-k o^,^. o'^,.oL a/vL,AJL L*'€o.l=o*',

t pt1.t\,l - ,r.t ktfCrcC-lt ,rff",

1lr* ; *1"*I aD-l-L Vtnr-- y. I&Igg}t1ry pt-+ t t+. h^^ne tu)g

op"xoLo'"5. ,q1;oetxa1hZ qArD ^rf .\ ^s-ar\,15

\ Jc ao.,.,.. d^o ,tt-r<r-k C'?S) i^/\ ?hc .//L!q-a^/^rr,&^- {oru^ lo arVOq

Z <urt I b') *L q'H Z * aLa,,"

tr- Zan+

d..^^

we- c,.r.^ utarlr, cr^l[- trrtv\tat) b^d Fl'a{ (ral V)t#

(r,t0\k') = ["w (vful rc>{xlua')

, la,o uLtu) <t'\r)(i"lu"')J

L)("u-r(-L )

\,

r?

Arx, * e

Page 18: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

ws &in^e, ft^e Fotrrr:*.n l"^or.^A 4"O,r^"^ q l4re tr 14" k ^A

u(?) = tJ* ubc\a

{a.A

L?x ca7)

cns)

{h. r^ we

u (9') *rP u (*) L

-*r{<- Vlnc U[a^^-cf' lt"^-iot ^ -**'t'c.r€- ^^"P'k^d

o+r *

oL

aul

a.. Qcl.u{ }ex'*^ cl*'nba lhz L{4'c'f q2

a. x.,ur-ll' *d Po-a*.:t"& M

$-he. poOr.r*r.ral-g

l-er^llaQ U G)

/\@ ?lrccs a-

a,\

ff'o efl-L 0(*) ools a^

rV ^chIJu(?l

;'^,ynrct\ u'\'v'J^,W t" 1., o- ;:l-@[<. I

*->at\c

A)\/\

p.rl"'"tr 1| .xr.ruo

V* 1r. L* zr{ ?-&&ln'g*h'* Ft^n ^^ at *'*\'"4- Q1 WL adtoy;6

o,r'^ o)-ldrro^ *l tLln A'w' a.^"oL )"da *.p,*^err'"L? oua-k4t ouu'ol-Uo-n

o,^ c *.J4 d u^+- aJ Qn^)r)'a , b ?1n* P

4 gox,k&r" n ocle,'e'-)*t''<tve4'

,t.,.,utv\ Vl^o fe t'o{ *+""*6e'r

I u(q) al*? a,.\J a-I 4uIw vr+

-{+L Z"vs

ID

Page 19: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

Ccrr*trec& z a-+ja^,\

I e^^,r^ol -rlill +: br( Y o.,r^llruX ^al h ea^';.l<ttt.l

tftt' gxr-^ Ii t ak ol^ , Re- ,c--Ol g\,@rD P)rLt g^^-.r^1.'8 al-.o*,. '-o'to'r-Ls'

we- ,*{"6.*[ **#. a. c/f6rro,tflb*^ {.G,Qu?) q(^r"^ uJe- otr#--^e- JL'

CO-r.,^,o.,.,.viga-Q- ,u'a,,o.,r,ua-..r[t-"^ arr'.rar,r-ola of' Lo E o4

P .z a*,a(at? )

to Ld ux, 'r-*^{'--i v1,^e l'{ o^.^^i 1[a"^"'at ",'a

* te, g) = qP t

ar^Gt t*"4 a€- Pr€"\^/-o t<-

@9- ec.,.-rvt"r', t/Jlo*lto-

(t=r)-

?, fl tt +..x--lots .n'G-

,olala*l-* lrr^,.^6 tlnc

a-l o

rl -- "Hf)"J-D?e,?o-DiIt

Ne.r-u cxO"al'r-'s{r'x' *J*e. 5c"\^n6 clu6teu

Z +^ z t4n,nw4^ ** a1)LT *,J et

\Ne -,f"-l V-C- -@-),,\r-e ^t€.tr . gt"e eav.t'+r,'t'l*az&

o.,'^,cl.crl ola" to *l- ;

TI;" "{G?,"7L+*

11,- aJ\^.o,o.^ke- f-o {; )^ U-o

(71')

Page 20: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

-f1""^^ uDe- ea^raL^.rel VL,^c hlart^^i-A,l-'..*,..*

H (+., T-) B I rr"^ (ctf-) c= t L!;Jt?^ Z t+;J.*,

{ f Ar,., * ^l'*' {*

,a trw

4/urs

a^"& 1Tn ho

,"!o-['ot ^

w--t\

T)Lskrs ^--L.^,d'd* 6

^),l^,A

r*,pa,t*oolc.kc,-

Cs z2

ur\ W 2't a*'lba'o.,.^-er ;( /La- h, (,t& -

csts).

l+ (t^, t P; ) = f A^,^{r-'p,,,-

MtN

L,oe

c.ro )

wt^.^21r. 'ir. , of ent')-.*-l ato a"ior'r.rs ' F Ltc.&Lfl ptVr"''Ok {^o._

e)

t 6tlnn

csr)

J* "- Lp* +*l = fr""^ (rq+

I t-, ;+i J

VoiL6.i OU'a;^6" V{'^e oa6^'^bo-(-

O""6tt uJg p i-.^2^^^

Irn^tdr* €>

C so)

[ {;l*e- (3?) , ut ul'l ft-*.11ri n(1i,^ -oLL'

+^4 U1+) szo)>

a-^^

Ja^i

tl^e |^rrxLk

{Arv, A^^^

\r t H,,, *

Ndorr, u)Q- Y* *^t tq a8-* O.^,^ol- 83 I

Page 21: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

f t tlJo - b.d# TuxaLas

we r'^'au) $ur^-^r,, to W,L @ bekaec.rn p *k"oU ,

alr- ftsp."-e Ulor-;Zlo,r-r^or^ ,,*% -Onoot- !i(^e

c-u; Ltvvl

H +

q1^e- tua-^d l<:u*, &xx,^:taer t&a ,*[<- o--l-to'^' b* ko'"u^ Tattc*J porc k€'{.ca a^'^d- Vhe {oe;r C tlz n te oa't"t"Pe'^"'ua-k |*

{*r.,*;co

I

z t-;t'J V (X-;, x,i )

h"re ,prc-v

C 521)

pd-'^4

( fs)

ke2

cs7)

l-4^e fiocl l'ta{ u)c qxA e@D"\)'

fwtrcs, rci ) -- v C)Q , r<a ) J'

ldP-*I ["'"'o 'b-t\ .^r/"krr<r.clL\a^^ rj lhc

v(r+i, & ) e e.l

Ut)ta r^bA

Or a'

I +c -*ilt^d<d- - LDlilxLj- ;L,^r,**;o^^

vlw;,*i ) -4 5 ('P., -?{pi)

Ne.r-r-r 2c* t,r t'o*trr\ +^@r^) ft u'*Jt Y|"u t'l-l "nhna-*-)o'-t'

tf I C v(t;,?t ),c{j

^ r^4r1.L- ?"s^ [-* a.r-k a"r.

ZJ

Page 22: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

4V^e .r*u6&--b"eg ,'.,-W or- Lcor^ ,- L'1 6) cDQ.li' irtst*?{.o,*e@d-1

!* uoc) *+ C*l * t*)

dlra "A,.n6r"la€A .\ttv,.v< ', ' ;! UCx-) $4:,ra,,ter {-Ltc

Itl

tl,.*gi::..i

r

l

l

)J\

t+trf $}ct. a+ luor>.L& *+xidatu

uau6 bc <t::cckl<t^

bc- u^n[-rr*+ve t-. gt'?a f f,,o"( Cs?)

trno*8

a-t

n-

v !z [** v(,(, rc ) p+cx t{t*1+Al*u')

slri^ n oL,n.-'oS[ *"utL ' g-l 4 h4 ra- evob lbv^; ,Ut- oce^^l*;*aFcar l^rtk r/. ,r^^ph6u*{ VO *lr'.1 tg'

L

l"lJI

/vvlc.e.^A , @ol'L |-t"e t'cka"" 3

\S) = *+c*) lo)

+ j-,- +&e ,rto[.

d^,t^ ; & o,,-^t w-px-t-)crt r'*la, A{' u)e u)@u.ol' LPPttl-

L+'^ *', \.o .or-^t'^tl<- Y utc *rp-* &{no"f

\f lfil = o-

[.[,tr'), ']rtUl], = S(d-g)

*c*l *+tu) = U t+cUr*Cr.) * t ti- t) csl)or

Page 23: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

lt^

L$ (u) +(r') | o) * $(x:tr) t o)*--

o

Jd4<- v(r,x-) +f C'c) + ,*, $rU) )o)

6 U-U) t o>

v tu> v(U,&) lU)ia.tts

dil n o. uJd - ;--b"ro-ko"t a,"ot- t{ xt^,a*'loL ^^o( b. l'c\4''tr '

alne **A av^koot 6 CS 8l ;

%j *{ w><jt+'}-^6t tL^c z-b'dfr ,.,JLk,a*"[i*-

(" r)

N)"\c \a.aul l-t''a k-uo a"'^"^ai ['^"0aA'o*" f'f^lto'6' qJu

+gt^l-' Nco^rJ t'{ 1f oc.[s a"''^ or"A ^l'*lt:- w;[h

f(r) tClc' ) .r,it;- vi& 4'

p^l*e!-ry'A il/b^

l- We

L 7,,. h''L'et,

A^,oH.u.ru ^*.Irt'&fo i^ t-{^e- ond"ttL * C'il * Cn ) ' a"'& *lO) *{@') '

lot o.^^r.,^,,Lh,.' lzr'l.- >L Vl'v' ; - Y1"""" f "'*-'k r'*^ UAe-

opoo-.L otltlx-. Firsl ,.' $-'a^ ,"- wc--tt r 'p tusL l-o*k* p'ol *4ailr

l,Vwo3, W ,*cocl.g. g,-,1 +' p,,l p1/*,-6 |elai".4. oJt;,| tr f.,r

t't^e- xopVs 6-<x'll' V1^r4 lhe '"U'"an"s '

j**' v ("/,r"' ) + + cq +1 L,"'t'l' t* l * c*>IZ1/

Page 24: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

Nh^, &+i ar-e VeD to cl^cu"^6e f-t> a)"\otl'^uL baa,a' Le*t" tJ > bc-

\aA^^(, baA/.h , ,l*^--ln V"4l

lv-) = Z 1"')(aelr)o<

wa Vp,aul rrrro-fto',,. qr*rr*i.<.rX Jt^ot""A (o'r-"^ Pil^c V*LS I La

*+tnl = I (o'lx-) ^-lOa

*ltcl Z Qd *) a"(al

Nbr, l.lJe. ?be l't^,^; (60) f Oo*^ t'[ pa,.^^c, J

I

zeJ

dr,Ld.q+' I" ts;r

v(K,*') <"/l)c>(1)uc') (x.'15)(x-'.ld+ Cd "' %t

(oe, pl x, >c'> < *-'zL' \a(n J )

hlritr- V(>c''2") <('PI*'%') E

a^.,\ -bltn-o'e I "ri;xzrtlat a^^d- n^o I

I(p$s "*o& !**' {v,PO-"

6

{ or , P) O l2c, 1&'> '

a- +ue h'o"',^ o{t1^rr* 0 bom"*-&t

& d il^'^'js

&r

\AJV.l,v^'4'

IOr; liz lrt, ?4'> < "t, i I

J Cecna^n

d,f )pUL<-truA 1

u)4

( 6')

L

D < ,, pl i lbr,6 ) ; "i a-c nr-or,p, $' d

J.L{

a)i/, lhe- o*chc>yteoL

Page 25: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

X^l pa'r.l.*l^af , lL( u), +<^ppott- t{"o.J tlne- borrrj *lelcr er,'cr. {t4re.

^/^o{/\^r^ lv4^^ Al-oI{A lta>. Mo're,ovcJ( , cDs * g"* po @A.l.li-vr\e- +'1"a1

v (ri, xj ) = v Lxt - xi) Gz1

e, LI &f-rrd" * rA o\r\ W^c Jiala.\,(.,r- bolrt e-e.,^ &4n pnkat^. d^^^,

v)?- gt

4 r., \r,''1 v \ q, ?t ) = !^d4L' < lL, t 'l? lr(,rc') 12c,24', ? , ?')

V t2c-'rL') (1a,l,r.t l)c, la') 1><,?"'lqr?')

vz

Jd.>1-

v(r-.,or) +J**'lana,*'

VL(u>c e k\ra') t (? x-. ?t*')e

&o.,.6e wa)e,.- g-bbA tP E = 24 - 2.-\ ' (l^rJ'^ ge-tIJJ?-No.u.l

1v,c I n l?, ?') = !** vca) iC?-to) x i (?'- r,') (w--E)DL

r.i"h h@A/ t't^r} a-f0-a'*tt s uJ' to u+pr@,b f'h. x- a',.d' 5 '*[tdi t o4A '

i(?+?'-t -t")N- ( L(?t-t'')3e )1 vLs) c(tz-.'^'l0l?.?'> tL

a1,E- Y- ^ lkfro-l ,r).. a \K-o*,.,r d^e'r S

"u(et1

L

Page 26: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

Wa ab\o d-e'dlr,\L

v(?) =

Vt u" uJe- i^r--* gzl

Iq -i9qe- to v(t)

v(1,'-?') Jq*g', le+r,r". aJ al.' 4?ta?

(e<)

cas>

-Jl&r, I )lrd)t/ tl 't t'l,E

+ IT?-?*L r.,wtq,qt

drrd^ Ta,f,^^ a )A)rr\,\. *d r.eal-{<- nry 'e^''c:.^V '

--€ \re- |*rreo,^,. Z po.x k..gl.lxoH*.

o.t

3'

V

5 b*^fl" Vro,n*t*VLn 5 r"t'r-tPe*'t>

M.n-e*l-9nnn. ^a" = ?t ?

V- + AaY,,rxts<,vahla'- "t s*o"@

(e e)

,O-tc^-a-\o.0-U *lrgL\ Dtte- P. l'{/"1t - O.vw€̂ ".l-^" I t- &a^rya

cr\t ^gd<v-- "-?tzt -- ?'* ?

Page 27: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

\rJe- ).l"rr* rrrxii<-

o., ehs,^Qw€ ?

v(r)4 , Jr., ",'.f

v(r)4f"l.Fo'.=

rq,tl rZV

n- I -\-7

LL 1, u,fl(e +)

3

a+ - -L.o^6l"Ior*'-t"* b o-*k^^a- tl"^)' t-'

O"r'L U- tla , 1 ,.r.t!- auur- da L^Ctl)r\e $,lA,t,r-r63

30: 6^rr{ -'/a-*,t-.,.-.,.gf€- O"",'-

L-L*J\ol*

u,.rcl,lrtotf 'tp'Um '

Wl",r^^ "f^-" P

,-rl\,^.utr o- - tl (-

(6, t) o8)

(,\r-,6, Lt,rt I C I?" t' q"' 6') =

r.ue Gt'x'^])'^^/\e 0 a"nn'*

R9 v--g1n} ff""ou6'1"' E?

,n rr,rJ WL -,r-*- ?o-dt'dt Vat't^ be,wner, ltl<-rt) ,

l= u)e ,"ne-eaL /-;t, U ) . No"'r'l I'l"e .^"6tl^'-oo JU^ t'"t'

<,t, tL-'lilq,,$') {ta 5a'6'

I a*p,o^^tt- c}J\J\ -l,p,/-rv,I

Jhe ^f;^ '+"APc-ttAO

(67) n lr^r-^'S ezeL loy.f,rff.

^,..'^t

(€s)+_)_ Z v(?)aqr_,7,,lvot- rV-,Qr, f

6rd'

O-rLuc' 41, ra- Xae rf , ['+

{

Page 28: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

'r,rJe J-c^^-{4 Pl^"^ dia6',rar,','*Lt "*AU

(q,- q )

a)\

z. e._

(,\" ^ ? )6-I

(g,,r) (rL,6')

11" l^/asl'r^^^1o'-+"-^'l- z-\tc,d'<4 ,^*l<-na,eho.^^ Llae UD(a)^*b

Idr.^^

\/ ( \-'*.) --I v, ''9'l

tr.rU.rrtp e aJLL et,e- eVaxgo^' q [)^<- Pat' k JrA . IL'o Co'':!o'",^V

CL)). {t^ bwortP- ) & eof I", ,tlprt U -a

n ^^n'A l-o J&oL blo <tsrV u';{l^ V^e

pnorg e6'u'r'"/

q l!**r* geBrLt-t- Pl'*t 'x21 ( +o)

(6e)

v (*t - 24)a

.o-V'-*-s

lw, - *zl

We- {i^^;l .P -" o

^r^k 7e,^c,i'io"

J;"*^J* ""-k

r'^t-rr-d- :<e-,c/dz JoL (e l2

ll

e-a^-^ +c^-t/r-!.* VVe ',t^& t*

Page 29: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

L.l- e-^"-frlc V(? ) (6 \). we- ho,l,'e ( *'a'u't eD erLt ll"n"6' -LAUA

-9r" rr-t

LD)

v(q') =

(o

J&.,.u v(,n)

J.).

&"-.. ft, rf _pr

ot \r e,a

f

lL^l J.Lo utY- ,\-A .tM

l*h-" 4 l^e- ? a"''r i"D

6c).j,.,s1 l^e o"r )' 4)",-

k'o,^,"^? &""ce c:

?,

rf

%d

?,,af,)t e ,t tll eoO--AJ'*

I'C^rr* Qr ' 'r = Qn

iT _19.^,r-^.."9 e-

r oe-r- o

v(eh)

Yt?,) =

to

aZ.n e

t?fu?i

ZT &ro

co

pc

I

@

J "Loj '{- e-z

ro

I-t

-'9rLdz ?-

@

l"un)o

of4' e>

- i?v-e-

.]qr

(

- ,7.* -qf )j e-

o

ql-aP. eiQr

t e.

Page 30: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

-

N"lt-

,%",a"€

['urJ' i€ u)e ^tL P -- o o'l

xl,^*^"5>a Oo oW'u'5: ;It5*oft'1,',;^ poJ we u.>o's'Q& o la l-o-t-"'"' o-

,i.t oi*la ,p +o fl be/ca""/'/'\ *'A

v(9,) = LJ!?; --'?-/

r@ )

l" ]; g-.,p)t^ @ (-.?-rb)ra(

aoLq - P

Zve?i

?!

I

,P - i7?

t

lLva (P +r

/) ^ "?

?) ( y'"?) )

]? q z

.r.,U*r,h (^,^r * e.-*

./ (?).2\v e-

?' ,?' (+r )

a>

,T1r* o^^rr,.ilr P-'o i "-coil) ^-'n9] Vr },,,o"rea, o-*ugl (-, ?=o, Dd n

a;w.L ),/deek'

3o

Page 31: -fgtlandi/courses/04---second-quantization.pdf · To 'Y-t"L ll,rn Eaf, E uitj- @.!.to pDb0r:-r"l,\ ou-,r.d.c,f-t+eu,*;"e-".[ € *r-"L.A r,tr[^icl p.rrow--c,La* o- s^ar€- -tr- rc;.-r-rs

a-

Fo, &^"^ , )t[" n* ,QnoL a"l ,'""-athtst c+ o""a-^7& ' l^. '',.k, a,c+;.,',,'" "t

ga^ "d ^,\erll.-{- pa-x r{'.!lA, Qx{n<- od* u "'' ts''l y o* hg-"t c-e'-1

.t-^i

Vo r-< a-

(-> o-

.1r\^'-r-r O- il

V(?) = LlT VO

f(,r)o

tLlne- ,rccL'c<.rs t! |L'e ai:'"^^' J^^ *l/\^^ @-t&* il)e f

I"(e;'.to) [' L $'r oa-rOn

IJ

G--

a*

d. fr7oD

-i1ne- e-

), f1rZn Vo l&.J

o

zf

d-r'o-

_.,?r

r a,.^^ (q")?

4r Vo

Y:e Jo

TL

)-,t-,.t (4?) a-+ c@. (q-E ) (+z I

v(? ) ^1r?

!.t^i^ L4,c.Vs tiL+ev (?')

?