One can coose te bass n te 'bg' space V n te form of symmetrzed products of sngle partcle wavefunctons ' p(x) drawn from an ortonormal complete set of

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1 8.54: Many-body penomena n condensed matter and atomc pyscs Last moded: September 4, 3 Lecture 3. Second Quantzaton, Bosons In ts lecture we dscuss second quantzaton, a formalsm tat s commonly used to analyze many -b o d y problems. Te ey deas of ts metod were developed, startng from te ntal wor of Drac, most notably, by Fo c and Jordan. In ts approac, one tns of mult-partcle states of bosons or fermons as sngle partcle states eac lled wt a certan numb e r of dentcal partcles. Te language of second quantzaton often allows to reduce te many-body problem to a sngle partcle problem dened n terms of 'quaspartcles,'.e. partcles 'dressed' by nteractons.. Te Fo c space Te many -b o d y problem s dened for N partcles (ere, bosons) descrbed by te sum of sngle-partcle Hamltonans and te two-body nteracton Hamltonan: H = N H (x a ) a= a6=b H () (x a x b) H (x) = ; r x U (x) H () (x x () ) = U (x ; x ) () m were x a are partcle coordnates. In some rare cases (e.g. for nuclear partcles) one also as to nclude te tree-partcle, and ger order multpartcle nteractons, suc a s a b c H (3) (x a x b x c), etc. Te system s descrbed by te many-body wavefuncton (x x ::: x N ), symmetrc wt respect to te permutatons of coordnates x a. Te symmetry requrement follows from paretcles ndestngusablty and Bose statstcs (.e., te wavefuncton nvarance under permutatons of te partcles). Te wavefuncton (x x ::: x N ) obeys te Scrodnger t = H. Snce te numb e r of partcles n typcal stuatons of nterest s extremely large, solvng ts equaton drectly presents a very ard problem. Tere are several approaces, owever, tat allow to gan nsgt. Te metod of second quatzaton, storcally te rst many-body tecnque developed n te 3's, wll be dscussed ere. We sall start wt denng te Fo c space, sometmes also called te 'bg' manypartcle space, M n O N o V = V N symm te sum of te N -t symmetrc p o wers of te sngle partcle Hlbert space V. It descrbes te states of a system contanng any numb e r of partcles N = 3 :::. (3)

2 One can coose te bass n te 'bg' space V n te form of symmetrzed products of sngle partcle wavefunctons ' p(x) drawn from an ortonormal complete set of states n V,! = m!m!::: m m :::(x x ::: x N ) = ' p (x )' p (x ):::' pn (x N ) (4) N! wt te sum taen over all p e r m utatons of te states ' p(x). Te numbers m p ndcate ow m a n y tmes te functon ' p(x) appears n te product. Te number of permutatons n te sum s equal to te numb e r o f w ays to combne N elements nto groups contang m, m, etc., elements eac (m m ::: = N ). Ts combnatoral factor, equal to N!=m!m!:::, denes te normalzaton factor n Eq.(4). One can cec tat te states (4) are ortogonal and form a complete set n V. As an example, consder free Bose partcles n a b o x L L L, n wc case te sngle partcle states can be cosen as egenstates of te sngle partcle problem E' (x) = ; r '. Assumng perodc boundary condtons, we ave egenstates of a plane wave m form ' n(r) = p exp ( n r) n = n n = ( n n n 3) (5) V L wt nteger n 3 and V = L 3. Te energes of tese states are E n = s ten spanned by te functons m. Te space V ' n(r) p (' n(r)' m(r ) ' m(r)' n(r )) ' n(r)' n(r ) ::: (6) correspondng to te no-partcle state, one partcle, two partcles, etc. Te energes of tese states are E n E n E m E n ::: (7) Note tat te structure of te two-partcle functons depends on weter te partcpatng sngle-partcle states are derent or te same. To mae progress, one can ntroduce te so-called numb e r representaton. Allow any total partcle numb e r N and focus on te dependence of te state on te occupaton numbers m. Ts dependence s captured most vvdly by te representaton n wc a n auxlary oscllator, along wt te creaton and annlaton operators, s assocated wt eac sngle partcle state. Te occupaton numbers are nterpreted n ts representaton as numb e r of quanta n eacy oscllator. Te corresponng Foc states, n te numb e r representaton, ave te form Y m jm m ::: = p a j (8) = m! were j s te no-partcle state, and m = N. Ts representaton accounts correctly for te symmetry propertes of te states (4) due to Bose statstcs. n

3 . Second-quantzed operators In te numb e r representaton, te many-body Hamltonan s represented by a polynomal n te operators a, a : () H = H a a j H m a a j a a m (9) wt te quanttes H, H () m m beng te one- and two-partcle matrx elements, H = ' (x)jh (x)j' j (x) = ' (x)h (x)' j (x)dx () = ' (x)' j (x )jh () (x x H )j' (x)' m(x ) = ' (x)' j (x )H () (x x )' (x)' m(x )dxdx m One can prove te equvalence of ts representaton to te orgnal many -b o d y problem formulated n terms of many-partcle wavefuncton (x x ::: x N ) by drectly evaluatng te matrc elements of te Hamltonan between all pars of many body states, and sowng tat n b o t representatons te results agree. Te combnatorcs nvolved n ts proof s combersom, albet completely stragtforward. Instead of revewng t ere, we refer to te boo by J. R. S c reer, \Te Teory of Superconductvty" tat contans an Appendx descrbng te analyss n some detal. Anoter proof can b e devsed usng functonal ntegral, and we sall tal about t later on. Te expressons (9), are true for any ortogonal set of functons ' (x). In te case wen tese functons are cosen to b e te egenstates of te sngle-partcle problem, te matrx elements of H vans b e t ween derent states, and te one-partcle part of te Hamltonan smples to ' (x)jh (x)j' j (x) = E () H = E a Snce n^ = a a s notng but te numb e r operator, te egenvalues of te one-partcle Hamltonan correspondng to te numb e r states (8) are E n n ::: = n n ::: jh jn n ::: = E n (3) In te above example of free bosons n a box, te states are labeled by dscrete momenta, and te expresson (3) becomes E n n ::: = E n (4) () If te bosons are nteractng va a two-body potental U = U (r ; r ), from Eqs.(9), we obtan te two-partcle Hamltonan of te form H () = 3 4 () ju 3 a j 3 4 a a a 3 a 4 (5)

4 were te matrx element n (5), evaluated on te plane wave states (5), as te form () ju j 3 4 = ; r; r 3 r 4 r 3 e U (r ; r)d 3 V rd r (6) Ts expresson can b e smpled and evaluated by coosng a = r ; r as an ntegraton varable nstead of r, after wc te ntegral n (6) factors as ( e 4 ; )a U (a)d 3 a e ; r; r 3 r 4 r d 3 r = U ~ ( ; 4 ) = 3 4 (7) ~ R were U () = e ;r U (r)d 3 r s te Fourer transform of te nteracton potental, and V = 3 4 = 3 4 = (8) = Fnally, t e two-body Hamltonan taes te form H () = U ~ ( ; 4 )a a a a (9) V 3 4 = 3 4 were te sum s taen over all ntegers parameterzng te plane wave states (5) subject to te constrant = 3 4. Ts constrant, as t s clear from te calculaton, arses due to translatonal nvarance of te system. yscally, t expresses te conservaton of momentum n two partcle scatterng. Te second-quantzed nteracton Hamltonan s wrtten n terms of te operators a, a wc remove or add partcles. One may tus b e worred by apparent partcle non- conservaton. After loong at t closer, owever, and tang nto account te commutaton relatons of a, a wt te numb e r operator ^n = a a, na ^ = a( n ^ ; ) na ^ = a ( n ^ ) () one can sow tat te total number of partcles N = a a commutes wt te Haml- tonan and s tus conserved. ^.3 Feld operator A v ery useful representaton of te second-quantzed many-body amltonan s provded by te eld operator, r s t ntroduced by Jordan, '(x) ^ = a ^ ' (x) ' ^ (x) = a ^ ' (x) were x labels conguraton space, e.g., x = r n D = 3 Te states ' (x) n Eq. can be te egenstates of a sngle-partcle problem, suc as, e.g., te plane waves of te above secton, or any oter convenent ortonormal bass set. 4

5 Te operators obey commutaton relatons ['(x) ^ '(x ^ )] = [' ^ (x) ' ^ (x )] = ['(x) ^ ' ^ (x )] = (x ; x ) () wc can b e proven by usng te commutaton relatons of a ^ and a ^ along wt te ortogonalty of te states ' (x). For example, ^ ' a ^ ] = ' (x) ' (x ) (3) [ '(x) ^ (x )] = ' (x) ' (x )[^ a = ' (x) ' (x ) = (x ; x ) (4) We note tat, altoug some partcular bass set ' (x) was employed to contruct te eld operators, ter propertes, suc as te commutaton relatons (), are nvarant wt respect to te coce of bass. Usng te eld operators, te second-quantzed problem (9), can b e expressed as a polynomal! H = ' ^ (x) ; r x U (x) '(x)dx ^ m ' ^ (x)' ^ (x )U(x ; x )'(x) ^ '(x ^ )dxdx (5) n wc te quadratc and te quartc parts descrbe nonnteractng partcles and ter nteracton, respectvely. It s sometmes elpful to tn of '(x) ^ as a \quantzed wavefuncton." In te eld operator representaton, te many b o d y problem starts loong very muc le a sngle partcle problem. Of course, ts smplcty s only apparent, snce we stll ave a quartc term n te Hamltonan, expressng te nteractons and leadng to \nonlnear" dynamcs. Not just te Hamltonan, but many oter quanttes also tae a smple form n terms of te eld operator. For example, partcle densty n(x) = (x ; x a ) b ecom es n(x) ^ = ' (x )j(x ; x )j' j (x )a a ' j = ^ (x)'(x) ^ (6) R due to ' (x )(x ; x )' j (x )dx = ' (x)' j (x). Smlarly, te partcle current operator s ^ j(x) = ' (x)r ' x '(x) ; r x (x) '(x) (7) m Te densty (6) and current ( 6 ) o b e y c o n tnuty relaton wrtten as an operator t n ^ rj ^ =. a 5