Second Quantization for Fermions

Transcription

1 9 Chaper Secod Quazao for Fermo Maro Pr Iuo Superor de Ceca y Tecología Nucleare, Ave Salvador Allede y Luace, Qua de lo Molo, La Habaa 6, Cuba. The objec of quaum chemry co of eracg may parcle yem of elecro ad ucle. A accurae decrpo of uch yem requre he oluo of he may-parcle Schrödger equao. I prcple, he -body wave fuco cofgurao pace coa all poble formao, bu a drec oluo of he Schrödger equao mpraccal. I herefore eceary o reor o approxmae echque, ad o wor wh he framewor of more covee repreeao of he quaum mechac operaor ad wave fuco: he Secod Quazao. I a relavc heory, he cocep of Secod Quazao eeal o decrbe he creao ad deruco of parcle. However, eve a o-relavc heory, Secod Quazao grealy mplfe he dcuo of may decal eracg parcle. Th formalm ha everal dc advaage: he ecod-quazed operaor corporae he ac (Ferm our cae) a each ep, whch cora wh he more cumberome. L.A. Moero, L.A. Díaz ad R. Bader (ed.), Iroduco o Advaced Topc of Compuaoal Chemry, 9-39, 3, 3 Edoral de la Uverdad de La Habaa, Havaa.

2 3 approach of ug aymmerzed produc of gle-parcle wave fuco. I alo allow u o cocerae o he few marx eleme of ere, hu avodg he eed for dalg drecly wh he may-parcle wave fuco ad he coordae of all he remag parcle. Fally, he Gree fuco ad dey marce, whch coa he mo mpora phycal formao cocerg o he groud-ae eergy, he eergy ad lfeme of exced ae, ad oher molecular propere, are ealy expreed h formalm. The Schrödger equao Ay problem o o-relavc elecroc rucure requre he oluo of he Schrödger equao, Ĥ Ψ EΨ () where Ĥ ad Ψ repree he Hamloa ad he wave fuco of he yem repecvely. I he Bor-Oppehemer approxmao, he elecroc Hamloa for aom ad molecule (he yem of our ere) ae he form N ZI Ĥ ˆ elec ĥ () r r r I I j j j j where ĥ he h-core operaor, whch coa he ec eergy ad he poeal eergy of eraco bewee ucle ad elecro. We hould oberve ha h erm compoed by he um of operaor volvg he coordae of he parcle oe a a me hece, belog o he group of he ymmerc oe-parcle operaor: Â (3) ĥ The ecod erm repree he poeal eergy of eraco bewee every par of parcle herefore; belog o he o-called ymmerc wo-parcle operaor, Bˆ rj ' rj (4) j. j Here, all par of parcle are coued oce, whch accou for he facor of ½, ad he double um ru over he dce ad j eparaely, excludg he value equal j. Thee wo d of operaor are he mo frequely oe almo all cae of ere quaum chemry. The Udguhed Prcple I well ow ha for a correc decrpo of a yem wh decal parcle o eough o olve for he Schrödger equao, eceary o corporae he ac of he parcle. I he cae of a aembly of fermo, he may-parcle wave fuco aumed o have he followg propery Ψ x,x,...,x,x,...,x Ψ x,x,...,x,x,..., (5) ( ) ( ) j j x where x deoe he coordae of he h parcle, cludg he paal coordae r ad ay dcree varable uch a he z compoe of p. Equao (5) how ha he wave fuco mu be aymmerc uder he erchage of he coordae of ay wo parcle. For a yem wh depede parcle, we ca ealy afy he aymmerc rerco ug he Slaer ormalzed deerma

3 3 ( x,x,...x ) where ( x), ψ ( x),..., ( x) ψ ( x) ψ( x)... ψ ( x) ( x ) ψ ( x )... ψ ( x ) ψ!... ψ Ψ (6) ( x ) ψ ( x )... ψ ( x ) ψ ψ are he orhoormal gle-parcle wave fuco occuped by all parcle. We hould oe ha here ex gle-parcle ae bu we do o ow exacly whch parcle occupe each ae. Moreover, afed rgorouly he Paul excluo prcple. Coequely, we ca coder a complee e of orhoormal me-depede gle-parcle wave fuco { ψ ( )},,,..., ; ψ ( x) ψ ( x) x j dx δj (7) ad he occupao umber for each ae ( ) mu be ae or. O he oher had, f we have a complee orhoormal e of oe-parcle fuco { ψ ( x) } he, he e of all deerma { Ψ I } bul o hee fuco, equao (6), coue alo a complee ad orhoormal -parcle ba. Therefore, we ca ow expad ay eracg may-body wave fuco a follow: Ψ I C Ψ (8) Th expreo compleely geeral ce mply he expao of he wave fuco a complee e of ae. From he Fr o he Secod Quazao Le u ow aalyze he mple poble fermo yem: wo elecro a ymmerc poeal well. We wll ae a gle-parcle fuco hoe obaed olvg he oe-parcle problem for h poeal, ad umbered he creag order of eergy. I The ae of he yem correpodg o Slaer deerma ca be wre a Ψ j { ψ( x) ψ j( x) ψ j( x) ψ( x )}, j ;, j,,3,... (9) ψ x are he gle-parcle fuco, whch afy equao (7). where { ( )} The ymmerc oe-parcle operaor ae he form  V( x) V( x ) () where V could be a exeral poeal, ad a example of marx eleme * * * * Ψ3 Â Ψ [ ( x ) ( x ) ( x ) ( x )]( V( x ) V( x )) ψ ψ3 ψ3 ψ () * [ ψ( x) ψ( x ) ψ( x) ψ( x )] dxdx ψ3( x) V( x) ψ( x)dx I he prevou reul, wa ae o accou he orhoormaly of he gleparcle fuco. I

4 3 Le u ow oberve ha for a ymmerc operaor, he marx eleme doe o chage f we hold oly aymmerc oe of wave fuco, elmag furher he ormalzao coeffce, ha, * * Ψ3 Â Ψ [ ψ( x) ψ3( x )]( V( x) V( x )) () * [ ψ( x) ψ( x ) ψ( x) ψ( x )] dxdx ψ3( x) V( x) ψ( x)dx The ex ep o roduce he marx oao. For he ae of mplcy, we wll rera our pace o hree ae V V V 3 ψ, ψ, ψ3, V V V V 3 (3) 3 V 3 V 3 V 3 where V ψ * ( x) V( x) ψ ( x)dx, AV V I h oao, he marx eleme dcued above expreed a V V V 3 Ψ3 Â Ψ V V V 3 3 V 3 V 3 V 3 (4) V V V 3 V V V 3 3 V 3 V 3 V 3 Here, he ub-dce ad defy he parcle. Noe ha he wave fuco o he lef, a well a he fr erm of he wave fuco o he rgh, whe he parcle defer creae, he occupao umber colum ge dow. We wll ay ha hee fuco are a ormal form. O he corary, he la erm of he fuco Ψ oormal. Performg he marx produc we ge Ψ Ψ 3 V (5) 3 Â Furhermore, we have o oe ha marx V dffer from V oly he parcle defer, ce boh marce are equal. I more covee o roduce a ew marx e defed by all eleme equal zero excep oe row ad colum, for example, e (6) Thu, we ca wre marx V he form 3 V, A a coequece, he oe-parcle operaor become V e (7)

5 33 3, 3, E V e V A, e E (8) Equao (8) how a mpora reul: marx E doe o deped o he parcle defer. Th a crucal po he whole reame, ead of performg a um over parcle, we ca well um over ae. The orgal wave fuco ca ow be rewre a follow ~ Φ Φ (9) Φ repree ha oly he fr ad he hrd ae are occuped, whle he fr ad he ecod ae occur ~ Φ. The wave uder he la fuco deoe ha aymmerzed. Th reul how ha wave fuco are characerzed by he occupao umber of each ae, whch are equal eher zero or oe. Thu, we have obaed a occupao umber repreeao. Le u ow cocerae our aeo he marx eleme of E. I eay o ee ha marx E ha o ac o a wave fuco a he rgh de wh a occuped ae, ad mlarly o a wave fuco a he lef de wh a occuped ae. I parcular, ( ) ~ E, 3 Φ Φ () Thu, covee o mplfy he oao by repreeg marx E a a produc of wo ew operaor,.e., E () where a operaor ha ahlae a parcle he ae o he rgh de, a log a deroy a parcle he ae o he lef oe. However, we have o coder a lle complcaed deal: ome marx eleme of E volve a mu g, for example ( ) ~,E 3 Φ Φ ()

6 34 The appearace of a phae facor, whch chage he g of marx eleme becaue of he o-ormal form of he ecod erm o he rgh de ha ha a occuped ae correpodg o a occuped ae o he lef. I geeral, we ca regard ha marx E move a parcle from he gle ae o he ae, ad he phae facor cocer o he umber of parcle eceary o permue h procedure. Coequely, we ca deerme he phae facor of ( Φ, E Φ ~ ) by he expreo ( ) where r he occupao umber of he ae r., r (3) r Icludg he phae facor o he ew operaor whch defe marx E oe ca ge r r r r r r r r r ( ) r, ( ) (4) ( ) ( ) ( ) ( ) ( ) ( ) (5) I he prevou reul, we have uppoed ha >, ad correpod o he occupao umber of he fal ae of he yem. Th value hould be zero ce he fal ae obaed afer he ahlao by he operaor of he gle ae he al ae of he yem. The cluo of he phae facor he operaor exclude he requreme o he aymmerzao of he wave fuco a he rgh de. Thu, we fally ca geeralze our reul, ad wre ou he marx eleme of he oe-parcle ymmerc operaor a: Φ..., Φ V (6), where {} deoe he occupao umber of gle-parcle ae he al wave fuco of he yem, whle { '} repree he occupao umber of hee oe-parcle ae cocerg he fal oe. Le u ow aalyze marx eleme of he ymmerc wo-parcle operaor (,x j) ' V( x, x j) Bˆ V x (7) j.j For h mple yem formed by wo parcle, he operaor (7) oly oe V, whch could be he Coulomb eraco.

7 35 The marx eleme for ace, * * Ψ3 Bˆ Ψ ψ3( x) ψ( x ) V( x, x ) ψ ( x) ψ( x ) dxdx * * ψ( x) ψ3( x ) V( x, x ) ψ ( x) ψ( x ) dxdx (8) Iroducg marx oao, mlarly we dd before for oe-parcle operaor, oe ca oba Bˆ V E 'E',,',' (9) where * * V ψ ( x) ψ ( x ) V( x, x ) ψ ( x) ψ ( x ) dxdx (3) Here he expreo E 'E' a fuco of he ew operaor ad whch hould be obaed. There are four poble d of produc of hee operaor our example, ha, ' ' 3 ' 3 ' ' 3 ' ' ' 3 (3) The calculao how ha he fr wo vara gve ame reul a la oe, bu precedg by a phae facor. The caue of h 3 V 3 V, 3 V 3 V (3) Tag o accou he phae facor oe ca wre: E 'E' ' ' (33) I fac, for our yem of wo parcle we have ( Φ, 3 ) Φ ( ) Φ, ( )( ) 3 Φ (34) ( Φ, 3 Φ ) ( Φ, ) 3 Φ The re of erm oly afford zero. The,, '' V Φ ' ' Φ { 3 V 3 V,,',' (35) 3 V 3 V } 3 V 3 V Geeralzg h reul, he marx eleme of wo-parcle ymmerc operaor ca be wre a Φ Φ..., '' V ' ' (36),,',' Creao ad deruco operaor The ju roduced operaor afy ome propere. Le u oe ha he aco of he par of operaor ad over a ame fuco dffer oly a g, due o phae facor aocaed o each operaor. Accordgly o h, operaor ad afy he acommuao rule

8 36 [, ], [ ] (37) (38) O he oher had, we ca oberve ha operaor, whch ahlae a parcle o he lef, gve he ame reul f we coder ha creae a parcle acg o he wave fuco a he rgh de. Hece, we wll call h d of operaor a creaor. Followg a mlar procedure le before, oe ca oba he la acommuao rule for hee operaor, ha,, or [, ] δ (39) Le u ow repree he occupao umber ae hrough he creao ad deruco operaor. For h purpoe, we roduce he vacuum wave fuco Φ..., whch here are o parcle. Accordgly, for every deruco operaor oe ca oba Φ (4)... Ay wave fuco of a yem ca be obaed creag parcle by he aco of operaor o he vacuum. Some example are: Φ 3Φ Φ Φ (4) Uually, o ga geeraly he occupao umber ae are repreeed a medepede abrac ae vecor α αα3...α (4) where he oao mea ha (4) a ae of parcle, whch, here a parcle he egeae α, aoher oe he egeae α, ec, of a complee orhoormal e of gleparcle ae { α }. The egeae of h e whch do o appear (4) are empy. Correpodgly, he vacuum ae repreeed a, ad he creao ad deruco operaor afy he followg equao: α, (43), α (44) I ealy ee ha he e of acommuao rule (37) (39) produce he correc ac:., herefore, whch preve wo parcle from occupyg he ame ae.., herefore ( ) ( ) or ( ). Th la relao mple ha he umber operaor for he h mode ˆ, ad ha he egevalue zero ad oe, a requred. Thu, he parcle umber operaor ˆ.

9 37 3. α α... α ( ) αα...(o α )... α α Ω..., { α α } α Ω Ω....α Example: The Harree-Foc eergy Nowaday, oe of he mo ueful mehod Quaum Chemry he Harree-Foc (HF) approxmao. I h mehod, he groud-ae fuco ae a a depede - parcle ae, whch he ecod quazao formalm ae he form Ψ HF αα α3... α, { α α } Ω (45)...α Tag o accou ha Hamloa () ow repreeed a Ĥ ĥ '' ' ', (46),,',' oe ca very ealy calculae he groud ae eergy. Ceraly, Ĥ ĥ '' ' ' (47),,,',' where ad ' ' are he o-called fr ad ecod order Reduced Dey Marce (- ad -RDM) repecvely. I he -RDM, he ae ha o be pree he e of he HF gle-parcle ae Ω o he corary, he operaor acg o he lef yeld zero. A he ame me, he operaor hould creae a parcle he rece ahlaed ae o oba aga he HF fuco, due o he orhogoaly of he oe-parcle ae. Thu, oe ge δδ, Ω (48) The uao for he -RDM very mlar: he operaor ad have o ahlae wo parcle ae belogg o he e Ω, whle operaor ad mu creae hee ae o geerae aga he HF fuco vew of he orhogoaly. There are wo dffere way of afyg h codo, ha afford he followg ' ' ( δ' δ' δ' δ' ) δjδ,, j Ω (49) Subug he reul (48) ad (49) obaed for he HF - ad -RDM o equao (47), oe ca oba Ĥ ĥ [ j j j j ] (5) Feld I ofe covee o form he lear combao of he creao ad deruco operaor ψˆ (x) ψ (x) (5),j * ψˆ (x) ψ (x) (5)

10 38 where he coeffce are he gle-parcle wave fuco ad he um over he complee e of gle-parcle quaum umber. Here, he dex may deoe ow he e of quaum umber {, z }. Thee quae ψˆ ad ψˆ are called feld operaor. They are operaor becaue hey deped o he creao ad deruco operaor. The feld operaor afy mple acommuao relao * ψˆ (x), ψˆ (x') ψˆ (x) ψˆ (x') δ(x x' (53) [ ] ) [ ψˆ (x), ψˆ (x')] [ ψˆ (x), ψˆ (x')] (54) Thee equale follow from he acommuao relao for he creao ad deruco operaor, ad he compleee of he gle-parcle wave fuco. The hamloa operaor () ca be rewre erm of hee feld operaor a follow: Ĥ dx ψˆ (x)h(x) ψˆ (x) dxdx' ψˆ (x) ψˆ (x')v(x,x') ψˆ (x') ψˆ (x) (55) Th expreo readly verfed ce he egrao over paal ad p coordae produce he correpodg marx eleme, leavg a um of hee oe mulpled by he approprae creao ad deruco operaor. Noe carefully he orderg of he la wo feld operaor he Coulomb poeal eergy, whch eure ha he hamloa herma. I h form, he hamloa ugge he ame of ecod quazao ce he above expreo loo le he expecao value of he hamloa ae bewee wave fuco. The quae ψˆ ad ψˆ are o wave fuco, however, bu feld operaor, hu ecod quazao he feld are he operaor ad he poeal ad ec eergy are ju complex coeffce. Clog Remar The ecod quazao a good repreeao of he quaum mechac operaor ad wave fuco hrough he creao ad deruco operaor. Srcly peag, here o really aoher quazao. I quaum formalm very ueful for reag may decal eracg parcle yem. The ma advaage of h formalm he guaraee of he udguhed prcple, whch mplcly he acommuao relao of he operaor ad. Thu, oe avod he cumberome wor of ug aymmerzed produc of gle-parcle wave fuco. Recommeded Bblography R. P. Feyma, Sacal Mechac, Readg Ma: Bejam (97). J. Avery, Creao ad ahlao operaor, McGraw-Hll (979). J.P. Blazo ad G. Rpa, Quaum Theory of fe yem, Cambrdge Ma: MIT Pre (986). Referece: P. A. M. Drac, Proc. Roy. Soc. (Lodo) 4A, 43 (97).

11 39 P. Jorda ad O. Kle, Z. Phy 45, 75 (97); P. Jorda ad E. P. Wger, Z. Phy 47, 63 (98); V. Foc, Z. Phy 75, 6 (93).