SummetrÐec kai Quarks Nikìlaoc A. Tetr dhc Iw nnhc G. Flwr khc
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Perieqìmena Eisagwgikèc ènnoiec 5. Eisagwg............................. 5.2 SummetrÐa Isospin......................... 0 2 StoiqeÐa JewrÐac Om dwn 3 2. SummetrÐec kai Om dec (Groups)................. 3 2.2 'Algebra miac Om dac....................... 7 2.3 'Algebra thc Stroform c..................... 9 2.4 Anaparast seic thcom dac................... 23 2.5 H Om da SU(2).......................... 28 2.6 SÔnjeshAnaparast sewn.................... 3 2.7 Peperasmènec om dec summetrðac P, C............. 36 2.8 SU(2)-Isospin........................... 37 2.9 H om da SU(3).......................... 39 2.0 Di gramma B rouc........................ 4 2. Isospin kai Strangeness...................... 42 3 Statikì Montèlo twn Quarks 49 3. KbantikoÐ ArijmoÐ twn Quarks.................. 49 3.2 Idiokatast seic (q q): Mesìnia.................. 54 3.3 Problèyeic gia ta mesìnia.................... 6 3.4 Idiokatast seic (qqq): Baruìnia................. 64 3.5 Analutikìc prosdiorismìc idiokatast sewn........... 75 3.6 Problèyeic gia ta baruìnia.................... 8 3.7 Magnhtikèc Ropèc......................... 87 3
4 PERIEQ OMENA
Kef laio Eisagwgikèc ènnoiec. Eisagwg ParathroÔme peiramatik ìti hm za tou prwtonðou kai tou netronðou eðnai perðpou Ðsec: m p m n. H sômptwsh aut mac odhgeð na jewr soume to prwtìnio p kai to netrìnio n wc dôo idiokatast seic (states) isospin enìc kai mìno swmatðou, tou noukleonðou. Gia par deigma, èqoume dôo idiokatast seic spin gia èna swm tio me spin S = /2: thn idiokat stash me probol spin ston z- xona m = +/2 (sômbolo ) kai thn idiokat stash me m = /2 (sômbolo ). Kai stic dôo katast seic, hm za tou swmatðou eðnai hðdia: m( ) =m( ). Oi dôo katast - seic spin gia to swm tio apoteloôn mia diplèta (doublet). Kat' analogða, èqoume dôo idiokatast seic isospin gia èna swm tio me isospin I = /2 (p.q. noukleìnio): thn idiokat stashme probol isospin ston z- xona I 3 =+/2 (sômbolo p) kaithnidiokat stashme I 3 = /2 (sômbolo n). Kai stic dôo katast seic, hm za tou swmatðou eðnai hðdia: m(p) m(n). Oi idiokatast seic isospin (p, n) sqhmatðzoun mia diplèta (isospin doublet). 'Estw èna sôsthma dôo noukleonðwn. K je noukleìnio èqei spin /2 me idiokatast seic spin (, ). SÔmfwna me thn prìsjesh stroform n, to sôsthma dôo noukleonðwn mporeð na èqei eðte S =(summetrik triplèta 2 +=3 - spin triplet), eðte S = 0(antisummetrik aplèta 2 0 + = - spin singlet). Oi idiokatast seic autèc eðnai: S =,M S = = spin triplet S =,M S =0 = ( + ) 2 S =,M S = = { spin singlet S =0,M S =0 = ( ) 2 5 (.)
6 KEF ALAIO. EISAGWGIK ES ENNOIES ParathroÔme ìti h triplèta eðnai summetrik kai h monadikìthta antisummetrik sthn enallag twn noukleonðwn () kai (2). Gia na katal xoume stic katast seic (.), arkeð na qrhsimopoi soume ton telest kat bashc (lowering operator) S, hdr shtou opoðou p nw sthn idiokat stash S M S eðnai: S SM S = S(S +) M S (M S ) S, M S. (.2) Kataskeu zoume thn fian terhfl kat stash S =,M S =, hopoða ja prèpei na eðnai grammikìc sunduasmìc ìlwn twn dunat n idiokatast sewn S =/2,M S S 2 =/2,M S2, twn dôo noukleonðwn (ìpou M S = ±/2), ètsi ste M S = M S + M S2, dhlad : S, M S = M M 2 /2,M /2,M 2. (.3) M,M 2 C (S) Oi suntelestèc C (S) M M 2 onom zontai suntelestèc Clebsch-Gordan kai brðskontai kataqwrhmènoi se pðnakec. Na shmeiwjeð, ìti /2, /2 kai /2, /2. Gia thn an terh kat stash S =,M S =, omonadikìc sunduasmìc twn M, M 2 (me epitreptèc timèc ±/2) pou dðnei M + M 2 =eðnai o profan c M = M 2 =/2 (summetrik kat stash). Epomènwc, S =,M S = = /2, /2 /2, /2. Gia na broôme thn amèswc qamhlìterh kat stash S =,M S =0, ja dr soume sthn S =,M S = me ton telest S. O telest c S gia ticidiokatast seic tou olikoô sust matoc dôo swmatidðwn eðnai S = S x is y =(S x, + S x,2 ) i(s y, + S y,2 )=(S x is y ) +(S x is y ) 2 = S, + S,2. H dr sh tou telest kat bashc ja gðnei wc ex c: S, =(S, + S,2 ) = (S, ) + (S,2 ). Apì thn sqèsh (.2), èqoume S,(i) = ( +) ( ) =, ìpou 2 2 2 2 i =, 2. Sunep c, h dr sh tou telest S dðnei S, = +. All, S, = ( + ) ( ), 0 = 2, 0. Epomènwc, lônontac wc proc thn, 0, èqoume to deôtero mèloc thc diplètac (se kanonikopoihmènh morf ):, 0 = ( + ). (.4) 2 Qrhsimopoi ntac thn Ðdia mèjodo mporoôme na prosdiorðsoume thn kat stash,. Sugkekrimèna, èqoume: S, 0 = 2, = 2 (S, ) + 2 (S,2 ) = 2, dhlad,, =. (.5)
.. EISAGWG H 7 ParathroÔme ìti, xekin ntac apì thn summetrik idiokat stash, kai dr ntac me ton summetrikì (wc proc ta dôo swm tia) telest S = S, + S,2, ja prokôyoun 2 +=3summetrikèc katast seic me to Ðdio S =, kai me ìlec tic dunatèc probolèc M S (triplèta). Den mporoôme na prosdiorðsoume tic katast seic S =0(dhlad thn aplèta (singlet) 0, 0 ) me apl qr sh tou S, ìpwc prohgoumènwc, diìti o telest c S 2 (Casimir operator) metatðjetai me ton S [, dhlad S, S 2] = 0, kai sunep c o S den apoteleð telest kat bashc gia thn idiotim S. Epistrèfoume sto an ptugma (.3) kai parathroôme ìti oi mìnec epitreptèc timèc gia ta M,M 2 ste na èqoun jroisma mhdèn, eðnai oi M =/2, M 2 = /2 kai h M = /2, M 2 =/2. Epomènwc, mporoôme na gr youme 0, 0 = C /2, /2 +C /2,/2, ìpou apomènei na prosdioristoôn oi suntelestèc Clebsch-Gordan tou anaptôgmatoc. To an ptugma thc parap nw kat stashc moi zei sthn, 0, diìti kai oi dôo apoteloôn grammikoôc sunduasmoôc twn idðwn anusm twn b shc. Epeid, ìmwc, oi dôo autèc idiokatast seic eðnai diaforetikèc (antistoiqoôn se diaforetikì S), ja eðnai orjog niec. H sqèshorjokanonikìthtac gr fetai genik S,M S, M = δ SS δ MM. Efarmìzoume thn apaðthsh orjogwniìthtac stic, 0 kai 0, 0 : 0, 0, 0 =0= 2 (C /2, /2 + C /2,/2 ), h opoða sunep getai, amèswc, C /2, /2 = C /2,/2. Dhlad, sthn sqèsh paramènei mìno mia pollaplasiastik stajer, h opoða prosdiorðzetai apì thn kanonikopoðhsh: 0, 0 = C /2, /2 ( ). Opìte, brðskoume kai thn kanonikopoihmènh morf thc antisummetrik c aplètac (singlet): 0, 0 = ( ). (.6) 2 To k je noukleìnio èqei isospin I =/2, stehi 3 = ±/2 na antistoiqeð sthn kat stash tou p kai tou n, antðstoiqa. Kat' analogða me to spin, oi dunatèc katast seic isospin dôo noukleonðwn ja perilamb noun mia summetrik triplèta (I =) kai mða antisummetrik aplèta (I =0). Sugkekrimèna, I =,I 3 = = pp isospin triplet I =,I 3 =0 = (pn + np) (.7) 2 { isospin singlet I =,I 3 = = nn I =0,I 3 =0 = (pn np) 2
8 KEF ALAIO. EISAGWGIK ES ENNOIES Par deigma An ta dôo noukleìnia brðskontai se katast seic qamhl n sqetik n energei n (sqetik c troqiak c stroform c L =0), na qrhsimopoihjeð h Arq tou Pauli gia na deiqjeð ìti to jroisma S + I eðnai perittìc akèraioc. Wc gnwstìn, to prìblhma twn dôo swm twn pou allhlepidroôn me dunamikì V ( r 2 r ) an getai ston prosdiorismì thc kðnhshc tou kèntrou m zac kai thc sqetik c kðnhshc, dhlad thc kðnhshc swmatðou (m zac Ðshc me thn anhgmènhm za) sto kentrikì dunamikì V (r) =V (r). Tìte, oiidiosunart seic thc stroform c L 2 sthn anapar stashjèshc eðnai apl c oi sfairikèc armonikèc Y m l (Ω), oi opoðec èqoun omotimða (parity) ( ) l. Profan c, h enallag twn swmatðwn () kai (2) se sfairikèc suntetagmènec sunep getai ton metasqhmatismì r r, Ω Ω. Epomènwc, Yl m ( Ω) = ( ) l Yl m (Ω). Gia qamhlèc enèrgeiec to sôsthma brðsketai sth jemeli dh st jmh (ground state) me L = M =0, ètsi ste Y0 0( Ω) = Y 0 0 (Ω). Autì shmaðnei ìti h qwrik kumatosun rthsh eðnai summetrik sthn enallag twn swmatidðwn. Gia na mhn parabi zetai hapagoreutik arq tou Pauli prèpei holik kumatosun rthsh tou sust matoc na eðnai antisummetrik sthn enallag twn dôo swmatðwn. H kumatosun rthsh tou sust matoc ja eðnai ψ(, 2) = qwrik S, S 3 I,I 3. Kat thn enallag, èqoume S, S 3 ( ) S S, S 3 kai I,I 3 ( ) I I,I 3. Epomènwc ψ(2, ) =( ) S+I ψ(, 2). H apaðthsh tou na eðnai antisummetrik holik kumatosun rthsh gr fetai ( ) S+I =, to opoðo sunep getai ìti o ekjèthc S + I eðnai perittìc akèraioc. H purhnik dônamhparamènei analloðwth(invariant) semetasqhmatismoôc isospin, dhlad eðnai anex rthth apì thn tim tou I 3 se k je pollaplèta ( se k je sônolo idiokatast sewn me kajorismèno I). H shmasða thc analloiìthtac aut c ja faneð, amèswc, exet zontac touc pur nec 6 2He, 6 3Li kai 6 4Be. Oi pur nec autoð mporoôn na jewrhjoôn wc dèsmia sust mata nn, np kai pp kollhmèna se èna swm tio lfa 4 2He to opoðo èqei I =0, dhlad : 6 He + nn 2 He =4 2 6 3Li = 4 2 He + np (.8) 6 4 Be =4 2 He + pp An k noume mia diìrjwshgia thn pwsh Coulomb metaxô twn prwtonðwn kai gia thn diafor m zac p n, prokôptei ìti oi m zec tou 6 2He kai tou 6 4Be eðnai Ðsec, en to lðjio, 6 3Li, mporeð na brejeð se dôo katast seic: mia me Ðdia m za me touc llouc dôo pur nec kai mia deôterh kat stash diaforetik c m zac. Oi diorjwmènec purhnikèc m zec sumfwnoôn me tic problèyeic gia mia triplèta I =(perilamb nei touc pur nec hlðou kai bhrullðou kaj c kai ton pur na lijðou sth diegermènh kat stash) kai mia aplèta I = 0(pou apoteleð thn
.. EISAGWG H 9 jemeli dh kat stash tou lijðou). To gegonìc ìti parousi zetai ekfulismìc (wc proc tic m zec), antikatoptrðzei to gegonìc ìti h purhnik dônamh eðnai analloðwth se strofèc isospin (dhlad, h purhnik dônamh sumperifèretai isìtima se ìla ta mèlhmiac pollaplètac adronðwn).
0 KEF ALAIO. EISAGWGIK ES ENNOIES.2 SummetrÐa Isospin Anafèrame parap nw ìti oi purhnikèc (isqurèc) dun meic eðnai analloðwtec se strofèc ston q ro tou isospin. H summetrða aut onom zetai summetrða isospin. H summetrða aut mporeð na gðnei perissìtero katanoht me b sh mia analogða. 'Opwc gnwrðzoume, up rqei aprosdioristða( eleujerða) prosanatolismoô enìc hlektronðou se kentrikì dunamikì. Dhlad, mia kat stash me kajorismènh stroform l perièqei (2l +) dunatèc idiokatast seic ( dunatoôc prosanatolismoôc), oi opoðec eðnai isodônamec metaxô touc, afoô h enèrgeia E = E nl exart tai mìno apì to metro l thc stroform c kai ìqi apì ton prosanatolismì (dhlad ton kbantikì arijmì m). H aprosdioristða tou prosanatolismoô sundèetai me th summetrða tou probl matoc se strofèc (rotational invariance): H Qamiltonian (ousiastik to dunamikì V (r) =V (r)) eðnai analloðwth se strofèc. H summetrða aut thc H odhgeð ston ekfulismì tou f smatoc. O ekfulismìc sunðstatai sto ìti (2l +) diaforetikèc idiokatast seic antistoiqoôn sthn Ðdia idiotim enèrgeiac. To ìti o ekfulismìc eðnai sunèpeia thc summetrðac prokôptei an jewr some ton telest U o opoðoc ja dr sei p nw stic katast seic tou sust matoc kai ja ektelèsei thn strof ( ìpoia llh pr xh summetrðac jel some). H kat stashmetasqhmatðzetai sômfwna me thsqèsh: ψ ψ = U ψ. (.9) H pijanìthta èna sôsthma pou perigr fetai apì thn kat stash ψ na brðsketai kai sthn kat stash φ prèpei na paramènei analloðwth k tw apì to metasqhmatismì. Autì sunep getai: ψ φ 2 = ψ φ 2 = ψ U U φ 2. (.0) Epomènwc o telest c U prèpei na eðnai monadiakìc: U U =. (.) An oi strofèc sto q ro isospin eðnai summetrðec tou sust matoc, ja prèpei h Qamiltonian na paramènh analloðwth k tw apì th dr sh tou telest U. Autì shmaðnei ìti tastoiqeða pðnaka φ H ψ prèpei na paramènoun amet blhta se strofèc isospin: φ H ψ = φ H ψ = φ U HU ψ, (.2) gia ìlec tic katast seic φ, ψ. Epomènwc, hqamiltonian ja prèpei na ikanopoieð thsqèsh: U HU = H. (.3)
.2. SUMMETR IA ISOSPIN Oi idiokatast seic thc Qamiltonian c H ikanopoioôn thn exðswsh: H dr shtou U apì ta arister sthn exðswshdðnei: H ψ = E ψ. (.4) UH ψ = E (U ψ ) =UHU (U ψ ) =H (U ψ ). (.5) Sunep c, h metasqhmatismènh kat stash ψ = U ψ prèpei na eðnai idiokat stashthc Qamiltonian c me thn Ðdia idiotim me thn ψ (ekfulismìc). Me to Ðdio akrib c skeptikì, kai epeid parathroôme ìti up rqei ekfulismìc maz n ( energei n) t xhc 3 se sôsthma dôo noukleonðwn, upojètoume thn Ôparxh tou (an logou me to spin) kbantikoô arijmoô isospin I. Gia dedomèno I up rqoun 2I + idiokatast seic, oi opoðec eðnai isodônamec metaxô touc afoô odhgoôn se ekfulismì (Ðsec m zec se k je triplèta me I =.). H isodunamða aut twn (2I +) idiokatast sewn isospin ( alli c o ekfulismìc maz n t xhc 2I +) ja prèpei kat' analogða me ta parap nw na eðnai apotèlesma k poiac eidik c summetrðac tou probl matoc. Aut den eðnai llhapì thn proanaferjeðsa analloiìthta twn purhnik n dun mewn se metasqhmatismoôc isospin. H analloiìthta twn purhnik n dun mewn stic allagèc tou I 3 (gia dedomèno k je for I), dhlad h adunamða di krishc apì tic purhnikèc dun meic twn katast sewn me diaforetik I 3 wc diaforetik n, onom zetai analloiìthta isospin(isospin invariance). Par deigma Na prosdiorisjeð o lìgoc twn energ n diatom n twn diasp sewn pp π + d kai np π 0 d. Jewr ste ìti to deutèrio (d) èqei I =0(aplèta) kai to piìnio (π) èqei I =(triplèta π +,π 0,π ). Gia thn pr th di spash pp π + d, to isospin twn proiìntwn ja eðnai I = I π + I d kai epeid I d =0,jaeÐnai I = I π. Sunep c I = I π =, ra kai I 3 = (diìti to π + eðnai to mèloc thc triplètac isospin tou pionðou me to megalôtero fortðo kai ra ja èqei I π + =+). Gia th deôterh di spash np π 0 d sthn telik kat stasheðnai I =kai I 3 =0. H energìc diatom eðnai an loghme to tetr gwno tou pl touc met bashc: σ T fi 2 f V i 2. (.6) Oi dôo diasp seic diafèroun mìno sto isospin. Sunep c, h telik f (final) kai arqik i (initial)katast shja eðnai ousiastik oi idiokatast seic isospin.
2 KEF ALAIO. EISAGWGIK ES ENNOIES Gia thn pr th di spash, h arqik idiokat stash isospin eðnai mia kat stash pp =,, en h telik eðnai π + d =,. Sunep c, h energìc diatom gia thn pr thdi spasheðnai σ, V, 2. (.7) Gia thdeôterhdi spash, harqik kat stasheðnai np = (, 0 0, 0 ), 2 en gia thn telik π 0 d =, 0. Sunep c, henergìc diatom gia thdeôterh di spasheðnai σ 2 (, 0 0, 0 ) V, 0 2 2 = 2, 0 V, 0 2. (.8) Shmei netai ìti sthn teleutaða exðswsh qrhsimopoi jhke h orjogwniìthta twn idiokatast sewn isospin. UpologÐzoume ton lìgo twn energ n diatom n: σ (pp π + d) σ 2 (np π 0 d) V, 2 =2, 2 =2, (.9), 0 V, 0 diìti, ìpwc proanafèrjhke, lìgw summetrðac isospin, oi purhnikèc dun meic den k noun di krish sthn I 3 -sunist sa tou isospin, me apotèlesma ta dôo eswterik ginìmena sthn (.9), epeid anafèrontai kai ta dôo se katast seic me I =,naeðnai Ðsa. Προκύπτει από την εξίσωση (.7) αν λύσουμε ως προς np.
Kef laio 2 StoiqeÐa JewrÐac Om dwn 2. SummetrÐec kai Om dec (Groups) H om da twn strof n (Rotation Group) apoteleðtai apì to sônolo twn strof n ston tridi stato q ro. An R kai R 2 eðnai strofèc (dhlad an koun sthn om da), tìte kai to ginìmenì touc R R 2 eðnai epðshc strof (opìte an kei epðshc sthn om da). Dhlad, to sônolo twn strof n eðnai fikleistìfl ston pollaplasiasmì ( sônjeshstrof n). MporeÐ kaneðc na parathr sei ìti up rqei monadiaðo stoiqeðo () sthn om da, h opoða antistoiqeð sthn strof kat gwnða mhdèn, ste na isqôei R = R = R gia k je stoiqeðo R thc om dac. Akìma, k je stoiqeðo R èqei kai antðstrofo R, ste na isqôei RR = R R =. Autì isqôei, profan c, kai gia tic strofèc, afoô to antðstrofo thc strof c gôrw panw èna epðpedo kat gwnða φ eðnai hstrof kat gwnða ( φ) sto Ðdio epðpedo. To ginìmeno den eðnai aparaðthta metajetikì, dhlad R R 2 R 2 R, ektìc pì tic strofèc sto Ðdio epðpedo. IsqÔei, ìmwc, p nta h prosetairistik idiìthta R (R 2 R 3 )=(R R 2 )R 3. H om da eðnai suneq c, upì thn ènnoia ìti k je strof perigr fetai apì èna sônolo suneq n paramètrwn (α,α 2,α 3 ), ìpwc p.q. oi treic gwnðec tou Euler. Oi par metroi autèc mporoôn na omadopoihjoôn wc sunist sec dianôsmatoc a =(α,α 2,α 3 ) me kateôjunsh k jeth sto epðpedo peristrof c kai mètro thgwnða peristrof c. EpÐshc, h om da twn strof n eðnai mia om da Lie, diìti k je peperasmènh strof (enn. kat peperasmènh gwnða) mporeð na prokôyei wc ginìmeno apeirost n strof n. Tìte h om da kajorðzetai pl rwc apì th sumperifor thc gôrw apì thn mon da (gôrw apì to tautotikì stoiqeðo). EÐnai fanerì ìti toapotèlesma enìc peir matoc den mporeð na exart tai apì ton prosanatolismì thc metrhtik c suskeu c. Epomènwc, oi strofèc prèpei na 3
4 KEF ALAIO 2. STOIQE IA JEWR IAS OM ADWN eðnai summetrðec tou sust matoc. Ex' orismoô, hfusik paramènei amet blhth se èna metasqhmatismì-summetrða. Epomènwc, prèpei oi pijanìthtec met bashc (kai oi mèsec timèc, wc upoperðptwsh) na paramènoun analloðwtec se strofèc. 'Estw ψ hidiokat stashenìc sust matoc, hopoða met apì mia strof twn suntetagmènwn metasqhmatðzetai sthn ψ = U ψ, ìpou U eðnai o telest c tou metasqhmatismoô. Ousiastik o U antistoiqeð sthn anapar stash thc om dac sto q ro twn katast sewn tou sust matoc. Sthn perðptwsh mac U = U(R) antistoiqeð sto stoiqeðo strof c R thc om dac. Hpijanìthta èna sôsthma pou perigr fetai apì thn idiokat stash ψ na brejeð sthn idiokat stash φ, prèpei na paramènei amet blhth se strofèc: φ ψ 2 = φ ψ 2 = φ U U ψ 2. (2.) Ta eswterik ginìmena paramènoun analloðwta gia k je φ, ψ idiokatast seic. Epomènwc, U U =, dhlad o telest c U eðnai monadiakìc (unitary). Oi unitary telestèc U(R ), U(R 2 ),... sqhmatðzoun om da me thn Ðdia akrib c dom me thn om da twn strof n, alli c, sqhmatðzoun mia monadiak anapar stashthc om dac twn strof n. Dhlad se k je strof R mporoôme na antistoiqðsoume ènan monadiakì telest U(R), o opoðoc dr ntac ston q ro twn idiokatast sewn ψ metasqhmatðzei tic idiokatast seic se strammènec ψ,oisuntetagmènec jèshc x twn opoðwn èqoun strafeð sômfwna me thn R. An oi strofèc R thc om dac eðnai summetrðec tou sust matoc, ja prèpei hqamiltonian na paramènhmonadiak analloðwth(unitarily invariant) Autì shmaðnei ìti ta stoiqeða pðnaka φ H ψ thc Qamiltonian c prèpei na paramènoun amet blhta se strofèc: φ H ψ = φ H ψ = φ U HU ψ, (2.2) gia ìlec tic katast seic φ, ψ. Epomènwc, h Qamiltonian ja prèpei -ìpwc eðdame kai se prohgoômenh par grafo- na paramènei monadiak analloðwth, dhlad U HU = H. H sqèshaut mporeð na pollaplasiasteð apì ta arister me U kai k nontac qr sh thc idiìthtac - orismoô U U =twn monadiak n telest n, na d sei [H, U] =0. Sunep c, h apaðthsh tou na mhn exart tai h Fusik apì to sôsthma suntetagmènwn (summetrða), epib llei o telest c U twn strof n na metatðjetai me thn H. All ìtan èna mègejoc metatðjetai me thn Qamiltonian tou probl matoc, diathreð thn mèsh tim tou qronik stajer. Epomènwc, h parap nw metajetik sqèsh upodhl nei thn Ôparxh miac diathroômenhc posìthtac. O Ðdioc o telest c U parìlo pou diathreðtai, den antistoiqeð se parathr simo fusikì mègejoc (observable) diìti eðnai monadiakìc (U = U ) kai ìqi ermitianìc (A = A). Sunep c, h mèsh tim kai oi idiotimèc tou den eðnai pragmatikoð arijmoð gia ìlec tic katast seic tou sust matoc. Ja doôme, ìmwc, ìti o genn torac tou U (o genn torac twn strof n) eðnai parathr simo mègejoc kai m lista diathroômeno.
2.. SUMMETR IES KAI OM ADES (GROUPS) 5 'Opwc anafèrame kai sta prohgoômena, se mia om da Lie ìlec oi idiìthtec mporoôn na exaqjoôn melet ntac apeirostoôc metasqhmatismoôc polô kont sto tautotikì stoiqeðo (). Gia par deigma, h strof kat apeirost gwnða ɛ gôrw apì ton xona z, U = U z (ɛ) se pr th t xh (wc proc ɛ) ja gr fetai: U(ɛ) =U(0) + ɛ ɛ ɛ=0 U + O(ɛ 2 ). Wc gnwstìn, k je monadiakìc telest c mporeð na grafeð sthgenik morf U = e ia(ɛ) ìpou A(ɛ) kat llhloc ermitianìc telest c. AnaptÔssontac thn èkfrash aut gia mikr ɛ èqoume U =+i A ɛ ɛ=0 ɛ + O(ɛ 2 ). ParathroÔme, loipìn, ìti o genn torac ja prèpei na ekfrasjeð wc Uz ɛ ɛ=0 = ij 3, ìpou J 3 kat llhloc ermitianìc telest c (ton opoðo onom zoume genn tora (generator) twn strof n gôrw apì ton xona z). Dhlad gr foume to an ptugma wc: U z = iɛj 3 + O(ɛ 2 ). (2.3) Apomènei mìno na prosdioristeð o telest c J 3, o opoðoc ja antistoiqeð se diathr simo kai parathr simo fusikì mègejoc (observable), mia kai eðnai ermitianìc. M lista, eðnai eôkolo na parathr soume ìti h apaðthsh na eðnai monadiakìc o U dðnei: U U =(+iɛj 3)( iɛj 3 )=+iɛ(j 3 J 3 )+O(ɛ 2 )=. (2.4) Gia na isqôei h exðswsh aut se pr th t xh wc proc ɛ, ja prèpei J 3 = J 3, dhlad ja prèpei o genn torac J 3 na eðnai ermitianìc. Ja melet soume thn dr sh thc strof c R sthn kumatosun rthsh (anapar stash jèshc) r ψ = ψ(r) enìc sust matoc. Up rqoun dôo isodônamoi trìpoi na ektelèsoume thn strof. EÐte strèfoume tic suntetagmènec krat ntac stajerì to sôsthma (passive), eðte strèfoume to sôsthma, krat ntac stajeroôc touc xonec (active). Shmei netai ìti strof twn axìnwn kat gwnða θ isodunameð me strof tou sust matoc kat gwnða ( θ). Strèfontac to sôsthma, h arqik kumatosun rthsh ψ(r) gðnetai (ψ(r)) = U(R)ψ(r) =ψ(r r). Ektel ntac mia apeirost strof gôrw apì ton z- xona, èqoume R r (x + ɛy, y ɛx, z). Epomènwc, to an ptugma Taylor gðnetai: U(R)ψ(r) =ψ(x + ɛy, y ɛx, z) =ψ(r) iɛ(/i)(x y y x ) ψ. (2.5) MporoÔme na xanagr youme thn teleutaða exðswshupì morf telest n: ( Uψ(r) = (iɛ) ) i (x y y x ) ψ. (2.6) Ο A πρέπει να είναι απαραίτητα ερμιτιανός ώστε να ισχύει η ιδιότητα-ορισμός U U = e ia e ia =.
6 KEF ALAIO 2. STOIQE IA JEWR IAS OM ADWN Opìte apì thn sôgkrishtwn exis sewn (2.6) kai (2.3), brðskoume thn èkfrash tou genn tora J 3. M lista, an parathr soume ìti (/i) j p j (ìpou j =, 2, 3), mporoôme na doôme ìti o genn torac gr fetai: J 3 = xp y yp x, (2.7) dhlad o genn torac J 3 twn strof n gôrw apì ton xona-z anagnwrðzetai wc h trðth sunist sa tou telest thc (troqiak c) stroform c. Sunep c, afoô o J 3 diathreðtai, kai oi idiotimèc thc J 3 ja diathroôntai kat thn kðnhsh kai ja apoteloôn kaloôc kbantikoôc arijmoôc. ParathroÔme, loipìn, ìti mia summetrða tou sust matoc odhgeð se diathroômenh posìthta. Sthn perðptws mac, h summetrða wc proc strofèc odhgeð sthn diat rhsh thc troqiak c stroform c (conservation of angular momentum). Mia peperasmènh strof kat gwnða θ (ìqi aparaðthta mikr ) mporeð na prokôyei wc diadoq n apeirost n strof n ɛ 0. Upojètoume ìti h gwnða θ apoteleðtai apì n apeirostèc strofèc kat ɛ, dhlad θ = nɛ. JewroÔme ìti n kai to ɛ 0 kat tètoio trìpo ste to ginìmenì touc θ na eðnai peperasmèno. MporoÔme na fiqtðsoumefl mia peperasmènhstrof kat θ ektel ntac diadoqik n apeirostèc strofèc kat ɛ: ( U(θ) =(U(ɛ)) n =( iɛj 3 ) n = i θ n 3) n J e iθj 3. (2.8) Sthn teleutaða èkfrash qrhsimopoi same to gnwstì ìrio ( + x/n) n e x ìtan n. Sunep c, k je strof gôrw apì èna aujaðreto xona n mporeð na grafeð sthn genik morf : U(θ) =exp( iθn J). (2.9) Gia strofèc gôrw apì touc xonec x kai y, oi ermitianoð genn torec ja eðnai oi antðstoiqec sunist sec thc stroform c J kai J 2.
2.2. ALGEBRA MIAS OM ADAS 7 2.2 'Algebra miac Om dac H dom miac om dac Lie kajorðzetai apì tic sqèseic met jeshc metaxô twn gennhtìrwn, dhlad apì thn 'Algebra twn metajet n (commutator algebra) thc om dac. Ac jewr soume mia om da me genn torec S i, ste k je stoiqeðo thc na mporeð na anaparastajeð apì ton telest U i (ɛ) =exp( iɛs i ). An ektelèsoume treic diadoqikèc pr xeic U i (ɛ)u j (θ)u i ( ɛ), tìte se pr th t xh wc proc ɛ ja èqoume: U tot = e iɛs i e iθs j e iɛs i =( iɛs i )e iθs j ( + iɛs i ). (2.0) Ac prospaj soume na skeftoôme tð ja sunèbaine apì fusik poyhan i = j. Tìte, profan c, h k je fistrof fl ja ginìtan gôrw apì ton Ðdio xona i = j, opìte to ginìmeno ja tan metajetikì. Tìte, loipìn, h pr th fistrof fl kat ɛ ja exoudetèrwne thn trðth fistrof fl kat +ɛ, opìte h ìlh diadikasða ja sunðstato se mða kai mìno fistrof fl kat θ (pou proèrqetai apì thn deôterh strof ). Sthn perðptwsh ìmwc pou i j, oi genn torec S i kai S j den metatðjentai. Sunep c, perimènoume to apotèlesma mac na eðnai arket diaforetikì apì to aplì e iθs j. Opwsd pote, ìmwc, ja prèpei to apotèlesm mac na ekfulðzetai sthn apl aut morf ìtan [S i,s j ]=0. Perimènoume, dhlad, se pr th t xh wc proc ɛ kai θ, na isqôei mia sqèsh thc morf c U tot = e iθs j + iɛ [S i,s j ] M gia k poion telest M, ste na katal gei sto swstì an ptugma ìtan oi genn torec metatðjentai. Pr gmati, an ektelèsoume tic pr xeic sthn (2.0) ja doôme ìti to apotèlesma mporeð na tejeð sthmorf : U tot = e iθs j + iɛ [ e iθs j,s i ] + O(ɛ 2 ). (2.) An to e iθs j anaptuqjeð gôrw apì to θ =0, dhlad e iθs j = iθs j θ 2 S 2 j +...,eðnai eôkolo na faneð ìti [e iθs j,s i ]= iθ[s j,s i ]+O(θ 2 ), opìte hsqèsh pou mantèyame sta parap nw ikanopoieðtai! M lista, h (2.) an grafeð gia ìrouc an terhc t xhc wc proc ɛ kai gia tuqaðo mesaðo telest Z antð tou exp ( iθs j ) gr fetai: e iɛs Ze iɛs = Z + iɛ[s, Z] (/2)ɛ 2 [S, [S, Z]] + O(ɛ 3 ), (2.2) kai onom zetai exðswsh Baker-Hausdorff. An, t ra, ektelèsoume prin apì tic treic autèc strofèc akìma mia pou na teðnei na fiexoudeter seifl thn exp( iθs j ), se mhdenik t xh wc proc ɛ kai θ ja p roume ton tautotikì telest, en oi diorj seic pr thc t xhc wc proc (ɛθ) ja eðnai an logec tou metajèth [S i,s j ]. Gia thn akrðbeia ja eðnai: U tot = e iɛs i e iθs j e iɛs i e iθs j =+(ɛθ)[s i,s j ]+..., (2.3)
8 KEF ALAIO 2. STOIQE IA JEWR IAS OM ADWN ìpou oi ìroi an terhc t xhc èqoun paraleifjeð. To ginìmeno (sônjesh) twn tess rwn fistrof nfl eðnai kai autì strof (kleistìthta thc om dac). Sunep c ja mporeð se pr tht xhwc proc (ɛθ) na grafeð sthn genik ekjetik morf : ( U tot =exp i(ɛ θ) ) Cɛ,θ k S k =+i(ɛ θ) Cɛ,θ k S k +.... (2.4) k k SugkrÐnontac tic (2.4) kai (2.3) se pr tht xhwc proc (ɛ θ), kai, jewr ntac ìti to i mporeð na aporrofhjeð apì touc suntelestèc Cij k, brðskoume thn sunj khkleistìthtac thc om dac: [S i, S j ]= k C k ij S k, (2.5) pou mac lèei polô apl ìti o metajèthc dôo gennhtìrwn eðnai grammikìc sunduasmìc twn gennhtìrwn thc om dac. Oi suntelestèc Cij k tou grammikoô sunduasmoô onom zontai suntelestèc dom c thc om dac (structure constants), apl lgebra thc om dac. HonomasÐa aut èqei no ma upì thn ènnoia ìti an s- to q rotwn gennhtìrwn miac om dac Lie orðsoume wc pr xhpollaplasiasmoô dôo anusm twn S i, S j ton metajèth [S i,s j ], tìte o q roc (mazð me thn pr xh tou pollaplasiasmoô) gðnetai lgebra (ètsi onom zetai halgebrik dom pou prokôptei an se èna anusmatikì q ro orðsoume pr xhpollaplasiasmoô), pou onom zetai lgebra Lie thc om dac. Ja doôme amèswc parak tw ìti hgn sh thc lgebrac thc om dac (dhlad twn suntelest n dom c) eðnai arket gia thn kataskeu anaparast sewn thc om dac.
2.3. ALGEBRA THSSTROFORM HS 9 2.3 'Algebra thc Stroform c Oi sqèseic met jeshc twn telest n thc stroform c (gennhtìrwn thc om dac twn strof n) eðnai gnwstèc apì thn Kbantomhqanik : [J i,j j ]=iε ijk J k. (2.6) Epomènwc, oi suntelestèc dom c gia thn lgebra twn strof n eðnai Cij k = iε ijk kai ìpwc proanafèrjhke, kajorðzoun pl rwc tic idiìthtec thc om dac. Epeid ta J i den metatðjentai metaxô touc, mìno oi idiotimèc enìc apì touc treic genn torec (èstw tou J 3 )eðnai kal orismènoi kbantikoð arijmoð. Mhgrammikèc sunart seictwn gennhtìrwn, oi opoðec metatðjentai me ìlouc touc genn torec thc om dac onom zontai telestèc Casimir. Gia thn om da twn strof n o J 2 = J 2 + J2 2 + J3 2 eðnaiomonadikìc telest c Casimir. Pr gmati, metatðjetai me ìlouc touc genn torec: [J 2,J i ]=[J j J j,j i ]=J j [J j,j i ]+[J j,j i ]J j = iε jik J j J k +iε jik J k J j = iε jik {J j,j k } =0. (2.7) Gia thn apìdeixh thc teleutaðac isìthtac arkeð na parathr soume ìti to - jroisma tou ginomènou miac antisummetrik c sta {j, k} posìthtac (ìpwc h ε jik ) me mia summetrik ìpwc o antimetajèthc {J j,j k } isoôtai p nta memhdèn (profanèc). Wc antimetajèthc twn telest n A kai B orðzetai o summetrikìc sunduasmìc {A, B} = AB + BA. Sunep c [J 2,J i ]=0,opìte o J 2 eðnai telest c Casimir, efìsonmetatðjetai me ìlouc touc genn torec. Epeid ta J i den metatðjentai metaxô touc den mporoôn na diagwniopoihjoôn tautìqrona. Sunep c, mìno ènac apì touc J i, èstw o J 3, mporeð na diagwniopoihjeð. Eped o J 2 metatðjetai me ton J 3 ja diagwniopoieðtai tautìqrona me autìn, kai oi idiotimèc tou mporoôn na qrhsimopoihjoôn gia ton qarakthrismì twn idiokatast sewn (oi idiotimèc tou eðnai kaloð kbantikoð arijmoð). Oi J 2, J 3 èqoun koinèc idiokatast seic, tic opoðec sumbolðzoume me jm. Oi idiotimèc touc eðnai: Arqik analôoume ton telest Casimir wc ex c: J 2 jm = λ 2 (j) jm (2.8) J 3 jm = m jm. (2.9) J 2 =(J 2 + J 2 2 )+J 2 3 =(J + ij 2 )(J ij 2 )+Q + J 2 3, (2.20) ìpou to Q eðnai diorjwtikìc telest c pou mpaðnei sthn exðswsh gia na isqôei to an ptugma (epeid ta J, J 2 den metatðjentai, h tautìthta thc an lushc dôo tetrag nwn se ginìmeno den isqôei ìpwc gðnetai stouc arijmoôc). EÔkola
20 KEF ALAIO 2. STOIQE IA JEWR IAS OM ADWN upologðzetai ìti (J +ij 2 )(J ij 2 )=J 2 +J 2 2 i[j,j 2 ], opìte faðnetai amèswc ìti Q = i[j,j 2 ]= J 3. Dhlad to Q leitourgeð wc antistajmistikìc ìroc gia na isqôei to an ptugma. An kalèsoume J ± = J ± J 2, h(2.20) gðnetai: Me entel c antðstoiqo trìpo brðskoume: J 2 = J + J + J 2 3 J 3. (2.2) J 2 = J J + + J 2 3 + J 3. (2.22) ParathroÔme ìti (J ± ) = J. Prosjètontac kai afair ntac kat mèlh tic (2.2) kai (2.22), lamb noume tic polô qr simec sqèseic: kai ParathroÔme epðshc, ìti: J 2 J 2 3 = 2 {J +,J }, (2.23) [J +,J ]=2J 3. (2.24) [J 3,J ± ]=±J ±, (2.25) opìte oi telestèc J ± eðnai telestèc an bashc kai kat bashc (raising - lowering operators). H dr sh tou J +, gia par deigma, mporeð eôkola na deiqjeð ìti metasqhmatðzei tic idiokatast seic se autèc pou antistoiqoôn se megalôterh kat idiotim. Gia na to diapist soume, afoô dr soume me thn J + p nw sthn idiokat stash jm, droôme sthn prokôptousa kat stashme ton J 3 gia na elègxoume an apoteleð epðshc idiokat stash: J 3 J + jm =([J 3,J + ]+J + J 3 ) jm =(J + + J + J 3 ) jm, (2.26) h opoða mporeð na grafeð kai wc J + (J 3 +) jm. An qrhsimopoi soume thn exðswsh idiotim n (2.9), èqoume (J 3 +) jm = (m +) jm. An epanal boume thn diadikasða gia ton telest kat bashc J èqoume telik : J 3 J ± jm =(m ± )J ± jm. (2.27) H teleutaða exðswsh mac lèei ìti an h jm eðnai idiokat stash tou J 3 me idiotim m, tìte kai oi J ± jm eðnai epðshc idiokat staseic tou idðou telest kai m lista meidiotimèc (m ± ). Autì shmaðnei ìti oi idiokatast seic J ± jm eðnai an logec twn j, m ±. ParathroÔme ìti h dr sh tou telest J ± aneb zei/kateb zei tic idiokatast seic kat mia idiotim. Epeid, ìpwc parathr same, oi idiokatast seic J + jm eðnai an logec twn j, m + mporoôme na prosdiorðsoume ton suntelest analogðac apì thn apaðthsh orjokanonikìthtac. JewroÔme thsqèshanalogðac: J + jm = c m j, m +. (2.28)
2.3. ALGEBRA THSSTROFORM HS 2 PaÐrnoume to ermitianì suzugèc thc parap nw sqèshc: jm J = c m j, m +. (2.29) Pollaplasi zoume apì arister thn (2.28) me thn (2.29): jm J J + jm = c m 2 j, m + j, m + = c m 2, (2.30) ìpou qrhsimopoi same thn apaðthsh oi idiokatast seic jm na eðnai orjokanonikopoihmènec. EÐnai eôkolo na lôsoume wc proc touc suntelestèc c m : c m = jm J 2 J 3 (J 3 +) jm = λ 2 m(m +). (2.3) Shmei ste ìti qrhsimopoi same thn (2.22) gia thn antikat stash tou J J + kai ìti akìma den èqoume prosdiorðsei tic epitreptèc timèc twn λ 2 kai m. P ntwc, mporoôme na gr youme to genikì apotèlesma pou ja prokôyei an epanal boume thn Ðdia diadikasða gia ton J : J ± jm = λ 2 m(m ± ) j, m ±. (2.32) Sto shmeðo autì mporoôme na arqðsoume na anarwtiìmaste poièc eðnai oi epitreptèc timèc twn m kai λ 2. Profan c hèkfrash(2.32) den ja isqôei an h upìrizhposìthta eðnai arnhtik. MporoÔme na parathr soume apì ton orismì tou telest Casimir ìti: jm J 2 J 2 3 jm = λ 2 m 2 = jm J 2 + J 2 2 jm > 0. (2.33) H parap nw exðswsh mac bebai nei ìti, epeid h posìthta J 2 + J 2 2 eðnai p ntote jetik 2, ja prèpei kai λ 2 m 2 > 0, dhlad ìti h m eðnai p nta fragmènh m < λ. MporoÔme, p ntwc, na doôme mèqri poi tim tou m isqôei hèkfrash(2.32). Profan c ja prèpei m(m+) λ 2. An onom soume j th mègisth tim tou m thn opoða epitrèpei h prohgoômenh anisìthta, tìte profan c ja èqoume thn isìthta λ 2 = j(j +), apì thn opoða prokôptei ìti h sunj khperiorismoô gia tic idiotimèc m eðnai: m(m +) j(j +). (2.34) 2 Ηποσότητα J 2 + J 2 2 είναι γνησίως θετική και ποτέ μηδέν, διότι αν αυτό συνέβαινε, θα σήμαινε αυτομάτως J 2 = J 2 2 =0, και επειδή ισχύει πάντα η ανισότητα Jj 2 J 2 j, ο παραπάνω μηδενισμός θα έπεται αναγκαστικά: J j = J 2 j =0. Αλλάαυτό ισοδυναμεί με μηδενισμό της αβεβαιότητας ( J j ) 2 = J 2 j Jj 2, για όλα τα j =, 2, 3, δηλαδή ταυτόχρονη γνώση των J, J 2 και J 3 με απόλυτη ακρίβεια, πράγμα αδύνατον, διότι οι τελεστές δεν μετατίθενται. Άρα θα είναι αναγκαστικά J 2 + J 2 2 > 0.
22 KEF ALAIO 2. STOIQE IA JEWR IAS OM ADWN Den gnwrðzoume akìma thn el qisthtim pou mporeð na p rei to m, an kai gia lìgouc summetrðac (hjetikìthta arnhtikìthta tou m exart ntai prwtðstwc apì ton prosanatolismì twn axìnwn, pou eðnai aujaðretoc) upopteuìmaste ìti ja eðnai ( j). H anisìthta pou prèpei na ikanopoieðtai sthn perðptwsh twn arnhtik n m apì thn (2.32) eðnai m(m ) j(j +). An prosjèsoume kai sta dôo mèlh thc anisìthtac to /4 kai qrhsimopoi soume tautìthtec tetrag nwn, eôkola brðskoume ( m 2) 2 ( j + 2) 2. ApotetragwnÐzoume, kai èqoume (jewr same j>0 wc mègisthtim tou m): j 2 m 2, (2.35) dhlad j m. Epomènwc, ta ìria gia to m pou epib llei h exðswsh(2.32) eðnai ta gnwst gia thn stroform : m = j, j +,...,j,j. (2.36) 'Ewc t ra jewroôsame apl c ìti j>0, qwrðc na anh suqoôme an to mègejoc autì eðnai suneqèc kbantismèno (kai poièc eðnai oi epitreptèc timèc tou). Xanagr foume tic exis seic idiotim n J 2 jm = j(j +) jm (2.37) J 3 jm = m jm, (2.38) me m j. Epeid hmet bashapì thn uyhlìterhkat stash m = j èwc thn qamhlìterh m = j (gia dedomèno j) kai antðstrofa gðnetai apì touc telestèc J ± kat mða mon da idiotim c m k je for, sumperaðnoume ìti prèpei h diatreqìmenh apìstash na eðnai akèraio pollapl sio bhm twn, dhlad akèraioc arijmìc. Autì shmaðnei ìti h apìstash j ( j) =2j ja prèpei na eðnai akèraioc arijmìc. H apaðthsh 2j =0,, 2, 3, 4, 5,... shmaðnei ìti to j mporeð na p rei mìno akèraiec kai hmiperittèc timèc j =0,,, 3, 2, 5,... 2 2 2. 'Opwc eðnai gnwstì apì thn Kbantomhqanik JewrÐa thc Stroform c, oi akèraiec timèc tou j antistoiqoôn sthn troqiak stroform, en oi hmiperittèc timèc antistoiqoôn sthn eswterik stroform tou swmatidðou (spin).
2.4. ANAPARAST ASEIS THS OM ADAS 23 2.4 Anaparast seic thc Om dac H jm idiokat stash metasqhmatðzetai se strofèc kat gwnða θ gôrw apì ton y- xona wc: U jm = e iθj 2 jm. (2.39) H nèa idiokat stash ja mporeð na anaptuqjeð stic idiokatast seic stroform c, dhlad na ekfrasjeð wc grammikìc sunduasmìc twn jm me to m na lamb nei ìlec tic dunatèc timèc apì j èwc j: e iθj 2 jm = m d (j) m m (θ) jm. (2.40) Gia na prosdiorðsoume ton suntelest analogðac d (j) m m, arkeð na jumhjoôme ìti o probolikìc telest c jm jm prob llei thn kat stash sthn opoða dra ston q ro twn (kanonikopoihmènwn) idiokatast sewn jm kai dðnei èna par llhlo nusma (diadikasða an logh me thn n (n x), ìpou to nusma tou q rou x prob lletai sthn kateôjunshtou monadiaðou n). EÐnai logikì ìtan h probol aut gðnei p nw sta anôsmata b shc enìc q rou kai ajroðsome anusmatik tic probolèc, na p roume telik to arqikì nusma pou prob lame. Dhlad eðnai profanèc ìti to jroisma twn probolik n telest n ìlhc thc b shc ja d sei ton tautotikì telest : = jm jm. K nontac qr sh m thc teleutaðac tautìthtac, èqoume: U jm = U jm = m jm jm U jm. (2.4) SugkrÐnontac thn (2.4) me thn (2.40) èqoume telik thn èkfrash gia touc suntelestèc: d (j) m m (θ) = jm U(θ) jm. (2.42) Epeid gia k je tim tou j èqoume kai diaforetik b sh, sumperaðnoume ìti k je tim tou j orðzei mia diaforetik anapar stash thc om dac. Se dedomènh anapar stash k je stoiqeðo thc om dac antistoiqeð se èna pðnaka (2j +) diast sewn (ìsec kai oi dunatèc timèc tou m, dhlad ìsa kai ta anôsmata thc b shc pou epilèqjhke). Epomènwc, o pðnakac pou orðzetai apì thn (2.42) apoteleð thn j-di stath anapar stash pðnaka thc om dac. Oi (2j +) idiokatast seic jm apoteloôn mia (2j +)-di stath b sh miac (2j +)-di stathc mh anag gimhc (irreducible) anapar stashc thc om dac twn strof n. Par deigma
24 KEF ALAIO 2. STOIQE IA JEWR IAS OM ADWN Ja prosdiorðsoume touc pðnakec strof n gôrw apì ton xona y gia tic anaparast seic j = /2 (jemeli dhc anapar stash thc SU(2)) kai j =. i) Anapar stash j =/2 Jèloume na prosdiorðsoume ta stoiqeða pðnaka d (/2) m m = m e iθj 2 m. Sthn anapar stash j = /2 (2 diast sewn), wc anôsmata b shc èqoume tic idiokatast seic /2, /2 =(, 0) T kai /2, /2 =(0, ) T. 3 ProsdiorÐzoume pr ta ton pðnaka (J 3 ), pou eðnai diag nioc me stoiqeða ±/2 sth b sh twn idiokatast se n tou: m J 3 m (J 3 )= ( ) /2 0 = 0 /2 2 σ 3, (2.44) ìpou σ 3 oantðstoiqoc pðnakac Pauli. Me ton Ðdio trìpo, mporoôme eôkola na kataskeu soume touc pðnakec (J ± ): (J + )= ( ) 0 0 0, (J )= ( ) 0 0. (2.45) 0 Apì tic sqèseic J =(J + + J )/2 kai J 2 =(J + J )/2i brðskoume eôkola touc pðnakec (J ) kai (J 2 ): (J )= 2 ( ) 0 = 0 2 σ, (J 2 )= 2 ( ) 0 i = i 0 2 σ 2. (2.46) 'Eqontac, plèon, ekfr sei ton pðnaka J 2 sunart sei tou σ 2, èqoume telik : ( ) d (/2) cos(θ/2) sin(θ/2) m m = e iθσ 2/2 =cos(θ/2) iσ 2 sin(θ/2) =. sin(θ/2) cos(θ/2) (2.47) Sthn teleutaða sqèsh qrhsimopoi same thn gnwst idiìthta tou pðnaka σ 2 : exp( iθσ 2 )=cosθ iσ 2 sin θ. Enac trìpoc na apodeiqjeð aut h sqèsh eðnai o ex c: U = e iθσ2/2 = i n ( ) n (θ/2)n σ n. (2.48) n! n=0 3 Χρησιμοποιούμε το συμβολισμό (, 0) T = ( 0). (2.43)
2.4. ANAPARAST ASEIS THS OM ADAS 25 QwrÐzoume to jroisma se rtiouc kai perittoôc ìrouc: U = n=0 ( ) n (θ/2)2n (2n)! σ 2n 2 + i n=0 Qrhsimopoi ntac thn sqèsh σ 2 2 =, èqoume telik : n+ (θ/2)2n+ ( ) (2n +)! σ2n+. (2.49) U =cos(θ/2) iσ 2 sin(θ/2). (2.50) ii) anapar stash j = Ja efarmìsoume kai p li ta prohgoômena, dhlad ja prosdiorðsoume touc pðnakec J 3, J ± kai J, J 2. O J 3 eðnai profan c diag nioc me idiotimèc, 0, : (J 3 )= 0, (2.5) ìpou ta stoiqeða pou paraleðpontai ennooôntai mhdenik. Gia ton prosdiorismì twn upoloðpwn pin kwn eðnai qr simo na upologðsoume ta akìlouja: J +, =0, J +, 0 = 2,, J +, = 2, 0 kai J, =0, J, = 2, 0, J, 0 = 2,. Epomènwc, oi pðnakec an bashc/kat bashc gðnontai: (J + )= 0 0 2 0 0, (J )= 0 0 0 2 0 0. (2.52) 0 0 0 0 0 Epomènwc, mporoôme amèswc na broôme ton J 2 (o J af netai wc skhsh): 0 i 0 (J 2 )= i 0 i. (2.53) 2 0 i 0 Gia na upologðsoume ton ekjetikì pðnaka e iθj 2 ja qrhsimopoi soume to qarakthristikì polu numo tou (J 2 ). To polu numo autì eðnai to polu numo pou èqei wc rðzec tic idiotimèc tou pðnaka. O sun jhc trìpoc eôreshc tou qarakthristikoô poluwnômou eðnai o akìloujoc: Fèrnoume arqik thn exðswsh idiotim n (prìkeitai gia exðswsh pin kwn, kai epeid den up rqei perðptwsh sôgqushc, aplopoioôme ton sumbolismì (J i ) J i, sumbolðzontac ènan pðnaka kai ton telest tou me to Ðdio sômbolo) sthn morf : (J 2 m 2 ) X =0. (2.54)
26 KEF ALAIO 2. STOIQE IA JEWR IAS OM ADWN Apì thn apaðthsh na up rqei lôsh, gia to idio nusma X, di forh thc tetrimmènhc ( thc mhdenik c), prèpei h orðzousa tou pðnaka (J 2 m 2 ) na mhdenðzetai. EnnoeÐtai ìti m 2 eðnai h idiotim tou J 2 (pou antistoiqeð sto idio nusma X), pollaplasiasmènhme ton monadiaðo pðnaka I, ste na eðnai dunat h afaðreshthc apì ton pðnaka J 2. Apì thn apaðthsh aut, katal goume se èna polu numo P (m 2 )=0, to opoðo èqei wc rðzec tic idiotimèc m 2, dhlad eðnai to qarakthristikì polu numo tou pðnaka. QwrÐc, fusik, na ektelèsoume kamða pr xh perimènoume na broôme wc idiotimèc tou J 2 tic idiotimèc tou J 3, dhlad thn tri da (, 0, ). Autì eðnai profanèc apì thn stigm pou parathr soume ìti h epilog mac na diagwniopoi soume ènan apì touc treic pðnakec J i (dhlad ton J 3 ) den mporeð na metab llei tic idiotimèc twn pin kwn autèc kaj' autèc. To mìno pou mporeð na k nei h diagwniopoðhsh eðnai na fèrei ton J 3 se diag nia morf, me tic idiotimèc tou wc diag nia stoiqeða. SÔmfwna me ta parap nw, den qrei zetai na ektelèsoume kami pr xh gia na doôme ìti to qarakthristikì polu numo tou J 2 eðnai to P (m 2 )=m 2 (m 2 2 ) = m3 2 m 2, to opoðo kataskeu zetai ètsi ste na èqei rðzec ta 0, ±. EÐnai eôkolo na doôme ìti ènac pðnakac p nta ikanopoieð mia exðswsh antðstoiqh me aut tou mhdenismoô tou qarakthristikoô tou polu numou. An o pðnakac ( telest c) A ikanopoieð thn exðswsh idiotim n A ξ = ξ ξ, tìte kai opoiad pote analutik sun rthsh tou A ikanopoieð thn exðswsh f(a) ξ = f(ξ) ξ. Xèrontac, loipìn, mia exðswshgia tic idiotimèc tou J 2, dhlad thn exðswsh pou orðzei o mhdenismìc tou qarakthristikoô poluwnômou P (m 2 )=0, mporoôme na kataskeu soume mia exðswsh pin kwn P (J 2 )=0. Sthn perðptws mac: (J 2 ) 3 = J 2. (2.55) H (2.55) sunep getai ìti k je trðth dônamh tou J 2 mporeð na anaqjeð se pðnaka sthn pr th dônamh. 'Ena an ptugma, loipìn, thc tuqoôshc analutik c sun rthshc f(j 2 ) se seir wc proc J 2, ja perièqei ìrouc to polô mèqri deutèrac t xewc wc proc J 2 kai ra h seir ja termatðzetai. EÐnai, loipìn, fanerì ìti h sun rthsh e iθj 2 ja èqei an ptugma to polô deôterhc t xhc wc proc J 2 to opoðo mporoôme amèswc na anazht soume: e iθj 2 = c 0 + c J 2 + c 2 (J 2 ) 2. (2.56) Prìkeitai gia mia exðswsh pin kwn me 3 agn stouc suntelestèc c 0,c,c 2, h opoða lônetai amèswc an parathr soume ìti h exðswsh aut pou ikanopoeðtai apì ton pðnaka J 2 ja ikanopoieðtai kai apì tic 3 idiotimèc tou, dðnont c mac èna
2.4. ANAPARAST ASEIS THS OM ADAS 27 sôsthma 3 exis sewn me 3 agn stouc, to opoðo epilôetai: e iθ = c 0 + c +c 2 2 (2.57) e 0 = c 0 + c 0+c 2 0 2 (2.58) e iθ = c 0 c +c 2 2, (2.59) H epðlushtou sust matoc dðnei wc apotèlesma: c 0 = (2.60) c = i sin θ (2.6) c 2 = ( cos θ). (2.62) Antikajist ntac tic timèc gia ta c µ (µ =0,, 2) sthn (2.56) brðskoume thn telik morf gia ton pðnaka strof c: e iθj 2 ( + cos θ) 2 2 sin θ ( cos θ) 2 = 2 sin θ cos θ 2 sin θ. (2.63) ( cos θ) 2 sin θ ( + cos θ) 2 2
28 KEF ALAIO 2. STOIQE IA JEWR IAS OM ADWN 2.5 H Om da SU(2) 'Opwc eðdame oi genn torec thc om dac twn strof n eðnai oi telestèc thc stroform c kai ikanopoioôn thn lgebra thc exðswshc (2.6). Xekin ntac apì aut thn lgebra kataskeôsame mh anag gimec 2j +-di statec anaparast seic thc om dac, me j =0,,, 3... H jemeli dhc anapar stasheðnai 2 2 h j =/2, me genn torec J i = σ i /2 touc pðnakec Pauli. Gia j =/2, kat ta gnwst, J + /2,, 2 =0, J /2, /2 =0 (2.64) J + /2, /2 = /2, /2, J /2, /2 = /2, /2. (2.65) Opìte, èqoume: J = ( ) 0 = σ 2 0 /2, J 2 = ( ) 0 i = σ 2 i 0 2 /2, J 3 = ( ) 0 = σ 2 0 3 /2. (2.66) Wc b shthc jemeli douc anapar stashc (j =/2) èqoume ticidiokatast - seic tou σ 3, dhlad ta anôsmata: ( /2, /2 = 0), /2, /2 = ( 0 ), (2.67) Aut qrhsimopoioôntia gia thn perigraf swmatðdiou me spin /2 ( = (, 0) T, = (0, ) T ). Oi pðnakec metasqhmatismoô gôrw apì ton i- xona U(θ i )=e iθ iσ i /2 eðnai monadiakoð. Apì thn idiìthta aut èpetai h ermitianìthta twn gennhtìrwn, dhlad twn pin kwn Pauli: σ i = σ i. To sônolo twn (2 2) monadiak n pin kwn sqhmatðzei om da, h opoða sumbolðzetai me U(2). Aut h om da eðnai profan c genikìterh apì to thn om da metasqhmatism n U(θ i )=e iθ iσ i /2, diìti oi pðnakec Pauli èqoun thn eidik idiìthta na eðnai iqnoi (traceless): tr(σ i )=0. H idiìthta aut twn pin kwn Pauli antikatoptrðzei mia polô shmantik idiìthta twn metasqhmatism n U(θ i )=e iθ iσ i /2 : h orðzousa tou pðnaka tou mesqhmatismoô isoôtai me thn mon da, dhlad det(e iθ iσ i /2 )=+. H idiìthta twn monadiak n metasqhmatism n me orðzousa + na èqoun iqnouc genn torec eðnai genik. Ac apait soume: det ( e iθs) =det ( Me iθs M ) =, (2.68) ìpou qrhsimopoi same thn gnwst idiìthta twn orizous n det(bab )= det(b)det(a)/ det(b) =det(a). EÐnai profanèc ìti Me A M = e MAM. H apìdeixh sunðstatai sto naanaptôxoume to ekjetikì: Me A M MA n M (MAM ) n = = = e MAM. (2.69) n! n! n=0 n=0
2.5. H OM ADA SU(2) 29 Efarmìzontac to parap nw sthn (2.68), èqoume: det ( e iθs) ( =det e iθmsm ), (2.70) ìpou o pðnakac M eðnai aujaðretoc. MporoÔme, loipìn, na ton epilèxoume ètsi ste na diagwniopoieð ton genn tora S, me diag nia stoiqeða tic idiotimèc tou. Epeid o pðnakac MSM eðnai plèon diag nioc, o ekjetikìc pðnakac ja eðnai epðshc diag nioc, me idiotimèc tic e iθs i, ìpou s i eðnai h i-idiotim tou S. Epeid h orðzousa enìc diag niou pðnaka eðnai apl c to ginìmeno twn idiotim n tou, ja èqoume telik det ( e iθs) = ( e iθs i =exp iθ ) s i =exp( iθ tr (S)), (2.7) i i opìte hapaðthsh na eðnai det(e iθs )=odhgeð anagkastik sthn tr (S) =0. Sthn perðptwsh twn pin kwn Pauli: tr(σ i )=0. To sônolo twn (2 2) unitary pin kwn me orðzousa + eðnai èna uposônolo tou U(2), to opoðo onom zetai SU(2) ( Special Unitary Group stic 2-diast seic). To SU(2), ìpwc èqoume dh anafèrei, ikanopoieð mia orismènh lgebra metajet n. H lgebra thc om dac SU(2) eðnai hðdia me thn lgebra twn gennhtìrwn J i, dhlad [J i,j j ]=iε ijk J k. Apì ta prohgoômena, eðnai fanerì ìti h SU(2) ikanopoieð thn Ðdia lgebra me thn om da twn strof n stic 3-diast seic. Oi anaparast seicthcsu(2) me j =0,,, 3,...antistoiqoÔn se, 2, 3, 4,... 2 2 diast seic, antðstoiqa. H anapar stash me j = /2 thc SU(2) basðzetai s- touc pðnakec tou Pauli. Aut eðnai h jemeli dhc anapar stash thc SU(2), en ìlec oi upìloipec anaparast seic thc mporoôn na kataskeuastoôn apì thn jemeli dh. Par deigma Ja deiqjeð ìti k je pðnakac 2 diast sewn mporeð na anaptuqjeð sthn b sh twn pin kwn Pauli kai tou tautotikoô. Ac upojèsoume ìti isqôei to zhtoômeno an ptugma gia ton tuqaðo pðnaka 2- diast sewn: M = m 0 + m i σ i, (2.72) ìpou uponoeðtai ajroistik sômbash. ArkeÐ na deðxoume ìti oi suntelestèc m 0,m i tou anaptôgmatoc arkoôn gia na kajorðsoun monos mantaton M. Lamb nontac to Ðqnoc thc exðswshc, èqoume: tr(m) =m 0 tr() = 2m 0, (2.73)
30 KEF ALAIO 2. STOIQE IA JEWR IAS OM ADWN ìpou l bame upìyin mac ìti oi pðnakec Pauli eðnai iqnoi kai ìti to Ðqnoc tou tautotikoô pðnaka stic 2-diast seic isoôtai me 2. Gia na prosdiorðsoume touc m i, pollaplasi zoume thn exðswshepð σ j kai lamb noume kai p li to Ðqnoc: EÐnai eôkolo na elegqjeð ìti Antikajist ntac sthn (2.74), brðskoume: tr(mσ j )=m i tr (σ i σ j ). (2.74) tr(σ i σ j )=2δ ij. (2.75) tr (Mσ j )=2m j. (2.76) Apì tic (2.73) kai (2.76) oi suntelestèc orðzontai monos manta gia k je pðnaka M. Sunep c, oi pðnakec Pauli mazð me thn mon da sunistoôn pl rhb sh ston q ro twn 2-di statwn pin kwn.
2.6. S UNJESH ANAPARAST ASEWN 3 2.6 SÔnjesh Anaparast sewn 'Ena sôsthma pou par getai apì thn sônjesh sôo susthm twn me stroform j A, j B, kai idiokatast seic j A m A, j B m B mporeð na perigrafeð an qrhsimopoi soume wc b sh to eujô ginìmeno twn idiokatast sewn twn dôo swmatðwn: j A j B m A m B j A m A j B m B j A m A j B m B. (2.77) 'Otan exet zoume k je swmatðdio qwrist, oi idiokatast seic stroform c tou kajenìc apoteloôn b seic se diaforetikoôc q rouc (oi q roi eðnai diaforetikoð diìti anafèrontai se diaforetik swmatðdia). H epilog thc sônjethc b shc (2.77) ousiastik isodunameð me ton orismì miac sônjethc anapar stashc, h opoða isoôtai me to eujô ginìmeno (direct product, ) twn dôo mh anag gimwn (irreducible) anaparast sewn. Holik stroform J = J A + J B ikanopoieð, profan c, thn lgebra stroform n [J i,j j ]=iε ijk J k. Epomènwc, h probol J 3 perimènoume na eðnai o monadikìc apì touc genn torec J A,i,J B,i,J i = J A,i + J Bi pou diagwniopoieðtai. Autì ofeðletai sthn dom thc om dac pou epitrèpei mìno se èna genn tora na eðnai diag nioc k je for. Ac doôme, kat' arq c, thn sônjesh stroform n apì thn optik twn anusmatik n q rwn. Sthn arq èqoume dôo anusmatikoôc q rouc, ènan gia k je stroform. O sônjetoc q roc pou prokôptei apì to eujô ginìmeno thc (2.77) eðnai ènac entel c kainoôrioc q roc o opoðoc den mporeð na prokôyei apì kanèna grammikì sunduasmì anusm twn twn arqik n q rwn. Oi arqikèc b seic twn dôo q rwn (idiokatast seic twn J A,3, J B,3 ), profan c, den mporoôn na par goun ton kainoôrio q ro (lème ìti mia b shpar gei ènan anusmatikì q ro me grammikoôc sunduasmoôc twn anusm twn thc). Gia par deigma, fantasteðte ta sun jhdianôsmata a tou tridi statou EukleideÐou q rou, pou par gontai apì thn monadiaða orjokanonik b sh ˆx i. EpÐshc, fantasteðte ton q rotwn tetragwnik oloklhr simwn sunart sewn f(x) (q ro Hilbert), ston opoðo mia orjokanonik b sh eðnai h e n (x) = 2π e inx. AutoÐ oi dôo q roi den epikalôptontai, dhlad h tom touc eðnai to kenì. EÐnai profanèc ìti h b sh kajenìc q rou den mporeð na par gei kanèna nusma tou llou q rou. Sunep c, an jel soume na perigr youme ton anusmatikì q ro pou prokôptei apì ton sunduasmì twn dôo aut n q rwn, ja prèpei na orðsoume mia nèa b sh anusm twn tou nèou q rou. H b sh tou nèou q rou pou prokôptei apì to eujô ginìmeno twn dôo arqik n q rwn ja eðnai e n (ˆx i)=e n (x) ˆx i. H nèa b shden an kei se kanènan apì touc dôo arqikoôc q rouc. Opoioid pote pðnakec telestèc eðqan orisjeð stouc dôo arqikoôc q rouc, eðqan orisjeð wc proc tic antðstoiqec b seic kai, sunep c, kamða apì tic anaparast seic autèc den diathreð thn morf thc ston nèo q ro
32 KEF ALAIO 2. STOIQE IA JEWR IAS OM ADWN (afoô èqei all xei h b sh). EÐnai logikì ìti h nèa b sh ja d sei nèa morf stic anaparast seic pou proôp rqan stouc arqikoôc q rouc. H analogða tou paradeðgmatoc autoô me thn sônjesh stroform n prèpei na eðnai faner. Epanerqìmaste sthn stroform, kai epanalamb noume ìti h olik stroform J ikanopoieð thn Ðdia gnwst lgebra me tic epimèrouc stroformèc, all se diaforetikì anusmatikì q ro! Sunep c, ston nèo anusmatikì q ro, oi arqikèc stroformèc den ufðstantai kan wc anôsmata kai up rqoun mìno sunduasmoð twn dôo stroform n. EpÐshc, efìson oi b seic den eðnai plèon oi arqikèc (idiokatast seic stroform c k je swmatidðou qwrist ), ja all xoun kai ìlec oi anaparast seic pðnaka twn sunistws n thc stroform c. Ac upojèsoume ìti to sôsthma brðsketai se mia idiokat stash thc olik c stroform c J 3. Tìte, h mèsh anamenìmenh tim thc J 3 ja eðnai mia idiotim thc. Antijètwc, hmèshanamenìmenhtim thc J A,3 den ja eðnai en gènei idiotim thc, diìti h J A,3 den mporeð na diagwniopoihjeð tautìqrona me thn J 3. Autì faðnetai apì to gegonìc ìti den metatðjentai: [J i,j A,j ]=[J A,i + J B,i,J A,j ]=iε ijk J A,k. Sunep c, oi m A, m B den eðnai kaloð kbantikoð arijmoð ston nèo q ro, afoô den mporoôn pia na qarakthrðzoun tic nèec kbantikèc katast seic. Dhlad, oi idiokatast seic tou J A,3 den eðnai anôsmata b shc ston nèo q ro, ra den mporoôme na gnwrðzoume thn tim tou J A,3 me apìluthakrðbeia tautìqrona me ta J 3 kai J 2. To Ðdio isqôei kai gia to J B,3. Apì tic metajetikèc sqèseic [J 2 A, J 2 ]=[J 2 B, J 2 ]=[J 2 A,J 3 ]=0,oitelestèc Casimir twn arqik n q rwn: J 2 A, J2 B exakoloujoôn na eðnai kaloð kbantikoð arijmoð ston nèo q ro (dhlad metatðjentai me touc diag niouc J 3 kai J 2 ). Sunep c, mia kbantik kat stashston nèo q ro qarakthrðzetai apì touc arijmoôc j A, j B (to opoðo eðnai logikì, diìti k pwc ja prèpei na kajorðzetai apì poioôc arqikoôc q rouc stroform n katal goume ston nèo), kai apì touc arijmoôc pou qarakthrðzoun thn olik stroform j, m. Autì eðnai apìluta logikì, an skefteð kaneðc ìti mia stroform (ìpwc h fikainoôriafl stroform pou proèkuye wc sônjeshstroform n) prèpei na perigr fetai apì èna j kai èna m (ìpwc kai k je llh stroform ) kai, epiplèon, prèpei na gnwrðzoume apì poioôc q rouc proèkuye (dhlad apì poièc anaparast seic thc SU(2)) ste na kajorðzetai monos manta o nèoc q roc. Epeid to j miac anapar stashc thc SU(2) kajorðzei, ìpwc èqoume dei, monos manta thn anapar stash, èpetai ìti kai to ginìmeno dôo kajorismènwn tètoiwn anaparast sewn ja èqei kajorismèna dunat apotelèsmata (dhlad kajorismènec prokôptousec anaparast seic). Epomènwc, to olikì j kajorðzetai apì ta j A kai j B kai mìno. EpÐshc, epeid oi J A,3, J B,3 den metatðjentai me thn diathroômenh J 3, eðnai profanèc ìti den ja metatðjentai oôte me thn Qamiltonian. Sunep c den ja par goun kan diathr simouc kbantikoôc arijmoôc. Epeid to ginìmeno (2.77) eðnai eujô ginìmeno dôo jemeliwd n anex rthtwn irreducible anaparast sewn (diast sewn (2j A +)kai (2j B +)), hprokôp-