ocw.aoc.ntua.gr · 2015. 5. 8. · kef laio 1 fusik sust mata sto prohgoÔmeno mèroc jewr same...

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Βαρελογιάννης Γεώργιος

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Keflaio 1

Fusik Sust mata

Sto prohgoÔmeno mèroc jewr same èna aplì pardeigma, tic pijanèc jèseicmiac mplac se èna apo ta kouti, kai orÐsame thn entropÐa pou metrei thnèleiyh plhroforÐac sto sÔsthma autì. Sta fusik sust mata pou mac endi-afèroun, th jèsh twn kouti¸n paÐrnoun oi dunatèc katastseic tou sust -matoc pou lègontai kai prospelsimec katastseic, kai to anlogotou erwt matoc se poiì koutÐ brÐsketai h mpla, eÐnai to er¸thma se poikatstash brÐsketai to fusikì sÔsthma pou mac endiafèrei.EÐnai profanèc ìti h angkh miac statistik c antimet¸pishc kai ra m mh-denik c entropÐac prokÔptei ìtan den eÐmaste ikanoÐ me bsh ta dedomèna poudiajètoume na gnwrÐzoume se poi apo tic prospelsimec fusikèc katast-seic brÐsketai to sÔsthma. Autì sumbaÐnei arket suqn dedomènou ìti tamakroskopik sust mata pou mac endiafèroun apoteloÔntai apo ènaterstio arijmì mikroskopik¸n bajm¸n eleujerÐac. EÐmaste loipìnupoqrewmènoi na katafÔgoume se mia statistik perigraf . H perigraf aut ègkeitai sta paraktw b mata:

• EntopÐzoume, me bsh ta dedomèna pou gnwrÐzoume gia to sÔsthma, ticprospelsimec katastseic tou sust matoc.

• Me bsh epÐshc ta dedomèna pou diajètoume, prospajoÔme na antis-toiqÐsoume mia katanom pijanot twn stic prospelsimec katastseic,dhlad na broÔme poi pijanìthta èqei to sÔsthma na brÐsketai se kjemia apo tic prospelsimec katastseic tou.

• Me bsh thn katanom pijanot twn tou prohgoÔmenou b matoc eÐmastese jèsh na upologÐsoume mèsec timèc fsik¸n posot twn kai ra me bshta dedomèna na knoume problèyeic.

3

4 Keflaio 1. Fusik Sust mata

Gia to pr¸to b ma basizìmaste sthn en gènei katanìhsh pou èqoume gia tosÔsthma, tìso gia touc mikroskopikoÔc tou suntelestèc ìso kai gia tou-c makroskopikoÔc periorismoÔc ìpwc gia pardeigma ì ìgkoc tou doqeÐoupou endeqìmena perièqei èna aèrio sÔsthma. Sthn perÐptwsh twn klassik¸nsusthmtwn, oi prospelsimec katastseic apartÐzoun to legìmeno q¸rotwn fsewn tou makroskopikoÔ sust matoc pou mac endiafèrei. Gia takbantik sust mata pou ja mac apasqol soun kata kÔrio lìgo, oiprospelsimec katastseic eÐnai autèc pou apartÐzoun to q¸ro Hilberttwn prospelsimwn katastsewn tou sust matìc. AfoÔ tautopoihjeÐ oprospelsimoc q¸roc twn fsewn, gia thn epilog thc katanom c pijan-ot twn pou antistoiqeÐ se èna sÔsthma se isorropÐa eÐmaste upoqrewmènoi naprosfÔgoume se èna axÐwma pou basÐzetai sthn empeirik mac katanìhsh: toaxÐwma Boltzmann - Gibbs . Me bsh thn epilog thc katanom c Boltzmann- Gibbs , gia orismènec posìthtec eÐnai arket eÔkolo na upologÐsoume ticmèsec timèc touc sunart sei twn dedomènwn tou sust matoc mia kai arkeÐ o up-ologismìc thc legìmenhc sunrthshc epimerismoÔ. Gia llec posìthteceÐnai pio dÔskoloc o upologismìc autìc kai prèpei upoqrewtik na orÐsoumetouc telestèc puknìthtac. O orismìc tou telest puknìthtac shmaÐneisthn ousÐa ton orismì thc katanom c pijanot twn pou epilèxame. Dedomènouìti oi telestèc autoÐ eÐnai aparaÐthtoi gia thn katanìhsh thc statistik c pnwsta kbantik sust mata, to ìlo prìblhma ja kwdikopoihjeÐ ston legìmenoformalismì twn telest¸n puknìthtac. Proèqei ìmwc o orismìc touq¸rou twn prospelsimwn katastsewn, gia ton opoÐo entopÐzoume orismènashmantik qarakthristik kurÐwc to legìmeno je¸rhma Liouville , kai sthsunèqeia ja orÐsoume touc telestèc puknìthtac.

1.1 Klassik Sust mata

Sta Klassik Sust mata, kje swmatÐdio perigrfetai apì èna dinusma jè-sewc r⃗ kai apì èna dinusma orm c p⃗. Dhlad apì 6 bajmoÔc eleujerÐac.AntÐstoiqa, an èqoume èna sÔsthma N swmatidÐwn, apaitoÔntai gia thn peri-graf tou sust matoc 6N bajmoÐ eleujerÐac. Gia na broÔme tic katastseicenìc sust matoc, orÐzoume to q¸ro twn fsewn. O q¸roc twn fsewneÐnai ènac 6N-distatoc uperq¸roc, o opoÐoc prokÔptei apìtouc 6N bajmoÔc eleujerÐac tou sust matoc. H katstash tousust matoc, th qronik stigm t = 0, apoteleÐ èna shmeÐo sto q¸ro twnfsewn. H troqi kje swmatidÐou, sto q¸ro twn fsewn, perigrfetai apìtic exis¸seic Hamilton:

1.2. Kbantik Sust mata 5

dr⃗idt

=∂H

∂p⃗i(1.1.1)

dp⃗idt

=∂H

∂p⃗i, (1.1.2)

ìpou o deÐkthc i anafèretai sto i-ostì swmatÐdio. H epÐlush tou parapnwsust matoc exis¸sewn eÐnai polÔ dÔskolh kai gia makroskopik sust mataen gènnei akatìrjwth. Upoqrewtik katafeÔgoume sth Statistik Fusik .

Je¸rhma Liouville

O ìgkoc tou q¸rou twn fsewn diathreÐtai apì th Qamiltonian dunamik

�i��

O ìgkoc tou q¸rou twn fsewn, diathreÐ thn puknìtht tou stajer ,parìlo pou to sq ma tou mporeÐ na allzei me thn prodo tou qrìnou.

• Wc prospelsimo q¸ro, onomzoume to q¸ro ston opoÐo mporeÐ na brejeÐh katstash tou sust matoc. O ìgkoc tou prospelsimou q¸rou twnfsewn sÔmfwna me to Je¸rhma Liouville sumperifèretai ìpwc o ìgkocenìc asumpÐestou ugroÔ.

• Wc prospelsimh katstash onomzoume kje shmeÐo tou prospelsi-mou q¸rou.

1.2 Kbantik Sust mata

Sta Kbantik Sust mata, kje katstash tou sust matoc ekfrzetai mèswenìc dianÔsmatoc | Ψ⟩. To dinusma | Ψ⟩ an kei sto q¸ro Hilbert, E , tousust matoc, o opoÐoc apoteleÐ ton prospelsimo q¸ro tou sust matoc. Oq¸roc Hilbert antistoiqeÐ sto q¸ro twn fsewn twn Klassik¸n Susthmtwn.To eswterikì ginìmeno sto q¸ro Hilbert, grfetai wc ⟨Φ | Ψ⟩, ìpou ta Φ kaiΨ, eÐnai tuqaÐa dianÔsmata tou q¸rou Hilbert.


Sta Kbantik Sust mata, kje mègejoc apoteleÐ ènan telest , o opoÐoc drasta dianÔsmata tou q¸rou Hilbert. Kje telest c Â, èqei èna sÔnolo idiodi-anusmtwn | ψn⟩ me idiotimèc αn. Dhlad isqÔei:

Â | ψn⟩ = αn | ψn⟩.

MporoÔme na qrhsimopoi soume ta idiodianÔsmata enìc telest (sun jwc thcQamiltonian c), wc bsh tou q¸rou Hilbert. Gia na problloume èna tuqaÐodinusma | Ψ⟩, sth bsh | ψn⟩, paÐrnoume to eswterikì ginìmeno ⟨ψn | Ψ⟩.Mlista, h posìthta | ⟨ψn | Ψ⟩ |2, ekfrzei thn puknìthta pijanìthtac nabrejeÐ to sÔsthma sthn idiokatstash | ψn⟩, ìtan ektelèsoume mÐa katllhlhmètrhsh sto sÔsthma.

1.3 Telestèc Puknìthtac

Oi Telestèc Puknìthtac eis qjhkan apì ton Von Neumann to 1950. OiTelestèc Puknìthtac, mporoÔn na parast soun èna statistikì meÐgma mikrokatastsewn.Se aut n thn perÐptwsh, den gnwrÐzoume epakrib¸c thn katstash tou sust -matoc. AntÐjeta, èqoume mÐa katanom pijanot twn kajar¸n mikrokatastsewn,me thn opoÐa proseggÐzoume pijanojewrhtik thn pragmatik katstash tousust matoc. To gegonìc ìti qrhsimopoioÔme mÐa katanom pijanot twn gia naperigryoume thn katstash tou sust matoc, mac upagoreÔei thn eisagwg enìc telest puknìthtac pijanot twn, ¸ste na ekfrsoume to statistikìmeÐgma twn mikrokatastsewn.

Telest c Puknìthtac KlassikoÔ Sust matoc

JewroÔme èna klassikì sÔsthma N swmatidÐwn, to opoÐo perigrfetai apìtic genikeumènec suntetagmènec qi kai tic genikeumènec ormèc pi, ìpou i =1, 2, . . . , 3N . O Telest c Puknìthtac ρ(qi, pi) eÐnai h puknìthta pijanìthtacna qarakthrÐzetai to sÔsthma apì tic dunatèc timèc (q, p), pou brÐskontai seènan stoiqei¸dh ìgko dτ , gÔrw apì to shmeÐo (qi, pi). O stoiqei¸dhc ìgkoctou q¸rou twn fsewn, dτ , dÐnetai apì th sqèsh:

dτ =1

N !

N∏i=1

d3qid3pi

~3(1.3.1)

1.3. Telestèc Puknìthtac 7

Idiìthtec tou Telest Puknìthtac:

• O Telest c Puknìthtac paristnei puknìthtac pijanìthtac, sunep¸cgia autìn ja isqÔei h sqèsh kanonikopoÐhshc :∫

Vdτρ(q, p) = 1. (1.3.2)

• H mèsh tim , enìc megèjouc A, mporeÐ na grafeÐ:

⟨A⟩ =∫VdτA(q, p)ρ(q, p) (1.3.3)

• 'Olh h plhroforÐa enìc statistikoÔ meÐgmatoc katastsewn, brÐsketaiston Telest Puknìthtac. H qronik exèlixh tou Telest Puknìthtac,dÐnetai apì th sqèsh:

∂ρ

∂t+ {H, ρ} = 0, (1.3.4)

ìpou {, } oi agkÔlec Poisson kai H h Qamiltonian tou sust matoc.

Telest c Puknìthtac KbantikoÔ Sust matoc

Kai to kbantikì sÔsthma mporeÐ na brÐsketai se èna statistikì meÐgmakatastsewn. JewroÔme san bsh tou q¸rou Hilbert, tic idiakatastseicenìc telest Â, pou dÐnontai apì th sqèsh:

Â | ψn⟩ = αn | ψn⟩.

H mèsh tim tou telest Â, ⟨Â⟩, dÐnetai apì th sqèsh:

⟨Â⟩ =∑n

Pnαn =∑n

Pn⟨ψn | Â | ψn⟩, (1.3.5)

ìpou Pn h pijanìthta na brejeÐ to sÔsthma sthn katstash ψn. Se autì toshmeÐo, orÐzoume ton Telest Puknìthtac ρ̂, mèsw thc sqèshc:

ρ̂ =∑n

Pn | ψn⟩⟨ψn | . (1.3.6)


Idiìthtec tou Telest Puknìthtac:

• To gegonìc ìti to jroisma twn pijanot twn isoÔtai me th monda,dhlad

∑n Pn = 1, sunepgetai thn akìloujh sqèsh gia to Ðqnoc tou

Telest Puknìthtac:Tr{ρ̂} = 1. (1.3.7)

• H mèsh tim , enìc telest Â, mporeÐ na grafeÐ:

⟨Â⟩ = Tr{ρ̂Â}. (1.3.8)

• 'Olh h plhroforÐa enìc statistikoÔ meÐgmatoc katastsewn, brÐsketaiston Telest Puknìthtac. H qronik exèlixh tou Telest Puknìthtac,dÐnetai apì th sqèsh:

i~∂ρ̂

∂t= [Ĥ, ρ̂], (1.3.9)

ìpou Ĥ h Qamiltonian tou sust matoc.

• MÐa teleutaÐa idiìthta, pou isqÔei kai sthn klassik perÐptwsh, eÐnaiìti gia dÔo diaforetikèc katanomèc ρ̂ kai ρ̂ ′, isqÔei h sqèsh:

−kBTr {ρ̂ ln ρ̂} ≤ −kBTr{ρ̂ ′ ln ρ̂ ′

}ìpou h isìthta isqÔei gia ρ̂ = ρ̂ ′.

(1.3.10)

1.4 Katanomèc IsorropÐac

'Ena sÔsthma, qarakthrÐzetai apì dÔo eÐdh dedomènwn:

• Epakrib¸c kajorismèna dedomèna

OrÐzoun ton prospelsimo q¸ro tou sust matoc, tìso se Kbantikìso se Klassik Sust mata. Gia pardeigma, jewroÔme ìti ìlec oimikrokatastseic eÐai isopÐjanec. Se aut n thn perÐptwsh, an èqoumeW isopÐjanec mikrokatastseic, o Telest c Puknìthtac grfetai:

ρ̂ =∑n

1

W| ψn⟩⟨ψn |

1.4. Katanomèc IsorropÐac 9

kai h entropÐa tou sust matoc, isoÔtai me

S = kB lnW

• Dedomèna StatistikoÔ Qarakt ra

Sth sugkerimènh perÐptwsh, h mèsh tim kpoiou megèjouc, eÐnai ka-jorismènh. Gia pardeigma, an to sÔsthma eÐnai se epaf me mÐa dex-amen jermìthtac, h enèrgei tou eÐnai kajorismènh. Dhlad isqÔei:U = Tr{ρ̂Ĥ}. Sth genik perÐptwsh, èqoume kpoio dedomèno thc mor-f c ⟨X̂i⟩ = Tr{ρ̂X̂i}, en¸ parllhla isqÔei Tr{ρ̂} = 1. Den uprqeièna monadikì ρ̂ pou na ikanopoieÐ touc parapnw periorismoÔc. Mlista,oi katastseic den eÐnai isopÐjanec diìti èqoume statistik katanom . Gi-a na epilèxoume to ρ̂ se aut n thn perÐptwsh, qrhsimopoioÔme to AxÐwmaBoltzman-Gibbs.

AxÐwma Boltzmann-Gibbs

O Telest c Puknìthtac ρ̂, ja prèpei na epilegeÐ apììlec tic katanomèc pou eÐnai sumbatèc me ta StatistikDedomèna, me krit rio th megistopoÐhsh thc entropÐac.Dhlad , th megistopoÐhsh tou ìrou S = −kBTr {ρ̂ ln(ρ̂)}.To sugkekrimèno krit rio, basÐzetai sto gegonìc ìtih epilog tou ρ̂, den prèpei na prosjètei perissìterhplhroforÐa, apì aut n pou pragmatik èqoume. IsodÔ-nama, ja prèpei na mh mei¸nei thn gnoia pou èqoume.

SÔmfwna me to axÐwma Boltzmann-Gibbs, o Telest c Puknìthtac upologÐze-tai apì th megistopoÐhsh thc entropÐac. Sth sugkekrimènh perÐptwsh, èqoumeepiplèon periorismoÔc. Autì mac anagkzei na qrhsimopoi soume th mèjodotwn pollaplassiast¸n Lagrange, ¸ste na enswmat¸soume touc periorismoÔc.OrÐzoume th sunrthsh:

S̄ = S −∑i

λiTr{ρ̂X̂i − λ0Tr {ρ̂}

}. (1.4.1)


Ta λi, eÐnai oi pollaplassiastèc Lagrange, oi opoÐoi antistoiqoÔn stoucperiorismoÔc ⟨X̂i⟩ = Tr{ρ̂X̂i}. H mèjodoc pou akoloujoÔme, eÐnai na up-ologÐsoume to ρ̂ apì th megistopoÐhsh thc S̄ kai èpeita na broÔme ta λi mèswtwn periorism¸n. H megistopoÐhsh thc S̄, odhgeÐ sth sqèsh:

dS̄dρ̂

= 0 ⇒

ln(ρ̂) +∑i

λiX̂i + λ0 + 1 = 0 (1.4.2)

Qrhsimopoi¸ntac thn parapnw sqèsh, mporoÔme na gryoume:

ρ̂ =1

Ze−

∑i λiX̂i , (1.4.3)

ìpou Z, h sunrthsh epimerismoÔ tou sust matoc. H morf thc sunrthshcepimerismoÔ, prokÔptei apì ton periorismì Tr{ρ̂} = 1. Eidikìtera, isqÔei:

Z = Tr{e−

∑i λiX̂i

}. (1.4.4)

Sthn perÐptwsh pou èqoume kajorismènh enèrgeia, dhlad sthn perÐptwsh touKanonikoÔ Sunìlou, h sunrthsh epimerismoÔ èqei th morf :

Z = Tr{e−λĤ

}=∑n

e−βEn , (1.4.5)

ìpou En, oi idiotimèc thc Qamiltonian c Ĥ. Epeid to Ðqnoc enìc telest eÐnaianexrthto thc bshc pou qrhsimopoioÔme, h parapnw sqèsh paramènei Ðdia,eÐte an qrhsimopoi soume san bsh tic idiokatastseic hc Qamiltonian c, eÐteìqi. H katanom sthn opoÐa katal xame, onomzetai katanom Boltzmann-Gibbs, kai isqÔei gia to Kanonikì SÔnolo.

⟨X̂i⟩ = −1

Z∂

∂λiTr{e−

∑i λiX̂i

}(1.4.6)

Z = Tr{e−

∑i λiX̂i

}(1.4.7)

⟨X̂i⟩ = −∂

∂λi(lnZ) (1.4.8)

1.5. Genik MejodologÐa 11

S = −kBTr{ρ̂ ln ρ̂} (1.4.9)

= −kBTr

{ρ̂

(lnZ +

∑i

λiX̂i

)}(1.4.10)

= −kB lnZ − kB∑i

λi∂ lnZ∂λi

(1.4.11)

dS = kB

(d lnZ +

∑i

λid⟨X̂i⟩+∑i

⟨X̂i⟩dλi

)(1.4.12)

d lnZ = −∑i

⟨X̂i⟩dλi (1.4.13)

dS = kB∑i

λid⟨X̂i⟩. (1.4.14)

Oi fusikèc metablhtèc thc entropÐac, eÐnai oi mèsec timèc twn X̂i. Oi λi, eÐnaioi epimèrouc pargwgoi thc entropÐac S/kB se sqèsh me tic mèsec timèc ⟨X̂i⟩.Autì shmaÐnei, ìti o pollaplassiast c Lagrange, λi, metr thn taqÔthtametabol c thc entropÐac wc proc th metabol thc mèshc tim c ⟨X̂i⟩.

∂S∂⟨X̂i⟩

= kBλi (1.4.15)

kB lnZ = S −∑i

⟨X̂i⟩∂S∂⟨X̂i⟩

(1.4.16)

H kB lnZ eÐnai o metasqhmatismìc Legendre thc entropÐac.

1.5 Genik MejodologÐa

Genik Mèjodoc gia sust mata se isorropÐa (mègisth en-tropÐa).

• OrÐzoume to q¸ro Hilbert twn prospelsimwn katastsewn.

• AnagnwrÐzoume tic �Stajerèc KÐnhshs� tou sust matoc kai sth sunèqeiatouc telestèc X̂i, pou antistoiqoÔn se autèc. Tètoiec stajerèc, eÐnaih enèrgeia, o arijmìc twn swmatidÐwn, ktl. Oi mèsec timèc ⟨X̂i⟩, eÐnaidedomèna pou qarakthrÐzoun to Makrskopikì SÔsthma se isorropÐa.


• Se kje X̂i antistoiqÐzoume èna λi. O Telest c Puknìthtac gia tosÔsthma se isorropÐa, dÐnetai apì thn katanom Boltzmann-Gibbs:

ρ̂ =1

Ze−

∑i λiX̂i . (1.5.1)

• Gia kje fusikì mègejoc Â, isqÔei ⟨Â⟩ = Tr{ρ̂Â}.

• Gia ta fusik megèjh pou diathroÔntai, to prìblhma eÐnai aploÔstero.UpologÐzoume kat�arq n, th sunrthsh epimerismoÔZ = Tr

{e−

∑i λiX̂i

}kai èpeita upologÐzoume to diathroÔmeno mègejoc pou mac endiafèrei,apì th sqèsh:

⟨X̂i⟩ = −∂

∂λiln (Z {λi}) (1.5.2)

• An endiaferìmaste gia th mèsh tim mÐac posìthtac Â, h opoÐa mporeÐna prokÔyei apì kpoia diathroÔmenh posìthta X̂i me parag¸gish wcproc kpoia parmetro ξ pou emfanÐzetai ston orismì thc, tìte h gn¸shthc Z sunart sei thc ξ arkeÐ gia ton prosdiorismì thc mèshc tim c touÂ kai isqÔei h sqèsh

⟨Â⟩ = ⟨∂X̂i∂ξ

⟩ = − 1λi

∂

∂ξln (Z {λi}) (1.5.3)

Na shmeiwjeÐ ìti, oi idiotimèc twn X̂i eÐnai sun jwc gnwstèc (toulqis-ton gia ta aploÔstera probl mata). Dedomènou ìti o upologismìc touÐqnouc enìc pÐnaka eÐnai anexrthtoc thc bshc sthn opoÐa anaparÐs-tatai o pÐnakac, epilègoume na ergastoÔme sth bsh ( anaparstash)sthn opoÐa ta X̂i eÐnai diag¸nia kai ra kai o telest c puknìthtac eÐnaidiag¸nioc. To Ðqnoc isodunameÐ sthn perÐptwsh aut me to jroismatwn antÐstoiqwn ekjetik¸n sta opoÐa antÐ gia ta X̂i brÐskontai oi an-tÐstoiqec idiotimèc touc.

• 'Otan h Qamiltonian tou sust matoc èqei th morf :

Ĥ =N∑i

Ĥi ìpou [Ĥi, Ĥj ] = 0 ∀ i, j (1.5.4)

To gegonìc ìti oi epimèrouc Qamiltonianèc metatÐjentai, shmaÐnei ìtito sÔsthm mac apoteleÐtai apì N anexrthta uposust mata. Ara

1.6. Kanonik SÔnola 13

o sunolikìc q¸roc Hilbert, E , tou sust matoc, prokÔptei apì to exw-terikì ginìmeno twn q¸rwn Hilbert twn uposusthmtwn, dhlad

E = E1 ⊗ E2 ⊗ . . .⊗ EN

. Autì èqei san apotèlesma h sunrthsh epimerismoÔ tou sust matoc,Z, na paragontopoieÐtai wc akoloÔjwc:

Z =N∏i

Zi (1.5.5)

Sun jhc eÐnai h perÐptwsh kat thn opoÐa ta N sust mata eÐnai ìmoiametaxÔ touc opìte jètontac Zi = Zo, èqoume gia th sunolik sunrthshepimerismoÔ thn apl morf :

Z = Z No (1.5.6)

• Tèloc, gia thn entropÐa isqÔei:

S = kB lnZ −∑i

∂

∂λi(kB lnZ) (1.5.7)

prgma pou anadeiknÔei pìso shmantik eÐnai h eÔresh thc Z{λi}.

1.6 Kanonik SÔnola

'Estw oi 3 Stajerèc KÐnhshc:

• Enèrgeia → X̂1 = Ĥ

• Arijmìc SwmatidÐwn → X̂2 = N̂

• 'Ogkoc → X̂3 = V

1)H,N ,V Kajorismèna Akrib¸c⇒Mikrokanonikì SÔnolo (p.q.idanikì aèrio).

Se aut n thn perÐptwsh èqoume W isopÐjanec prospelsimec katastseic. Oarijmìc W aut¸n twn katastsewn, exarttai apì tic timèc twn diathroÔmen-wn posot twn. O Telest c Puknìthtac, èqei th morf :

ρ̂ =∑n

| ψn⟩1

W⟨ψn | . (1.6.1)


H entropÐa dÐnetai apì th sqèsh:

S = kB lnW. (1.6.2)

Oi posìthtec ∂S∂U ,∂S∂N ,

∂S∂V eÐnai oi SuzugeÐc Metablhtèc twn U , N , V se

sqèsh me thn entropÐa S.

2)N kai V kajorismèna akrib¸c, H kajorismèno kat Mèso'Oro ⇒ Kanonikì SÔnolo (p.q. jermikì sÔsthma).

O Telest c Puknìthtac, èqei th morf :

ρ̂ =1

ZCe−λHĤ =

1

ZCe−βĤ . (1.6.3)

H sunrthsh epimerismoÔ, dÐnetai apì th sqèsh:

ZC = Tr{e−βĤ

}. (1.6.4)

Epilègoume san bsh tou q¸rou Hilbert, ta idiodianÔsmata | ψn⟩ thc Qamil-tonian c. IsqÔoun:

Ĥ | ψn⟩ = En | ψn⟩

e−βĤ | ψn⟩ = e−βEn | ψn⟩

ZC = Tr{e−βĤ

}=∑n

⟨ψn | e−βĤ | ψn⟩ =∑n

e−βEn

λi = β (1.6.5)

X̂i = Ĥ (1.6.6)

⟨X̂i⟩ = ⟨Ĥ⟩ (1.6.7)

⟨Ĥ⟩ = U = −∂ lnZC(β)∂β

(1.6.8)

H entropÐa tou sust matoc, dÐnetai apì th sqèsh:

S(U) = kB lnZC + βkBU (1.6.9)

H metablht ∂S∂U , eÐnai h Suzug c Metablht thc enèrgeiac wc proc thn en-tropÐa.

∂S∂U

= kBβ. (1.6.10)

1.6. Kanonik SÔnola 15

Oi diakumseic thc enèrgeiac, gÔrw apì th mèsh tim thc, dÐnontai apì thsqèsh:

⟨(⟨Ĥ⟩ − U)2⟩ = ∂2 lnZC∂β2

=−kB

∂2S/∂U2(1.6.11)

3)V kajorismèno akrib¸c. H kai N kajorismèna kat mèsoìro. Megalo-Kanonikì SÔnolo.

⟨Ĥ⟩ = Tr{ρ̂Ĥ}= U ⟨N̂⟩ = Tr

{ρ̂N̂}= N (1.6.12)

ρ̂ =1

ZGe−βĤ−λN N̂ (1.6.13)

ZG = Tr{e−βĤ−λN N̂

}(1.6.14)

U = ⟨Ĥ⟩ = −∂ lnZG∂β

(1.6.15)

N = ⟨N̂⟩ = −∂ lnZG∂λN

(1.6.16)

S = kB lnZG + kBβU + kBλNN. (1.6.17)

β =1

kB

∂S∂U

(1.6.18)

λN =1

kB

∂S∂N

(1.6.19)

4)N kajorismèno akrib¸c. Ta H kai V eÐnai kajorismèna katmèso ìro. Isojermikì-Isobarikì SÔnolo.

ρ̂ =1

ZIe−βĤ−λV V (1.6.20)

ZI =∫

dV e−λV V Tr{e−βĤ

}= metasqhmatismìc Laplace thc ZC (1.6.21)

Sugkentrwtik ja èqoume: β = 1kBT , λN = −µ

kBTkai λV =

pkBT

.


1.7 ParadeÐgmata Tupik¸n Ask sewn

H apìdeixh twn idiot twn twn Telest¸n Puknìthtac thc paragrfou 1.3 ka-j¸c kai twn sqèsewn pou prokÔptoun apì to axÐwma Boltzmann−Gibbs twnparagrfwn 1.4 - 1.7 apoteloÔn shmantikèc ask seic. Tupik paradeÐgmataeÐnai ta akìlouja:

΄Ασκηση

Αποδείξτε ότι για μια ποσότητα X̂i για την οποία υπάρχειστατιστικός περιορισμός (διατηρήσιμη ποσότητα) αρκείη συνάρτηση επιμερισμού για τον υπολογισμό της μέσης

τιμής από τη σχέση⟨X̂i

⟩= −∂ lnZ∂λi .

Απάντηση

⟨X̂i

⟩= tr

{ρ̂X̂i

}= tr

e−∑

i λiX̂iX̂i

tr{e−

∑i λiX̂i

} = tr

∂

∂λi

(−e−

∑i λiX̂i

)tr{e−

∑i λiX̂i

} =

=

− ∂∂λi tr{e−

∑i λiX̂i

}tr{e−

∑i λiX̂i

} Z=tr{e−∑i λiX̂i}= = − ∂Z

∂λi

1

Z= −∂ lnZ

∂λi

΄Ασκηση

Αποδείξτε ότι για τη μέση τιμή μιας ποσότητας Â πουπροκύπτει από μια πρώτη παραγώγιση μιας διατηρήσιμηςποσότητας X̂i ως προς μια παράμετρο ξ επίσης αρκεί η

συνάρτηση επιμερισμού και ισχύει η σχέση⟨Â⟩= − 1λi

∂ ln Z∂ξ .

Απάντηση

Μπορούμε για παράδειγμα να ξεκινήσουμε από το δεύτερο μέρος τηςισότητας και να καταλήξουμε στο πρώτο. Εχουμε λοιπόν

− 1λi

∂ ln Z

∂ξ= − 1

λi

1

Z

∂Z

∂ξ= − 1

λi

1

Z

∂

∂ξ

(tr{e−

∑i λiX̂i

})=

= − 1λi

1

Ztr

{∂

∂ξ

(e−

∑i λiX̂i

)}= − 1

λi

1

Ztr

{∂X̂i∂ξ

(−λi)(e−

∑i λiX̂i

)}=

1.7. ParadeÐgmata Tupik¸n Ask sewn 17

=1

Ztr

{∂X̂i∂ξ

(e−

∑i λiX̂i

)} Â= ∂X̂i∂ξ= tr

{Âe−

∑i λiX̂i

Z

}= tr

{ρ̂Â}⇒

⇒⟨Â⟩= tr

{ρ̂Â}= − 1

λi

∂ ln Z

∂ξ

Keflaio 2

Nomoi thc Jermodunamik ckai Statistik Fusik

Mhdenikìc Nìmoc thc Jermodunamik c:

DÔo sust mata se jermik isorropÐa me èna trÐto, mènounse jermik isorropÐa metaxÔ touc, an ta fèroume se epaf .

Apì ton parapnw nìmo mporoÔme na orÐsoume ta jermìmetra.

2.1 Jermik Epaf sto Kanonikì SÔnolo

'Estw dÔo sust mata α kai β se isorropÐa. An eÐnai jermik monwmèna, eÐ-nai anexrthta metaxÔ touc. Se aut n thn perÐptwsh, h enèrgeia tou kjesust matoc eÐnai mÐa statistik kajorismènh posìthta. Gia to kje sÔsthma,mporoÔme na eisgoume ènan pollaplassiast Lagrange, o opoÐoc ja sundèe-tai me ton periorismì, thc Ôparxhc statistik kajorismènhc enèrgeiac. HQamiltonian tou sunìlou twn dÔo anexrthtwn susthmtwn, grfetai wcex c:

Ĥ = Ĥα + Ĥβ . (2.1.1)

Epeid ta dÔo sust mata eÐnai anexrthta, oi epimèrouc Qamiltonianèc metatÐ-jontai, dhlad [Ĥα, Ĥβ ]. Gia ta dÔo sust mata, isqÔoun oi sqèseic:

ρ̂α =1Zα e

−βαĤα ρ̂β =1Zβ e

−ββĤβ

Zα = Tr{e−βαĤα

}Zβ = Tr

{e−ββĤβ

}Uα = ⟨Ĥα⟩ = − ∂∂βα lnZα Uβ = ⟨Ĥβ⟩ = −

∂∂ββ

lnZβ

(2.1.2)

19

20 Keflaio 2. Nomoi thc Jermodunamik c kai Statistik Fusik

Gia ton olikì Telest Puknìthtac, isqÔei:

ρ̂α+β = ρ̂α ⊗ ρ̂β =e−βαĤα−ββĤβ

Zα · Zβ(2.1.3)

Fèrnoume ta dÔo sust mata se jermik epaf . Plèon sth Qamiltonian tousust matoc, ja uprqei ènac ìroc allhlepÐdrashc V̂ . Dhlad :

Ĥ = Ĥα + Ĥβ + V̂ . (2.1.4)

H allhlepÐdrash V̂ , exarttai apì ta qarakthristik twn dÔo epimèrouc susth-mtwn. EÐnai polÔ asjen c kai den allzei th fusik tou kje sust matoc.Eidikìtera,isqÔei V̂

2.2. Pr¸toc Nìmoc thc Jermodunamik c 21

Genik:Se kje suzug metablht , antistoiqÐzoume mÐa diathr simh posìthta, h opoÐ-a, ìtan jètoume se epaf ta dÔo sust mata ¸ste na mporoÔn na antallsounth sugkekrimènh diathr simh posìthta, ja teÐnei na gÐnei Ðsh sta 2 sust mata.Gia pardeigma, an dÔo sust mata se elafr epaf mporoÔn na antallxounswmatÐdia, tìte to λN deÐqnei thn tsh tou sust matoc na apwlèsei swmatÐdi-a kai oi metabolèc tou λN qarakthrÐzei thn antallag swmatidÐwn akrib¸cìpwc to β thn antallag enèrgeiac.

2.2 Pr¸toc Nìmoc thc Jermodunamik c

Arq Diat rhshc thc Enèrgeiac

Kat th metbash enìc sust matoc, apì mÐa katstash 1 semÐa katstash 2, isqÔei:

U2 −U1 = W +Qìpou U h eswterik enèrgeia tou sust matoc, W to èrgo pouprokÔptei kat th metbash kai Q to posì thc jermìthtac pouantallsetai kat th metbash.

U = Tr{ρ̂Ĥ}⇒ (2.2.1)

dU = Tr{Ĥdρ̂

}+ Tr

{ρ̂dĤ

}(2.2.2)

↑ ↑ (2.2.3)dQ dW (2.2.4)

H jermìthta Q sqetÐzetai me th metabol twn pijanot twn tou sust matoc,en¸ to èrgo W sqetÐzetai me th metabol twn energeiak¸n epipèdwn. Jew-roÔme ìti h bsh tou q¸rou Hilbert tou sust matoc, eÐnai oi idiokatastseicthc Qamiltonian c | ψn⟩. Epiplèon, jewroÔme ìti h katstash tou sust -matoc, eÐnai mÐa upèrjesh twn | ψn⟩. H eswterik enèrgeia tou sust matoc,U , dÐnetai apì th mèsh tim thc Qamiltonian c:

U = ⟨Ĥ⟩ = Tr{ρ̂Ĥ}=∑n

PnEn. (2.2.5)

H sqèsh (2.2.1), qrhsimopoi¸ntac thn parapnw sqèsh, aplopoieÐtai wc ex c:

dU =∑n

(dPn)En +∑n

Pn(dEn) (2.2.6)


dQ =∑n

EndPn (2.2.7)

dW =∑n

PndEn. (2.2.8)

'Ergo:Ac upojèsoume mh apeirostèc metabolèc (dhlad meglec allagèc). Jew-roÔme ìti uprqei mÐa energ Qamiltonian Ĥeff , h opoÐa perigrfei th metabol

kai den eÐnai h olik Qamiltonian tou sust matoc. H Qamiltonian Ĥeffanafèretai sto sÔsthma pou energeÐ h Ĥeff sthn opoÐa oi parmetroi pouexart¸ntai apì exwterikoÔc pargontec antikajÐstantai apì paramètrouc ξαtwn opoÐwn h qronik exrthsh eÐnai elegqìmenh (p.q. gia kÔlindro autokin -tou eÐnai o ìgkoc). Gia kÔlindro autokin tou h parmetroc ξα eÐnai o ìgkocV (t). En gènei:

dHeff =∑α

Fαξα, (2.2.9)

ìpou Fα eÐnai oi genikeumènec dunmeic. Ta Fα eÐnai oi suzugeÐc posìthtectwn ξα kai eÐnai parathr shmec posìthtec. Ta ξα onomzontai genikeumènecmetatopÐseic. Gia pardeigma to ξα mporeÐ na eÐnai to magnhtikì pedÐo kai toFα h magnhtik rop . Genik, gia kje metabol isqÔei:

dW = Tr{ρ̂Ĥ}=∑α

⟨Fα⟩dξα, (2.2.10)

ìpou ⟨Fα⟩ h mèsh genikeumènh dÔnamh pou antistoiqeÐ sth genikeumènh metatìpishξα.

PÐesh = −⟨∂H

∂V

⟩≡ P = −Tr

{ρ̂∂Ĥ

∂V

}(2.2.11)

Pardeigma: 'Ergo exwterikoÔ magnhtikoÔ pedÐou: dW = ⟨M⃗⟩ · dB⃗.

ShmeÐwsh:Oi antallagèc èrgou eÐnai antistreptèc! An ξα → −ξα tìtedW → −dW . Autì den isqÔei gia tic antallagèc jermìthtac.

2.3. DeÔteroc Nìmoc thc Jermodunamik c 23

2.3 DeÔteroc Nìmoc thc Jermodunamik c

O deÔteroc nìmoc thc Jermodunamik c, sqetÐzetai me th mh antistreptìthtatwn metabol¸n twn makroskopik¸n susthmtwn. Arqik diatup¸jhke apìton Clausius.

O DeÔteroc Nìmoc kat ton Clausius:

H aujìrmhth antallag jermìthtac metaxÔ dÔo susthmtwn èqeimìno mÐa for. Apì to jermì sto yuqrì. Oi antallagèc thcjermìthtac eÐnai mh antistreptèc. Den mporoÔme na ftixoumemÐa mhqan h opoÐa ja metatrèpei ìlh th jermìthta se èrgo.

O DeÔteroc Nìmoc kat touc Kelvin-Ostwald:

To èrgo mporeÐ pnta na metatrapeÐ se jermìthta. 'Omwc den mpo-roÔme na kataskeusoume mÐa suneq epanalambanìmenh kÐnhsh(kuklik ) pou ja par gage ènan kleistì kÔklo èrgou, paÐrnontacjermìthta apì mÐa mìno phg me omoiogen jermokrasÐa.

Se mÐa sqedìn statik (quasi-static) metabol , ¸ste to sÔsthma eÐnai pntase jermik isorropÐa, isqÔei:

dS = δQT

(2.3.1)

ìpou S h jermodunamik entropÐa. Se kje llh perÐptwsh metabol c (p.q.apìtomh),apì mÐa katstash isorropÐac 1, se mÐa nèa katstash isorropÐac 2, h metabol eÐnai mh antitrept kai isqÔei:

S2 − S1 >∫ 21

δQ

T(2.3.2)

kai odhgeÐ se aÔxhsh thc entropÐac.ShmeÐwsh: H jermodunamik entropÐa eÐnai prosjetik .Apì thn poyh thc statistik c:

Q = Tr{dρ̂Ĥ

}⇒

βdQ = d(Tr{βρ̂Ĥ

})− Tr

{βρ̂dĤ

}= d

(Tr{βρ̂Ĥ

})+

1

Zd(Tr{e−βĤ

})(2.3.3)


ìpou qrhsimopoi same th sqèsh:

Z = Tr{e−βĤ

}(2.3.4)

Telik, qrhsimopoi¸ntac touc orismoÔc thc eswterik c enèrgeiac U kai thcsunrthshc epimerismoÔ Z, prokÔptei:

βdQ = d (βU) + d (lnZ) (2.3.5)

Sthn perÐptwsh tou KanonikoÔ Sunìlou:

kβdQ = d(kβU) + d (k lnZ) (2.3.6)= d(kβU + k lnZC) (2.3.7)

kβdQ = dSΣτατιστική (2.3.8)

Mèsw thc sqèshc:δQ = dSΘερμοδυναμική · T (2.3.9)

Jètontac

β =1

kT(2.3.10)

TautopoioÔme th Jermodunamik EntropÐa me th Statistik EntropÐa :

S = −kTr {ρ̂ ln ρ̂} = k lnZ + kβU (2.3.11)

Jermodunamik Dunamik: Posìthtec twn opoÐwn ta akrìtata, dÐnountic katastseic isorropÐac tou sust matoc.

Metablhtèc U,N, V T, V,N T, V, µ T, P,N

SÔnola Mikrokanonikì Kanonikì Megalokanonikì Isobarikì-

Isojermikì

Sunrthsh W ZC = ZG = ZI =∫dV

EpimerismoÔ Tr{e−βĤ

}Tr

{e−(βĤ−µN̂)

}Tr

{e−β(Ĥ+pV )

}Jermodunamikì EntropÐa El. Enèrgeia Meglo Dunamikì El. EnjalpÐa

Dunamikì S = k lnW F = −kT lnZC Ω = −kT lnZG G = −kT lnZISqèsh me F = U − TS Ω = U − µN − TS G = U + PV − TS

thn entropÐa

Diaforik TdS = dU+ dF = −SdT+ dΩ = −SdT+ dG = −SdT+Morf −µdN + PdV +V dP + µdN −PdV + µdN +V dP +Ndµ

2.4. TrÐtoc Nìmoc thc Jermodunamik c 25

2.4 TrÐtoc Nìmoc thc Jermodunamik c

EÐnai adÔnato na petÔqoume thn apìluth mhdenik jermokrasÐa.

Apì thn poyh thc Statistik c, h Statistik entropÐa eÐnaimhdèn sto apìluto mhdèn.

2.5 Sqèseic Maxwell sth Jermodunamik

Ac upojèsoume ìti èqoume mÐa monadiaÐa mza ousÐac. Oi metablhtèc P, V, T, Sden eÐnai anexrthtec. Wc epÐ to pleÐston, dÔo apì autèc mporoÔn na jew-rhjoÔn wc anexrthtec. Gia pardeigma:

dU = TdS − PdV (2.5.1)

dU =

(∂U

∂x

)y

dx+

(∂U

∂y

)x

dy (2.5.2)

ìpou o deÐkthc deÐqnei poia metablht diathreÐtai stajer kat th metabol .

Gia tic meiktèc deÔterec parag¸gouc, isqÔei h sqèsh:

∂2U

∂x∂y=

∂2U

∂y∂x(2.5.3)

Gia pardeigma: (∂T

∂V

)S= −

(∂P

∂S

)V

(2.5.4)

H parapnw exÐswsh, apodeiknÔetai mèsw thc sqèshc:

dU =

(∂U

∂S

)V

dS +(∂U

∂V

)SdV = TdS − PdV (2.5.5)

Sunep¸c, ja isqÔoun:(∂U

∂S

)V

= T kai

(∂U

∂V

)S= −P (2.5.6)

Mèsw thc isìthtac twn meikt¸n parag¸gwn:


∂2U

∂SV ∂VS=

∂2U

∂VS∂SV(2.5.7)

apodeiknÔetai h zhtoÔmenh sqèsh.

Keflaio 3

Efarmogèc kai Probl mata

3.1 Paramagnhtismìc

To fainìmeno tou paramagnhtismoÔ, perigrfei èna sÔsthma diakrit¸n swmatidÐ-wn iìntwn pou fèroun magnhtik rop , h opoÐa prosanatolÐzetai parllhlase èna exwterik efarmozìmeno magnhtikì pedÐo. H kje rop allhlepidrme to exwterikì magnhtikì pedÐo en¸ jewroÔme ìti den uprqei allhlepÐdrashmetaxÔ twn magnhtik¸n rop¸n.

�

Sq¨ma 3.1: QwrÐc thn efarmog magnhtikoÔ pedÐou (arister) oi magnhtikècropèc twn iìntwn èqoun tuqaÐa pìlwsh opìte h mèsh magn tish eÐnai mhdenik .Met thn efarmog tou magnhtikoÔ pedÐou (dexi), oi ropèc peristrèfontai kaigÐnontai parllhlec ( sqedìn parllhlec) tou pedÐou opìte h mèsh magn tisheÐnai peperasmènh

27

28 Keflaio 3. Efarmogèc kai Probl mata

Proèleush thc magnhtik c rop c enìc iìntoc

'Ena iìn, apoteleÐtai apì ton pur na kai èna sÔnolo hlektronÐwn. H magnhtik rop tou iìntoc prokÔptei apì ta hlektrìnia thc exwterik c tou stibdac. Hphg thc magnhtik c rop c, eÐnai h olik troqiak stroform kai to olikì spintwn hlektronÐwn thc exwterik c stibdac. Eidikìtera, h magnhtik rop enìciìntoc, to opoÐo qarakthrÐzetai apì touc kbantikoÔc arijmoÔc thc troqiak cstroform c L, tou olikoÔ spin S kai sunolik c stroform c J , dÐnetai apì thsqèsh:

µ⃗ = −gµBJ⃗ (3.1.1)

ìpou g eÐnai o pargontac Lande:

g =3

2− L(L+ 1)− S(S + 1)

2J(J + 1)(3.1.2)

kai µB h magnhtình tou Bohr:

µB =e~2mc

(3.1.3)

Oi eswterikèc, sumplhrwmènec stibdec, den suneisfèroun sth magnhtik rop tou iìntoc diìti qarakthrÐzontai apì mhdenik troqiak stroform kai mh-denikì spin.

H eÔresh twn kbantik¸n arijm¸n L, S, J enìc iìntoc, basÐzetai sthn katanom twn hlektronÐwn sta dunat troqiak thc exwterik c stibdac, me bsh toucKanìnec tou Hund.

Kanìnec Hund

1. Ta hlektrìnia topojetoÔntai sta troqiak me tètoio trìpo ¸ste naexasfalÐzetai to mègisto dunatì olikì spin, S =

∑i si.

2. Ta hlektrìnia topojetoÔntai sta troqiak me tètoio trìpo ¸ste naexasfalÐzetai h mègisth dunat olik troqiak stroform , L =

∑i li.

3. An ligìtera apì ta mis troqiak thc exwterik c stibdac katalam-bnontai apì hlektrìnia, J =| L − S |. An perissìtera apì ta mis- troqiak thc exwterik c stibdac katalambnontai apì hlektrìnia,J = L+ S.

3.1. Paramagnhtismìc 29

ParadeÐgmata

Iìn V 3+ Fe 2+

Arijmìc 3d

katastsewn

2 6

Kateileimmènec

Katastseic

lz = 2

lz = 1

lz = 0

lz = −1

lz = −2

6

6

lz = 2

lz = 1

lz = 0

lz = −1

lz = −2

?6

6

6

6

6

1os Kanìnac

2os Kanìnac

3os Kanìnac

L = 2 + 1 = 3 L = 5− 3 = 2

S = 12 +12 = 1 S =

52 −

12 = 2

J =| L− S |= 2 J = L+ S = 4

΄Ασκηση

΄Εστω ένα σύστημα N ανεξάρτητων σωματιδίων με σπιν1/2 παρουσία ενός μαγνητικού πεδίου B, σε θερμοκρασίαT . Να βρεθεί η εντροπία S, η ελεύθερη ενέργεια F, ηεσωτερική ενέργεια U και η μέση τιμή της μαγνήτισης.

Απάντηση

Η Χαμιλτονιανή του συστήματος γράϕεται:

Ĥ = −∑i

µ⃗i · B⃗ (3.1.4)

Επειδή η Χαμιλτονιανή έχει αυτή τη μορϕή, τα σωματίδια είναι ανεξάρτητακαι ο χώρος Hilbert του ολικού συστήματος προκύπτει από το εξωτερικόγινόμενο των χώρων Hilbertτων σωματιδίων. Δηλαδή:


E = E1 ⊗ E2 ⊗ · · · ⊗ EN (3.1.5)

Η Χαμιλτονιανή του κάθε σωματιδίου μπορεί να γραϕεί στη μορϕή:

Ĥo = −µBσ⃗ · B⃗ (3.1.6)

όπου σ οι πίνακες του Pauli. Για τους πίνακες σ ισχύουν:

σ 2x = σ2

y = σ2

z = 1 (3.1.7)

σxσy = −σyσx = iσz (3.1.8)

σzσx = −σxσz = iσy (3.1.9)

σyσz = −σzσy = iσx (3.1.10)

Οι πίνακες του Pauli, είναι 2× 2 πίνακες, οι οποίοι στη βάση όπου είναιδιαγώνιος ο σz, γράϕονται:

σx =

(0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

)(3.1.11)

Η Χαμιλτονιανή (3.1.6) είναι διαγώνια για την παραπάνω βάση των πινάκ-ων του Pauli, αν θεωρήσουμε ότι B⃗ = Bẑ. Συγκεκριμένα, θα έχει τημορϕή:

Ĥo =

(−µBB 0

0 µBB

)(3.1.12)

Κάθε υποσύστημα, διαθέτει δύο ενεργειακά επίπεδα:

E1 = +µBB (3.1.13)

E2 = −µBB (3.1.14)

Zo = Tr{e−βĤo

}= eβµBB + e−βµBB = 2 cosh

(µBB

kBT

)(3.1.15)

Ο Τελεστής Πυκνότητας ρ̂, δίνεται από τη σχέση:

ρ̂ =e−βĤ

Tr{e−βĤ

} = e−βĤe−βµBB + eβµBB

=1

e−βµBB + eβµBB

(eβµBB 0

0 e−βµBB

)(3.1.16)


Από την παραπάνω σχέση, μπορούμε να υπολογίσουμε τη μέση τιμή τουσπιν, κατά τη διεύθυνση z. Δηλαδή:

⟨σz⟩ = Tr {ρ̂σz} =eβµBB − e−βµBB

eβµBB + e−βµBB= tanh

(µBB

kBT

)(3.1.17)

Η συνάρτηση επιμερισμού του ολικού συστήματος, δίνεται από τη σχέση:

Z =N∏i=1

Zi = (Zo)N =[2 cosh

(µBB

kBT

)]N(3.1.18)

F = −kBT lnZ = −NkBT ln[2 cosh

(µBB

kBT

)](3.1.19)

F = U − TS ⇒

S = −∂F∂T

(3.1.20)

= NkB

[2 cosh

(µBB

kBT

)− µBBkBT

tanh

(µBB

kBT

)](3.1.21)

U = F + ST

= −NµBB tanh(µBB

kBT

)(3.1.22)

Γνωρίζουμε τη θερμοκρασία του συστήματος κατά μέσο όρο. Η ενέργειατου συστήματος, είναι μία διατηρήσιμη ποσότητα. Γενικά, η μέση τιμήμίας ποσότητας A, η οποία προκύπτει από μία διατηρήσιμη ποσότητα Xi,δίνεται από τη σχέση:

⟨A⟩ =⟨∂Xi∂ξi

⟩= − 1

λi

∂

∂ξiln(Z(ξ)) (3.1.23)

Στην περίπτωσή μας, η ενέργεια είναι μία διατηρήσιμη ποσότητα. Αντικα-θιστώντας στην παραπάνω σχέση, τις ακόλουθες μεταβλητές:

A → Μαγνήτιση (3.1.24)Xi → Ενέργεια (3.1.25)ξi → Μαγνητικό Πεδίο (3.1.26)

M =N∑i

µi (3.1.27)


⟨M⟩ = − 1β

∂

∂Bln(Z(ξ))

= −∂F∂B

= −NµB tanh(µBB

kT

)(3.1.28)

΄ΑσκησηΓενικέυουμε την προηγούμενη άσκηση στην περίπτωσηκατά την οποία η μαγνητική ροπή του κάθε ιόντος παίρνειοποιαδήποτε τιμή J.Απάντηση

Sq¨ma 3.2: Gia dedomènh tim J èqoume 2J+1 diaforetikèc timèc thc probol ckat ton xona Oz (poÔ orÐzetai apì to magnhtikì pedÐo) kai ra 2J + 1diaforetikèc timèc thc enèrgeiac allhlepÐdrashc me to magnhtikì pedÐo.

Θεωρούμε ότι η μαγνήτιση του κάθε ιόντος δίνεται από τη σχέση:

m = gµBSz (3.1.29)

όπου µB η μαγνητόνη του Bohr και

µB =e~2mc

(3.1.30)


Ενώ με g συμβολίζουμε τον παράγοντα Lande.

Η συνάρτηση επιμερισμού του συστήματος, δίνεται από τη σχέση:

Z = [Zi]N (3.1.31)

Η Χαμιλτονιανή του συστήματος, δίνεται από τη σχέση:

Ĥ = −∑i

µ⃗i · B⃗ (3.1.32)

Zi =J∑

Sz=−J

eβMB (3.1.33)

M =∑i

µi(Sz) (3.1.34)

Z =

[J∑

Sz=−J

eβgµBBSz

]N(3.1.35)

Χρησιμοποιούμε την ταυτότητα:

n∑k=−n

xk = x−n2n∑ρ=0

xρ = x−nx2n+1 − 1x− 1

=xn+1/2 − x−(n+1/2)

x1/2 − x−1/2

(3.1.36)

Τελικά η συνάρτηση επιμερισμού, παίρνει τη μορϕή:

Z =

sinh(

βgµB(2J+1)B2

)sinh

(βgµBB

2

)N (3.1.37)

Από την παραπάνω σχέση, μπορούμε να υπολογίσουμε οποιαδήποτε θερ-μοδυναμική ποσότητα μας ενδιαϕέρει. Η μαγνήτιση, δίνεται από τηνακόλουθη σχέση:

⟨M⟩ = −kT ∂∂B

ln(Z) = NgµBJBJ(βgµBJB) (3.1.38)

όπου B η συνάρτηση Brillouin, η οποία δίνεται από τη σχέση:

BJ(x) =2J + 1

2Jcoth

[2J + 1

2Jx

]− 1

2Jcoth

( x2J

)(3.1.39)

Στην περίπτωση J = 1/2, ισχύει:


B1/2 = tanh(x) (3.1.40)

Στην περίπτωση του κλασσικού ορίου J → ∞, ισχύει:

BJ→∞ = L(x) = coth(x)1

x(3.1.41)

Η παραπάνω συνάρτηση, ονομάζεται συνάρτηση Langevin. Αν θεωρή-σουμε x

3.2. ArmonikoÐ Talantwtèc 35

Για τον κάθε κβαντικό αρμονικό ταλαντωτή έχουμε

ZQuantum =∑n

e−βEn (3.2.2)

=∞∑

n=0

e−β(n+1/2)~ω (3.2.3)

= e−β~ω/2∞∑

n=0

{e−β~ω

}n(3.2.4)

=e−β~ω/2

1− e−β~ω(3.2.5)

=1

2 sinh( ~ωkT

) (3.2.6)Για έναν κλασσικό αρμονικό ταλαντωτή θα ίσχυε:

ZClassic = lim~ω/kT→0

ZQuantum (3.2.7)

Για τους Ν ανεξάρτητους αρμονικούς ταλαντωτές έχουμε

Z = [ZQuantum]N =[2 sinh

(~ω2kT

)]−N(3.2.8)

Η εσωτερική ενέργεια προκύπτει εύκολα

U = − ∂∂β

lnZ = N ∂∂β

ln [sinh (β~ω/2)] = N~ω2

coth

(~ω2kT

)⇒

(3.2.9)

U = N

(~ω2

+~ω

e~ω/kT − 1

)(3.2.10)

Η θερμοχωρητικότητα, δίνεται από την ακόλουθη σχέση:

C =

(∂U

∂T

)= Nk

(~ωkT

)2e~ω/kT[

e~ω/kT − 1]2 (3.2.11)

Από την ελεύθερη ενέργεια

F = −kT lnZ = NkT ln[2 sinh

(~ω2kT

)]= N

[~ω/2 + kT ln

(1− e−~ω/kT

)](3.2.12)


βρίσκουμε την εντροπία

S = −∂F∂T

= Nk

[~ω2kT

coth

(~ω2kT

)− ln

(2 sinh

(~ω2kT

))](3.2.13)

Keflaio 4

Kbantik Sust mataTautìshmwn SwmatidÐwn

Ta tautìshma swmatÐdia, sta plaÐsia thc kbantomhqanik cperigraf c, den eÐnai diakrÐsima.

'Estw dÔo swmatÐdia me spin 1/2, twn opoÐwn oi probolèc tou spin èqounantÐjeth for. Dhlad , to èna eÐnai sthn katstash |↑⟩ kai to llo sthnkatstash |↓⟩. Se autìn ton orismì uprqei aujairesÐa sthn epilog thcpnw kai thc ktw kateÔjunshc. Ta dÔo dianÔsmata katstashc, sqhmatÐ-zoun èna didistato q¸ro, pou prokÔptei apì to exwterikì ginìmeno twn dÔokatastsewn: |↑⟩ ⊗ |↓⟩ kai |↓⟩ ⊗ |↑⟩. Kje dinusma autoÔ tou q¸rou, eÐnaimÐa upèrjesh twn dÔo katastsewn |↑⟩ kai |↓⟩. Dhlad , kje tuqaÐo dinusmagrfetai sth morf :

| ψ⟩ = α |↓↑⟩+ β |↑↓⟩ (4.0.1)

Oi katastseic thc parapnw morf , mporoÔn na anaparast soun thn Ðdiafusik katstash → Ekfullismìc Antallag c.

Pardeigma

'Estw ìti èqoume èna sÔsthma tri¸n tautìshmwn anexrthtwn swmatidÐwn.Kje swmatÐdio, dhmiourgeÐ èna q¸ro Hilbert, Ei. O q¸roc Hilbert tou olikoÔsust matoc, prokÔptei apì to exwterikì ginìmeno twn tri¸n èpimèrouc q¸rwn.

E = E1 ⊗ E2 ⊗ E3 (4.0.2)

37

38 Keflaio 4. Kbantik Sust mata Tautìshmwn SwmatidÐwn

JewroÔme touc ermitianoÔc telestèc B̂i oi opoÐoi droun stouc q¸rouc Hilbert,Ei, twn opoÐwn oi idiokatastseic | bn⟩i apoteloÔn bseic twn q¸rwn Ei. Hbsh tou q¸rou E , prokÔptei apì to exwterikì ginìmeno twn epimèrouc bsewn{| bn⟩i}.

{| bn⟩1 ⊗ | bn⟩2 ⊗ | bn⟩3} (4.0.3)Ta dianÔsmata tou q¸rou E ,

| bn⟩1 ⊗ | bn⟩2 ⊗ | bn⟩3 ≡| b 1n ; b 2n ; b 1n ⟩ (4.0.4)

eÐnai koin idiodianÔsmata twn epektsewn sto q¸ro E twn telest¸n B̂i. Homoiìthta twn swmatidÐwn DEN EPITREPEI na metr soume xeqwristtic mèsec timèc twn telest¸n B̂i. MporoÔme na metr soume thn posìthta Bgia kajèna apì ta trÐa swmatÐdia, qwrÐc na xèroume to swmatÐdio. Upojè-toume ìti mÐa mètrhsh mac èdwse touc kbantikoÔc arijmoÔc: bα, bβ kai bγ . Hkatstash tou sust matoc, mporeÐ na eÐnai mÐa apì tic akìloujec:

| bα; bβ ; bγ⟩ | bγ ; bα; bβ⟩ | bβ ; bγ ; bα⟩| bα; bγ ; bβ⟩ | bγ ; bβ ; bα⟩ | bβ ; bα; bγ⟩

Uprqoun 3!=6 katastseic pou prokÔptoun apì tic dunatèc metajèseic. Opoios-d pote grammikìc sunduasmìc touc, orÐzei ènan upìqwro to q¸ro Hilbert, E .Qreiazìmaste èna sumplhrwmatikì axÐwma, to opoÐo ja mac epitrèyei na arjeÐh aoristÐa pou prokÔptei apì ton ekfulismì antallag c.

AxÐwma

'Otan èna sÔsthma apoteleÐtai apì poll anexrthta tautìsh-ma swmatÐdia, mìno merik dianÔsmata (katastseic) tou q¸roupou prokÔptei apì to ginìmeno Kronecker twn q¸rwn katastsewntou kje swmatidÐou mporoÔn na perigryoun tic fusikèckatastseic tou sust matoc. Aut ta dianÔsmata kats-tashc, exart¸ntai apì th fÔsh twn swmatidÐwn. EÐte eÐnaientel¸c summetrik, eÐte entel¸c antisummetrik sthn an-tallag dÔo swmatidÐwn. Onomzoume Mpozìnia ta swmatÐdi-a gia ta opoÐa ta fusik dianÔsmata katstashc eÐnai sum-metrik sthn antallag dÔo swmatidÐwn. Onomzoume Fer-miìnia ta swmatÐdia gia ta opoÐa ta fusik dianÔsmata kats-tashc eÐnai antisummetrik. Oi fusikèc katastseic, apoteloÔnmìno èna polÔ mikrì mèroc tou q¸rou Hilbert tou sust matoctou sust matoc twn poll¸n swmatidÐwn. Sugkentrwtik è-qoume:

39

SwmatÐdia Spin Fusikèc KatastseicMpozìnia Akèraio SummetrikècFermiìnia Hmiakèraio Antisummetrikèc

Pardeigma

JewroÔme èna sÔsthma tautìshmwn swmatidÐwn. To swmatÐdio (1) brÐsketaisthn katstash | ϕ⟩ kai to swmatÐdio (2) sthn katstash | χ⟩. H katstashtou olikoÔ sust matoc, dÐnetai apì thn katstash | ψ⟩ =| ϕ1;χ2⟩.

• An ta swmatÐdia eÐnai mpozìnia, tìte h katstash tou sust matoc prèpeina eÐnai summetrik kat thn antimetjesh twn dÔo swmatidÐwn. 'Ara,h katstash | ψ⟩ prèpei na summetrikopoihjeÐ. JewroÔme ènan telest summetrikopoÐhshc Ŝ, tou opoÐou h drsh sthn katstash | ψ⟩, èqei toakìloujo apotèlesma:

| ψ⟩symmetric = Ŝ | ψ⟩ =1

2(| ϕ;χ⟩+ | χ;ϕ⟩) (4.0.5)

An oi katastseic ϕ kai χ eÐnai orjog¸niec metaxÔ touc tìte apì thnkanonikopoÐhsh thc ψ, prokÔptei:

| ψ⟩symmetric =1√2(| ϕ;χ⟩+ | χ;ϕ⟩) (4.0.6)

• An ta swmatÐdia eÐnai fermiìnia, tìte h katstash tou sust matocprèpei na eÐnai antisummetrik kat thn antimetjesh twn dÔo swmatidÐ-wn. 'Ara, h katstash | ψ⟩ prèpei na antisummetrikopoihjeÐ. JewroÔmeènan telest antisummetrikopoÐhshc Â, tou opoÐou h drsh sthn kats-tash | ψ⟩, èqei to akìloujo apotèlesma:

| ψ⟩antisymmetric = Â | ψ⟩ =1

2(| ϕ;χ⟩− | χ;ϕ⟩) (4.0.7)

An oi katastseic ϕ kai χ eÐnai orjog¸niec metaxÔ touc tìte apì thnkanonikopoÐhsh thc ψ, prokÔptei:

| ψ⟩antisymmetric =1√2(| ϕ;χ⟩− | χ;ϕ⟩) (4.0.8)

Sthn perÐptwsh pou oi kbantikèc katastseic, stic opoÐec brÐskontai ta dÔoswmatÐdia, eÐnai Ðdiec, dhlad | ϕ⟩ ≡| χ⟩, isqÔoun:


• An ta swmatÐdia eÐnai mpozìnia:

| ψ⟩ = 1√2(| ϕ;ϕ⟩+ | ϕ;ϕ⟩) (4.0.9)

Den parousizetai kanèna prìblhma.

• An ta swmatÐdia eÐnai fermiìnia:

| ψ⟩ = 1√2(| ϕ;ϕ⟩− | ϕ;ϕ⟩) = 0 (4.0.10)

Se aut n thn perÐptwsh, katal goume sthn apagoreutik arq tou Pauli.Dhlad , dÔo tautìshma fermiìnia den gÐnetaina katalboun thn Ðdia kbantik katstash.

Prèpei na anafèroume, ìti ta domik sustatik thc Ôlhc eÐnai ta fermiìnia.Ta mpozìnia paÐzoun to rìlo twn forèwn twn allhlepidrsewn metaxÔ twnfermionÐwn.

Pardeigma

'Estw trÐa tautìshma swmatÐdia, ta opoÐa brÐskontai stic katastseic ϕ, χ, η.H katstash tou sust matoc, ekfrzetai mèsw tou dianÔsmatoc katstashc| ψ⟩ =| ϕ1;χ2; η3⟩.

• An ta swmatÐdia eÐnai mpozìnia, tìte:

| ψ⟩symmetric = Ŝ | ψ⟩ =1

3!

∑metajèseic

| ψ⟩ =

=1

6( | ϕ1;χ2; η3⟩+ | ϕ1; η2;χ3⟩+ | χ1;ϕ2; η3⟩+

+ | η1;χ2;ϕ3⟩+ | χ1; η2;ϕ3⟩+ | η1;ϕ2;χ3⟩ ) (4.0.11)

An oi katastseic ϕ, χ kai η eÐnai metaxÔ touc orjog¸niec, mporoÔme nakanonikopoi soume thn katstash ψ, mèsw thc sqèshc ⟨ψ | ψ⟩, katal -gontac sth sqèsh:

| ψ⟩symmetric =1√6( | ϕ1;χ2; η3⟩+ | ϕ1; η2;χ3⟩+ | χ1;ϕ2; η3⟩+

+ | η1;χ2;ϕ3⟩+ | χ1; η2;ϕ3⟩+ | η1;ϕ2;χ3⟩ )(4.0.12)

4.1. Formalismìc Arijm¸n Katlhyhc 41

Sthn perÐptwsh pou | χ⟩ ≡| η⟩, prokÔptei:

| ψ⟩ = 1√3(| ϕ;χ;χ⟩+ | χ;ϕ;χ⟩+ | χ;χ;ϕ⟩) (4.0.13)

• An ta swmatÐdia eÐnai fermiìnia, tìte:


3!

∑metajèseic α

Eα | ψ⟩ =

ìpou Eα = 1 an èqoume rtio arijmì metajèsewn kai Eα = −1 an è-qoume perittì arijmì metajèsewn. Gia pardeigma:| ϕ1;χ2; η3⟩ = − |ϕ1; η2;χ3⟩. H antisummetrikopoÐhsh tou dianÔsmatoc katstashc, m-poreÐ na gÐnei mèsw mÐac orÐzousac Slater. Dhlad :


3!

∣∣∣∣∣∣| ϕ1⟩ | χ1⟩ | η1⟩| ϕ2⟩ | χ2⟩ | η2⟩| ϕ3⟩ | χ3⟩ | η3⟩

∣∣∣∣∣∣ ==

1√3!( | ϕ1;χ2; η3⟩− | ϕ1; η2;χ3⟩− | χ1;ϕ2; η3⟩+

− | η1;χ2;ϕ3⟩+ | χ1; η2;ϕ3⟩+ | η1;ϕ2;χ3⟩ ) (4.0.14)

Sthn perÐptwsh | ϕ⟩ ≡| χ⟩ h orÐzousa mhdenÐzetai. ProkÔptei dhlad hapagoreutik arq tou Pauli.

Ta parapnw efarmìzontai me parìmoio trìpo kai se sust mata meN tautìsh-ma swmatÐdia.

4.1 Formalismìc Arijm¸n Katlhyhc

MÐa metjesh twn swmatidÐwn DEN EPHREAZEI mÐa katstash entel¸csummetrik entel¸c antisummetrik . Sunep¸c, antÐ na exetzoume se poikats-tash brÐsketai kajèna apì ta N swmatÐdia, eÐnai protimìtero na metrme pìsaswmatÐdia brÐskontai se kje dunat katstash. Dhlad jèloume na peri-gryoume to sÔsthma me dianÔsmata katstashc th morf c:

| nψ1 ;nψ2 ; . . . ;nψM ⟩ (4.1.1)

ìpou oi metablhtèc nψi dhl¸noun pìsa swmatÐdia katalambnoun thn en-ergeiak katstash | ψn⟩. Gia pardeigma, ac jewr soume èna sÔsthma tri¸n


tautìshmwn swmatidÐwn, to opoÐo qarakthrÐzetai apì thn katstash | ϕ;ϕ;χ⟩.Se aut n thn perÐptwsh, uprqoun dÔo diaforetikèc dunatèc katastseic ϕkai χ. Oi antÐstoiqoi arijmoÐ katlhyhc eÐnai Ðsh me: nϕ = 2 kai nχ = 1.Sta plaÐsia tou formalismoÔ twn katastsewn tou arijmoÔ katlhyhc, todinusma katstashc tou sust matoc mporeÐ na grafeÐ wc | 2; 1⟩. Ta nèaaut dianÔsmata eÐnai pio praktik kai mporoÔn na apotelèsoun th bsh twnfusik¸n katastsewn (Bsh tou Fock).

• Eswterikì Ginìmeno

JewroÔme dÔo dianÔsmata: | n1, n2, . . . , nq, . . .⟩ kai | n1 ′, n2 ′, . . . , nq ′, . . .⟩.To eswterikì ginìmeno wn dÔo parapnw dianusmtwn, eÐnai diforo toumhdenìc, mìno ìtan isqÔei nq = nq ′ gia kje k. Dhlad , oi katastseicme diaforetikoÔc arijmoÔc katlhyhc eÐnai orjog¸niec.

• An ta swmatÐdia eÐnai mpozìnia, tìte oi arijmoÔ katlhyhc mporoÔn naproun tic timèc 0, 1, 2, . . . ,∞, arkeÐ na ikanopoieÐtai h sqèsh

∑k nk =

N , ìpou N o olikìc arijmìc swmatidi¸n.

• An ta swmatÐdia eÐnai fermiìnia, tìte oi arijmoÔ katlhyhc mporoÔn naproun tic timèc 0, 1, ikanopoi¸ntac tautìqrona th sqèsh

∑k nk = N ,

ìpou N o olikìc arijmìc swmatidi¸n.

• To kenì | 0⟩ orÐzetai wc h katstash gia thn opoÐa ìloi oi arijmoÐkatlhyhc eÐnai Ðsoi me to mhdèn. Dhlad to kenì eÐnai h katstashìpou den uprqei kanèna swmatÐdio sto sÔsthma.

Se autì to shmeÐo mporoÔme na orÐsoume to q¸ro Fock, wc :

HFock = H1 ⊕H2 ⊕ · · · (4.1.2)

ìpou me Hn sumbolÐzoume to q¸ro Hilbert pou dhmiourgeÐtai apì tic dunatèckatastseic pou èqoun n swmatÐdia. H kataskeu tou q¸rou Fock onomzetaiDeÔterh Kbntwsh. Sth DeÔterh Kbntwsh, asqoloÔmaste me ton arijmìtwn swmatidÐwn pou katalambnoun tic energeiakèc katastseic.

MerikoÐ Telestèc sthn Anaparstash Fock.O telest c tou arijmoÔ twn swmatidÐwn, eÐnai diag¸nioc.

N̂ =∑q

n̂q (4.1.3)

4.1. Formalismìc Arijm¸n Katlhyhc 43

Sthn perÐptwsh mh allhlepidr¸ntwn swmatidÐwn, h Qamiltonian grfetai:

Ĥ =∑q

Eqn̂q (4.1.4)

Lìgw tou ìti h Qamiltonian eÐnai diag¸nia, oi idiokatastseic thc, antis-toiqoÔn stic fusikèc katastseic tou sust matoc. Oi parapnw sqèseic eÐnaiqr simec ìtan ergazìmaste sto Megalo-Kanonikì SÔnolo, tautìshmwn mhallhlepidr¸ntwn swmatidÐwn. Se aut n th bsh mporoÔme akìmh na melet -soume swmatÐdia me metablhtì arijmì swmatidÐwn. Sthn perÐptwshN tautìshmwnmh allhlepidr¸ntwn swmatidÐwn, o telest c puknìthtac èqei th morf :

ρ̂ =∏q

ρ̂q (4.1.5)

ìpou

ρ̂q =1

Zqe−βEqn̂q−λN n̂q ≡ 1

Zq(Xq)

n̂q (4.1.6)

Z =∏q

Zq (4.1.7)

Sth bsh tou q¸rou Fock, mporoÔme na eisgoume touc telestèc dhmiourgÐackai katastrof c.

Gia ta mpozìnia:

O Telest c Katastrof c âq, katastrèfei èna swmatÐdio to opoÐo brÐsketaisthn katstash | q⟩.

âq | n1, n2, . . . , nq, . . .⟩ =√nq | n1, n2, . . . , (nq − 1), . . .⟩ (4.1.8)

O Telest c DhmiourgÐac â†q dhmiourgeÐ èna swmatÐdio sthn katstash | q⟩.

â †q | n1, n2, . . . , nq, . . .⟩ =√nq + 1 | n1, n2, . . . , (nq + 1), . . .⟩ (4.1.9)

[âq , âq′ ] = 0 (4.1.10)

[â †q , â†q′ ] = 0 (4.1.11)

[âq , â†q′ ] = δq,q′ (4.1.12)


Gia fermiìnia:

O Telest c Katastrof c ĉq, katastrèfei èna swmatÐdio to opoÐo brÐsketaisthn katstash | q⟩.

ĉq | n1, n2, . . . , nq, . . .⟩ =√nq | n1, n2, . . . , (nq − 1), . . .⟩ (4.1.13)

O Telest c DhmiourgÐac ĉ †q dhmiourgeÐ èna swmatÐdio sthn katstash | q⟩.

ĉ †q | n1, n2, . . . , nq, . . .⟩ =√nq + 1 | n1, n2, . . . , (nq + 1), . . .⟩ (4.1.14)

{ĉq , ĉq′} = 0 (4.1.15){ĉ †q , ĉ

†q′ } = 0 (4.1.16)

{ĉq , ĉ †q′ } = δq,q′ (4.1.17)

ParathroÔme ìti {ĉ †q , ĉ †q } = 0 ⇒(ĉ †q)2

= 0. Dhlad , mÐa energeiak

katstash, den mporeÐ na katalhfjeÐ me pnw apì èna hlektrìnia. ProkÔpteiloipìn h apagoreutik arq tou Pauli.

Kje dinusma tou q¸rou Fock, mporeÐ na prokÔyei apì th drsh twn telest¸ndhmiourgÐac sthn katstash tou kenoÔ | 0⟩, dhlad :

| n1, n2, . . . , nq, . . .⟩ =∞∏i=1

(c †i

)ni| 0⟩ (4.1.18)

Sth bsh Fock, oi telestèc arijmoÔ katlhyhc, grfontai:

n̂q = ĉ†q ĉq (4.1.19)

H Qamiltonian tou sust matoc, grfetai:

Ĥ =∑q

Eq ĉ†q ĉq (4.1.20)

EÐnai parìmoio me thn perÐptwsh tou armonikoÔ talantwt , ìpou Ĥ = (â†â+1/2)~ω.

4.2. Megalo-Kanonikèc Sunart seic Katanom c twn Kbantik¸n AerÐwn 45

4.2 Megalo-Kanonikèc Sunart seic Katanom -c twn Kbantik¸n AerÐwn

Ta Ĥ kai N̂ eÐnai diag¸nia sth bsh Fock. H sunrthsh epimerismoÔ, dÐnetaiapì th sqèsh:

ZG =∑{nq}

e−β∑

q(Eqnq−µnq) =∑{nq}

∏q

(Xq)nq =

∏q

∑{nq}

Xnqq

(4.2.1)ìpou Xq = e−βEq+βµ kai µ to qhmikì dunamikì, to opoÐo prokÔptei apì tonpollaplassiast Lagrange, λN = −µ/kT .

Pardeigma

'Estw èna fermionikì sÔsthma me 2 katastseic (p.q. spin ↑, ↓).

Ja qrhsimopoi soume th sqèsh (4.2.1), qrhsimopoi¸ntac q = 1, 2 kai nq =0, 1.

ZG =∑{nq}

∏q

Xnqq (4.2.2)

=

1∑n1=0

1∑n2=0

X n11 Xn2

2 (4.2.3)

= X 01 X0

2 +X0

1 X1

2 +X1

1 X0

2 +X1

1 X1

2 (4.2.4)

= = (X 01 +X1

1 )(X0

2 +X1

2 ) (4.2.5)

=∏q=1,2

(X 0q +X1

q ) (4.2.6)

=∏q=1,2

1∑nq=0

(X

nqq

)(4.2.7)

Blèpoume ìti epibebai¸netai pl rwc h sqèsh (4.2.1).H megalo-kanonik sunrthsh epimerismoÔ, enìc mpozonikoÔ aerÐou, dÐnetaiapì th sqèsh:

ZG =∏q

∑nq=0,1,2,...∞

Xnq

q

=∏q

1

1− e−β(Eq−µ)(4.2.8)


H megalo-kanonik sunrthsh epimerismoÔ, enìc fermionikoÔ aerÐou, dÐnetaiapì th sqèsh:

ZG =∏q

∑nq=0,1

Xnq

q

=∏q

(X 0q +X

1q

)=∏q

(1 +Xq) =∏q

(1 + e−β(Eq−µ)

)(4.2.9)

Suntelest c Katlhyhc fq

Ex orismoÔ fq = ⟨n̂k⟩.

fq = Tr {ρ̂qn̂q} = −1

β

∂

∂EqlnZG (4.2.10)

fq = −1

β

∂

∂Eq

∑q

ln(1± e−β(Eq−µ)

)±1(4.2.11)

Katanom Bose-Einstein:

f BEq =1

eβ(Eq−µ) − 1(4.2.12)

Katanom Fermi-Dirac:

f FDq =1

eβ(Eq−µ) + 1(4.2.13)

To meglo dunamikì Ω:

Ω = − 1βlnZG = ±

1

β

∑q

ln (1± fq) (4.2.14)

Gia fermiìnia - kai gia mpozìnia +.

Mèsoc arijmìc swmatidÐwn:

N = ⟨N⟩ =∑q

fq (4.2.15)

Eswterik Enèrgeia:

U =∑q

Eqfq = ⟨Ĥ⟩ (4.2.16)


EntropÐa:

S = −k∑q

[fq ln fq ± (1− (∓)fq) ln(1∓ fq)] (4.2.17)

To pnw eÐnai gia fermiìnia kai to ktw gia mpozìnia.

'Orio meglwn ìgkwn:

∑q

ϕ(Eq) = limV→∞

∫dED(E)ϕ(E)

D(E) eÐnai h puknìthta katastsewn kai ϕ(E) o arijmìc katastsewnsto disthma E ≤ Eq ≤ E + dE.

Arijmìc SwmatidÐwn:

N =

∫f(E)D(E)dE (4.2.18)

Eswterik Enèrgeia:

U =

∫f(E)ED(E)dE (4.2.19)

Meglo Dunamikì:

Ω = ± 1β

∫ln(1± f(E))D(E)dE (4.2.20)

Pardeigma

EleÔjera swmatÐdia se koutÐ ìgkou V . Parabolikèc energeiakèc z¸nec. Seaut n thn perÐptwsh, h puknìthta katastsewn èqei th morf :

D(E) =V

(4π)2(2m)3/2

~3E1/2 (4.2.21)

JewroÔme èna aèrio eleÔjerwn hlektronÐwn se èna alkalikì mètallo. Kjeiìn suneisfèrei èna eleÔjero hlektrìnio. Wc gnwstìn to spin tou hlektronÐoueÐnai 1/2.Mèsw thc sqèshcN = ⟨N̂⟩ =

∑q fq, mporoÔme na upologÐsoume th jermokrasÐ-

a Fermi tou metllou.


TFermi = TF =µ0k

=~2

2mk

(3π2

V

N

)2/3(4.2.22)

'Opou µ0 ≡ Enèrgeia Fermi se mhdenik jermokrasÐa. H taqÔthta Fermi,dÐnetai apì th sqèsh:

vF =

(2kTFm

)1/2∼ 106m/sec gia ta perissìtera mètalla (4.2.23)

fq = Θ(µ− Eq)−π2

6(kT )2δ(Eq − µ) . . . (4.2.24)

N =2V

(2π)3

∫d3qfq =

8π

3

V

(2π)31

~3(2m)3/2µ3/2 +

π

3

V

~3(2m)3/2(kT )2µ−1/2

(4.2.25)

µ = kTF

[1− π

2

12

(T

TF

)2+ . . .

](4.2.26)

E = 35NEF

[1 +

5π2

12

(T

TF

)2+ . . .

](4.2.27)

Ω =8π

3

V

m(2π)3

∫ ∞0

d3q(~q)4fq (4.2.28)

= −16π15

V

~3(2m)3/2µ5/2 − 2π

3

3

V

~3(2m)3/2(kT )2µ1/2 (4.2.29)

S = −∂Ω∂T

≃ 4π3

3

V

~3(2m)3/2k(kTF )

1/2kT (4.2.30)

U = Ω+ µN + TS = −3Ω/2 (4.2.31)

CV = T

(∂S∂T

)N

∼ V mk2

3~2

(3π2

N

V

)1/3T (4.2.32)

CV = γT (4.2.33)


γ = k2D(EF )/3 (4.2.34)

H stajer γ onomzetai stajer Sommerfeld. H qarakthristik idiìth-ta enìc metllou, eÐnai h grammik exrthsh thc jermoqwrhtikìthtac me thjermokrasÐa. Dhlad : T ↑→ CV ↑.

Parrthma aþ

Pollaplassiastèc Lagrange

'Estw ìti èqoume mÐa sunrthsh 2 metablht¸n f(x, y). jèloume na broÔme taakrìtata aut c thc sunrthshc. To diaforikì aut c thc sunrthshc, dÐnetaiapì th sqèsh:

df =∂f

∂xdx+

∂f

∂ydy. (aþ.0.1)

Sthn perÐptwsh pou oi metablhtèc x, y eÐnai anexrthtec metaxÔ touc, h eÔreshtwn akrottwn èggutai ston tautìqrono mhdenismì twn parag¸gwn:

∂f

∂x= 0

∂f

∂y= 0 (aþ.0.2)

Sthn perÐptwsh pou epijum¸ na brw ta akrìtata thc f(x, y), pnw se kpoiakampÔlh Γ, oi metablhtèc x kai y den eÐnai anexrthtec. Sugkekrimèna, up-ìkeintai se ènan periorismì g(x, y) = C (C stajer), pou proèrqetai apì thmorf thc Γ. Sth sugkekrimènh perÐptwsh, h eÔresh twn akrottwn, sthrÐze-tai stic dÔo akìloujec sqèseic:

df =∂f

∂xdx+

∂f

∂ydy = 0 (aþ.0.3)

dg =∂g

∂xdx+

∂g

∂ydy = 0 (aþ.0.4)

H pr¸th proèrqetai apì to mhdenismì tou diaforikoÔ thc f sta akrìtata,en¸ h deÔterh apeujeÐac apì th sqèsh g(x, y) = C. H mèjodoc gia na broÔmeta akrìtata se aut n thn perÐptwsh, eÐnai h mèjodoc twn Pollaplassiast¸nLagrange.

51

52 Parrthma aþ. Pollaplassiastèc Lagrange

aþ.1 MejodologÐa Pollaplassiast¸n Lagrange

IsqÔoun:

df = 0 (aþ.1.1)

dg = 0 (aþ.1.2)

MporoÔme na pollaplassisoume th sqèsh (aþ.1.2) me ènan pragmatikì arijmìλ, ¸ste na isqÔei:

λdg = 0 (aþ.1.3)

Prosjètwntac tic sqèseic (aþ.1.1) kai (aþ.1.3), prokÔptei h sqèsh:

df + λdg = 0 (aþ.1.4)

analutik (∂f

∂x+ λ

∂g

∂x

)dx+

(∂f

∂y+ λ

∂g

∂y

)dy = 0 (aþ.1.5)

O λ eÐnai o Pollaplassiast c Lagrangekai h eisagwg tou aposkopeÐ sthnaposÔzeuxh twn metablht¸n x kai y. Gia na eÐnai oi metablhtèc x kai yanexrthtec, prèpei na isqÔoun tautìqrona oi sqeseic:

∂f

∂x+ λ

∂g

∂x= 0 (aþ.1.6)

kai

∂f

∂y+ λ

∂g

∂y= 0 (aþ.1.7)

en¸ isqÔei pntote h exÐswsh periorismoÔ

g(x, y) = C (aþ.1.8)

Apì tic treic parapnw sqèseic, mporoÔme na upologÐsoume tic akrìtatectimèc x0 kai y0 kaj¸c kai ton Pollaplassiast Lagrange λ.

aþ.1. MejodologÐa Pollaplassiast¸n Lagrange 53

Pardeigma 1

H jermokrasÐa enìc shmeÐou (x, y), dÐnetai apì th sqèsh T (x, y) = xy. NabreÐte th jermokrasÐa twn dÔo jermìterwn shmeÐwn pnw sto monadiaÐo kÔklo.H exÐswsh tou monadiaÐou kÔklou, dÐnetai apì th sqèsh:

x2 + y2 = 1 (aþ.1.9)

apì thn parapnw sqèsh, prokÔptei o periorismìc g gia tic metablhtèc x kaiy. Dhlad :

g(x, y) = x2 + y2 = 1 (aþ.1.10)

IsqÔoun epiplèon oi sqèseic:

dT =∂T

∂xdx+

∂T

∂ydy = ydx+ xdy (aþ.1.11)

kai

dg =∂g

∂xdx+

∂g

∂ydy = 2xdx+ 2ydy (aþ.1.12)

Eisgwntac ton Pollaplassiast Lagrange, λ, prokÔptei to sÔsthma:

∂T

∂x+ λ

∂g

∂x= 0 (aþ.1.13)

∂T

∂y+ λ

∂g

∂y= 0 (aþ.1.14)

x2 + y2 = 1 (aþ.1.15)

Analutik

y + λx = 0 (aþ.1.16)

x+ λy = 0 (aþ.1.17)

x2 + y2 = 1 (aþ.1.18)

y = −2λx (aþ.1.19)

x+ 2λ(−2λx) = x(1− 4λ2) = 0 ⇒ (aþ.1.20)


λ = ±4 (aþ.1.21)

xmax = ymax = ±1√2

(aþ.1.22)

Telik, brÐskoume ìti Tmax =12 , xmax = ymax = ±

1√2, λ = ±12 . H mèjodoc

twn Pollaplassiast¸n Lagrange, efarmìzetai kai se peript¸seic me pollècmetablhtèc kai pollèc exis¸seic periorismoÔ, arkeÐ o arijmìc twn exis¸sewnperiorimoÔ na eÐnai mikrìteroc apì ton arijmì twn metablht¸n.

Pardeigma 2

JewroÔme thn sunrthsh f(x, y, z), h opoÐa upìkeitai stouc periorismoÔc:

g(x, y, z) = C1 (aþ.1.23)

kai

h(x, y, z) = C2. (aþ.1.24)

se aut n thn perÐptwsh,orÐzoume dÔo pollaplassiastèc Lagrange, ¸ste naisqÔei:

df + λ1dg + λ2dh = 0 (aþ.1.25)

ProkÔptei to sÔsthma:

∂f

∂x+ λ1

∂g

∂x+ λ2

∂h

∂x= 0 (aþ.1.26)

∂f

∂y+ λ1

∂g

∂y+ λ2

∂h

∂y= 0 (aþ.1.27)

∂f

∂z+ λ1

∂g

∂z+ λ2

∂h

∂z= 0 (aþ.1.28)

g(x, y, z) = C1 (aþ.1.29)

h(x, y, z) = C2 (aþ.1.30)

aþ.1. MejodologÐa Pollaplassiast¸n Lagrange 55

Pardeigma 3

JewroÔme èna sÔsthma pou apoteleÐtai apì N swmatÐdia. Kje swmatÐdio,mporeÐ na rÐsketai se èna apì ta R epÐpeda enèrgeiac. Kje epÐpedo enèrgeiacèqei enèrgeia Ei, ìpou i = 1, . . . , R. Kje energeiakì epÐpedo mporeÐ nakatalhfjeÐ apì perissìtera tou enìc swmatÐdia. Jètoume ni ton arijmì twnswmatidÐwn pou brÐskontai sto i-ostì energeiakì epÐpedo. H olik enèrgeia,E, tou sust matoc eÐnai stajer . Poi eÐnai h katanom twn swmatidÐwnsenergeiak epÐpeda, h opoÐa megistopoieÐ th sunrthsh:

P =N !

n1!n2! · · ·nR!. (aþ.1.31)

Oi gnwstoi tou probl matoc eÐnai ta ni, upì touc periorismoÔc:{ ∑Ri niEi = E = stajer∑Ri ni = N = stajer

(aþ.1.32)

Lìgw thc morf c thc sunrthshc P , arkeÐ na elaqistopoi soume ton parono-mast thc P . MporeÐ na deiqjeÐ, ìti o paronomast c n1!n2! · · ·nR èqei to Ðdioelqisto me th logarijmik sunrthsh ln(n1!n2! · · ·nR). Sunep¸c, mporoÔmena melet soume thn antÐstoiqh logarijmik sunrthsh, antÐ thc n1!n2! · · ·nR,¸ste na broÔme ta zhtoÔmena elqista. QrhsimopoioÔme th logarijmik sunrthsh,diìti mporoÔme na spsoume to ginìmeno twn paragontik¸n, se ajroÐsmata.'Eqoume loipìn:

f(n1, n2, . . . , nR) = ln (n1!n2! · · ·nR) = ln(n1!) + ln(n2!) + · · ·+ ln(nR!)(aþ.1.33)

Epeid o arijmìc twn swmatidÐwn eÐnai megloc, kat sunèpeia kai ta ni, m-poroÔme na proseggÐsoume touc logarÐjmouc me bsh thn prosèggish Stirling:

ln(n!) = n lnn− n (aþ.1.34)

Qrhsimopoi¸ntac thn parapnw prosèggish, prokÔptei:

f(n1, n2, . . . , nR) =∑i

ni ln(ni)−N (aþ.1.35)

Prèpei na elaqistopoi soume thn f , lambnontac upìyh touc periorismoÔc(aþ.1.32). Akolouj¸ntac th mèjodo twn Pollaplassiast¸n Lagrange, eisgoumetouc pollaplassiastèc λ1 kai λ2. ProkÔptei to sÔsthma:


∂f∂n1

+ λ1∂g∂n1

+ λ2∂h∂n1

= 0

∂f∂n2

+ λ1∂g∂n2

+ λ2∂h∂n2

= 0

∂f∂n3

+ λ1∂g∂n3

+ λ2∂h∂n3

= 0

· · ·

∂f∂nR

+ λ1∂g∂nR

+ λ2∂h∂nR

= 0

∑Ri ni = N∑Ri niEi = E

(aþ.1.36)

Oi pr¸tec R exis¸seic, mporoÔn na grafoÔn sth genik morf :

∂f

∂nk+ λ1

∂g

∂nk+ λ2

∂h

∂nk= 0 ìpou k = 1, . . . , R (aþ.1.37)

Antikajist¸ntac ta f, g, h, prokÔptei h sqèsh:

ln(nk) + 1 + λ1Ek + λ2 = 0 → (aþ.1.38)

nk = Ce−λ1Ek (aþ.1.39)

ìpou oi stajerèc C kai λ1, prosdiorÐzontai apì touc periorismoÔc:

R∑i

ni = CR∑i

e−λ1Ei = N (aþ.1.40)

kai

R∑i

niEi = C

R∑i

e−λ1EiEi = E (aþ.1.41)

Parrthma bþ

Armonikìc Talantwt c

bþ.1 Monodistatoc Kbantikìc Armonikìc Ta-lantwt c (Algebrik Mèjodoc)

Grfoume th Qamiltonian tou armonikoÔ talantwt (??), wc ex c:

Ĥ0 =P̂ 2

2m+

1

2mω2X̂2 (bþ.1.1)

OrÐzoume touc adistatouc telestèc:

x̂ =

√mω

~X̂ (bþ.1.2)

p̂ =1√mω~

P̂ (bþ.1.3)

Ĥ =1

~ωĤ0 (bþ.1.4)

Qrhsimopoi¸ntac touc parapnw telestèc, h Qamiltonian grfetai:

Ĥ =1

2

(x̂2 + p̂2

)̸= 1

2(x̂+ ip̂) (x̂− ip̂) (bþ.1.5)

Gia touc telestèc x̂ kai p̂, isqÔei:

[x̂, p̂] = i~ (bþ.1.6)

OrÐzoume dÔo nèouc telestèc:

57

58 Parrthma bþ. Armonikìc Talantwt c

â =1√2(x̂+ ip̂) (bþ.1.7)

â† =1√2(x̂− ip̂) (bþ.1.8)

Gia touc telestèc â kai â†, isqÔei:[â, â†

]= 1 (bþ.1.9)

H Qamiltonian grfetai:

Ĥ = â†â+1

2(bþ.1.10)

Se autì to shmeÐo, orÐzoume ton telest , N̂ , o opoÐoc onomzetai telest carÐjmhshc kai dÐnetai apì th sqèsh:

N̂ = â†â (bþ.1.11)

O telest c N̂ , eÐnai ermitianìc, dhlad gi�autìn isqÔei:

N̂ † = N̂ (bþ.1.12)

Oi telestèc â kai â† den eÐnai ermitianoÐ kai epomènwc den antistoiqoÔn sefusikèc posìthtec. O telest c N̂ eÐnai ermitianìc. Sunep¸c, paristnei mÐafusik posìthta.

H Qamiltonian mkporeÐ na grafeÐ wc ex c:

Ĥ = N̂ +1

2(bþ.1.13)

Epiplèon, isqÔoun oi sqèseic:

[N̂ , â

]= −â (bþ.1.14)[

N̂ , â†]

= â† (bþ.1.15)[Ĥ, N̂

]= 0 (bþ.1.16)

'Epeid oi telestèc Ĥ kai N̂ metatÐjentai, èqoun koinèc idiosunart seic.

bþ.1. Monodistatoc Kbantikìc Armonikìc Talantwt c (Algebrik Mèjodoc)59

N̂ | ψn⟩ = n | ψn⟩ (bþ.1.17)Ĥ | ψn⟩ = (n+ 12) | ψn⟩ (bþ.1.18)

Oi idiotimèc tou telest N̂ eÐnai tètoiec, ¸ste na isqÔoun:

â | ψn⟩ = c− | ψn−1⟩ (n > 0) (bþ.1.19)â† | ψn⟩ = c+ | ψn+1⟩ (bþ.1.20)

Mac endiafèrei h drsh tou telest â, sthn idiokatstash | ψ0⟩.

â | ψ0⟩ = 0 (bþ.1.21)

â† → Telest c DhmiourgÐac (bþ.1.22)â → Telest c Katastrof c (bþ.1.23)

Me ton telest â†, pernme apì thn idiotim thc enèrgeiac En sthn idiotim En+~ω ⇒ DhmiourgoÔme èna kbnto enèrgeiac. Me ton telest â, katastrè-foume èna kbnto enèrgeiac ~ω, pern¸ntac apì thn idiotim En sthn idiotim En − ~ω.Ta energeiak epÐpeda tou armonikoÔ talantwt den eÐnai ekfulismèna. Kjeidiotim thc Qamiltonian c eÐnai diaforetik . Sunep¸c, oi idiokatastseic thcQamiltonian c, mporoÔn na apotelèsoun bsh tou q¸rou Hilbert. Gia autèctic idiosunart seic, isqÔoun:

â | ψ0⟩ = 0| ψ1⟩ = c1â† | ψ0⟩

⟨ψ1 | ψ1⟩ = 1

⇒| ψ1⟩ = â† | ψ0⟩ (bþ.1.24)| ψ2⟩ = c2â† | ψ1⟩ = c2â†â† | ψ0⟩ = c2

(â†)2

| ψ0⟩ (bþ.1.25)

Qrhsimopoi¸ntac th sqèsh kanonikopoÐhshc

⟨ψ2 | ψ2⟩ = 1 (bþ.1.26)

prokÔptei:

| ψ2⟩ =1√2

[â†]| ψ0⟩ (bþ.1.27)


H genik morf , dÐnetai apì th sqèsh:

| ψn⟩ =1√nâ† | ψn−1⟩ =

1√n!

[â†]n

| ψ0⟩ (bþ.1.28)

H drsh twn telest¸n â kai â† stic idiokatastseic | ψn⟩, eÐnai h akìloujh:

â† | ψn⟩ =√n+ 1 | ψn+1⟩ (bþ.1.29)

â | ψn⟩ =√n | ψn−1⟩ (bþ.1.30)

Gia to fusikì telest jèshc X̂, èqoume:

X̂ | ψn⟩ =√

~2mω

[√n+ 1 | ψn+1⟩+

√n | ψn−1⟩

](bþ.1.31)

Gia to fusikì telest orm c P̂ , èqoume:

P̂ | ψn⟩ = i√m~ω/2

[√n+ 1 | ψn+1⟩ −

√n | ψn−1⟩

](bþ.1.32)

To apotèlesma eÐnai ìti se kje mètrhsh èqoume upèrjesh twn katastsewn| ψn+1⟩ kai | ψn−1⟩.

Sthn anaparstash jèshc:

ψn(x) = ⟨x | ψn⟩ (bþ.1.33)

⟨x | â | ψ0⟩ = 0 ⇒ (bþ.1.34)(mω

~x+

d

dx

)ψ0(x) = 0 (bþ.1.35)

ψ0(x) =(mωπ~

)1/4e−

mω2~ x

2(bþ.1.36)

ψn(x) =

[1

2nn!

(~mω

)n]1/2 (mωπ~

)[mω~x− d

dx

]e−

mω2~ x

2(bþ.1.37)

Polu¸numa Hermite.

Gia pardeigma:

ψ1(x) =

[4

π

(mω~

)3]1/4xe−

mω2~ x

2(bþ.1.38)

ψ2(x) =(mω4π~

)1/4 (2mω

~x2 − 1

)e−

mω2~ x

2(bþ.1.39)

bþ.2. Isotropikìc tridiastatoc kbantikìc armonikìc talantwt c 61

bþ.2 Isotropikìc tridiastatoc kbantikìc armonikìctalantwt c

Ĥ =ˆ⃗p 2

2m+

1

2mω ˆ⃗r 2 (bþ.2.1)

Sthn perÐptwsh anisotropikoÔ talantwt , to dunamikì ja eÐqe th morf :

V̂(ˆ⃗r)=m

2

(ω 2x x̂

2 + ω 2y ŷ2 + ω 2z ẑ

2)

(bþ.2.2)

Sthn perÐptwsh pou exetzoume, h Qamiltonian èqei thn akìloujh morf :

Ĥ =1

2m

(p̂ 2x + p̂

2y + p̂

2z

)+

1

2mω

(x̂ 2 + ŷ 2 + ẑ 2

)(bþ.2.3)

Ĥx =p̂ 2x2m

+1

2mωx̂ 2 (bþ.2.4)

O q¸roc Hilbert tou tridistatou armonikoÔ talantwt , Er⃗, prokÔptei apìto exwterikì ginìmeno twn q¸rwn Hilbert twn monodistatwn armonik¸n ta-lantwt¸n twn axìnwn x, y, z.

Er⃗ = Ex ⊗ Ey ⊗ Ez (bþ.2.5)

O Ĥx, eÐnai mÐa epèktash ston Er⃗ enìc telest pou dr mìno sto q¸ro Ex.Apì to prohgoÔmeno keflaio, isqÔoun:

Ĥx | ψnx⟩ =(nx +

1

2

)~ω | ψnx⟩ ìpou | ψnx⟩ ∈ Ex (bþ.2.6)

Ĥy | ψny⟩ =(ny +

1

2

)~ω | ψny⟩ ìpou | ψny⟩ ∈ Ey (bþ.2.7)

Ĥz | ψnz⟩ =(nz +

1

2

)~ω | ψnz⟩ ìpou | ψnz⟩ ∈ Ez (bþ.2.8)

nx, ny, nz = 0, 1, . . . (bþ.2.9)

Oi idiosunart seic tou Ĥ, eÐnai thc morf c:

| Ψnx,ny ,nz⟩ =| ψnx⟩ | ψny⟩ | ψnz⟩ =| ψnx⟩⊗ | ψny⟩⊗ | ψnz⟩ (bþ.2.10)

Ĥ | Ψnx,ny ,nz⟩ =(nx + ny + nz +

3

2

)~ω | Ψnx,ny ,nz⟩ (bþ.2.11)


Oi idiosunart seic (idiodianÔsmata) thc Ĥ eÐnai exwterik ginìmena twn idio-sunart sewn twn Ĥx, Ĥy kai Ĥz.

Ta energeiak epÐpeda tou isotropikoÔ armonikoÔ tridistatou talantwt ,eÐnai thc morf c:

En =

(n+

3

2

)~ω n = 0, 1, . . . (bþ.2.12)

Telestèc Katastrof c Telestèc DhmiourgÐac

âx =√

mω2~ x̂+

i√2m~ω

p̂x â†x =

√mω2~ x̂−

i√2m~ω

p̂x

ây =√

mω2~ ŷ +

i√2m~ω

p̂y â†y =

√mω2~ ŷ −

i√2m~ω

p̂y

âz =√

mω2~ ẑ +

i√2m~ω

p̂z â†z =

√mω2~ ẑ −

i√2m~ω

p̂z

Profan¸c:

[p̂x, p̂y] = [p̂x, p̂z] = [p̂z, p̂y] = 0 (bþ.2.13)

[ x̂, ŷ ] = [ x̂, ẑ ] = [ ẑ, ŷ ] = 0 (bþ.2.14)

Kat sunèpeia, oi monadikoÐ mh mhdenikoÐ metajètec, metaxÔ twn âi kai â†i eÐnai

oi akìloujoi: [âx, â

†x

]=[ây, â

†y

]=[âz, â

†z

]= 1 (bþ.2.15)

DÔo telestèc me diaforetikoÔc deÐktec x, y, z metatÐjentai, diìti droun sediaforetikoÔc q¸rouc.

IsqÔei:

bþ.2. Isotropikìc tridiastatoc kbantikìc armonikìc talantwt c 63

âx | Ψnx,ny ,nz⟩ = [âx | ψnx⟩]⊗ | ψny⟩⊗ | ψnz⟩ (bþ.2.16)=

√nx | Ψnx,ny ,nz⟩ (bþ.2.17)

| Ψnx,ny ,nz⟩ =1√

nx!ny!nz!

(â †x

)nx (â †y

)ny (â †z

)nz| Ψ0,0,0⟩ (bþ.2.18)

âx | Ψ0,0,0⟩ = ây | Ψ0,0,0⟩ = âz | Ψ0,0,0⟩ = 0 (bþ.2.19)

⟨r⃗ | Ψnx,ny ,nz⟩ = ⟨x | ψnx⟩⟨y | ψny⟩⟨z | ψnz⟩ = ψnx(x)ψny(y)ψnz(z)(bþ.2.20)

⟨r⃗ | Ψ0,0,0⟩ =(mωπ~

)3/4e−

mω2~ (x

2+y2+z2) (bþ.2.21)

Oi idiotimèc En = (n+ 3/2) ~ω thc Qamiltonian c eÐnai ekfulismènec giatÐìla ta kets thc bshc

{| Ψnx,ny ,ny⟩

}pou eÐnai tètoia ¸ste na ikanopoioÔn th

sqèsh n = nx + ny + nz, eÐnai idiodianÔsmata thc Ĥ me thn idiotim En =(n+ 3/2)~ω.O bajmìc ekfulismoÔ gn thc En, eÐnai Ðsoc me ton arijmì twn sunìlwn{nx, ny, nz}, pou eÐnai tètoia ¸ste n = nx + ny + nz. Sugkekrimèna:

gn =(n+ 1)(n+ 2)

2(bþ.2.22)

Mìno to epÐpedo E0 = 3~ω/2 den eÐnai ekfulismèno.

Χρηματοδότηση •  Το παρόν εκπαιδευτικό υλικό έχει αναπτυχθεί

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ocw.aoc.ntua.gr · 2015. 5. 8. · kef laio 1 fusik sust mata sto prohgoÔmeno mèroc jewr same...

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