SEMFE EMP 2011-2012 - mycourses.ntua.grmycourses.ntua.gr/courses/SEMFE1048/document/StatSemfeVarelogian… · SEMFE EMP 2011-2012 -STATISTIKH FUSIKH: Mèroc A - ... Oson afor sto

SEMFE EMP 2011-2012

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STATISTIKH FUSIKH: Mèroc A

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StoiqeÐa JewrÐac thc PlhroforÐac -

- Statistik EntropÐa

Tupolìgio FormalismoÔ Kbantomhqanik c

UpenjÔmish Axi¸mata Kbantomhqanik c

Exwterikì Ginìmeno Q¸rwn Katast�sewn

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Anapl. Kaj. Ge¸rgioc Barelogi�nnhc

12 DekembrÐou 2011

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Kef�laio 1

StoiqeÐa JewrÐacPlhroforÐac - Statistik EntropÐa

KatafeÔgoume se mia statistik perigraf enìc sust matoc miacseir�c gegonìtwn ìtan eÐnai adÔnath h akrib c perigraf lìgw èlleiyhc plhro-forÐac. QwrÐc thn elleÐpousa plhroforÐa ( �gnoia) h statistik perigraf ja tan peritt .

• ElleÐpousa plhroforÐa �gnoia eÐnai h diafor� an�mesasthn plhroforÐa pou èqoume kai thn plhroforÐa pou jaèprepe na èqoume gia na eÐnai dunat mia akrib c kai ìqistatistik perigraf .

Otan perigr�foume statistik� èna sÔsthma eÐnai shmantikì na gnwrÐzoumesusthmatik� pìso upoleÐpetai h plhroforÐa pou diajètoume apo aut pou jaepètrepe mia akrib perigraf .

• AntikeÐmeno thc jewrÐac thc plhroforÐac eÐnai akrib¸ch susthmatik posotikopoÐhsh thc elleÐpousac plhro-forÐac.

Ja parousi�soume ed¸ mìnon basik� stoiqeÐa thc jewrÐac plhroforÐac èqon-tac stìqo ton orismì thc legìmenhc Statistik c EntropÐac

• Statistik entropÐa eÐnai h posìthta pou katametr� thnelleÐpousa plhroforÐa.

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4 Kef�laio 1. StoiqeÐa JewrÐac PlhroforÐac - Statistik EntropÐa

Argìtera ja gÐnei h tautopoÐhsh thc statistik c entropÐac me thn posìth-ta pou eÐqe dh onomasteÐ entropÐa sta plaÐsia thc jermodunamik c twn ar-q¸n tou dek�tou en�tou ai¸na. Sthn tautopoÐhsh aut wfeÐletai h onomasÐ-a entropÐa gia th sun�rthsh pou metr� thn elleÐpousa plhroforÐa. EÐnailoipìn profanèc ìti h statistik entropÐa den eÐnai mia fusik posìthta. Hstatistik entropÐa èqei qarakthristika ta apoÐa eÐnai koin� eÐte èqoume nak�noume me fusik� statistik� sÔnola eÐte me mh fusik� statistik� sÔnolagia par�deigma ston tomèa thc oikonomÐac. Stìqoc tou parìntoc kefalaÐoueÐnai na anadeÐxei sqhmatik� p¸c merikoÐ aploÐ majhmatikoÐ periorismoÐ toucopoÐouc wfeÐlei na upakoÔei opoiad pote majhmatik sun�rthsh perigr�feithn elleÐpousa plhroforÐa, epib�lloun sth sun�rthsh aut th logarijmik dom . SumperaÐnoume loipìn ìti:

• H Ôparxh twn logarÐjmwn kai twn ekjetik¸n sunart -sewn pou sunant�me sthn statistik fusik kai th jer-modunamik eÐnai apotèlesma thc genik c jewrÐac thcplhroforÐac kai den mac ta epib�llei h fÔsh.

Dhlad log�rijmoi kai ekjetik� qarakthrÐzoun kai tic statistikèc analÔseic�llwn mh fusik¸n susthm�twn. Ja anadeÐxoume sto deÔtero kef�laio poi�eÐnai h �mesh sqèsh thc an�lushc enìc �ploÔ empeirikoÔ problhmatoc ìpwcautì thc mp�llac me ta M kouti� pou jewroÔme ed¸, me ta fusik� prob-l mata ta opoÐa eÐmaste upoqrewmènoi na antimetwpÐsoume me mia statistik prosèggish.

1.1 Par�deigma Statistik c An�lushc: Se poiì

koutÐ eÐnai h mp�la ?

Ja eisag�goume thn ènnoia thc elleÐpousac plhroforÐac kai th statistik en-tropÐa pou th metr�ei se èna aplì paradeÐgma katast�sewn pou emperièqoun�gnoia. Ja upojèsoume ìti èqoume M ìmoia kouti� mèsa sta opoÐaèqei topojethjeÐ mia mp�la. Den gnwrÐzoume se poiì koutÐ brÐsketai hmp�la. Jèloume na orÐsoume thn sun�rthsh h opoÐa posotikopoieÐ thn �gnoi�mac. Sthn aploÔsterh perÐptwsh den up�rqei kamÐa �llh plhroforÐa plhn tougegonìtoc ìti h mp�la brÐsketai mèsa se èna apì ta kouti�. Sthn perÐtwshaut ja perigr�youme th genik strathgik mia statistik c antimet¸pishctou probl matoc kai ja anadeiqjoÔn ta basik� qarakthristik� thc statis-tik c entropÐac ìpwc prokÔptoun apì aplèc empeirikèc apait seic mac apì thsun�rthsh pou posotikopoieÐ thn �gnoia. Sth sunèqeia mporoÔme na doÔme

1.1. Par�deigma Statistik c An�lushc: Se poiì koutÐ eÐnai h mp�la ? 5

thn pio genik kai pio endiafèrousa perÐptwsh sthn opoÐa endèqetai na èqoun-me k�poia prìsjeth statistik plhroforÐa (gia par�deigma eÐnai pio pijanì nabrÐsketai h mp�la sta pr¸ta kouti� apì arister� proc ta dexi�). H teleutaÐaaut perÐptwsh èqei poll� koin� me tic statistikèc analÔseic pou antisoiqoÔnsta pio koin� jermodunamik� probl mata.

1.1.1 AploÔsterh perÐptwsh: GnwrÐzoume mìnon ìti hmp�la eÐnai mèsa se èna apì ta M kouti�.

Den dÐnetai kammÐa prìsjeth plhroforÐa pèran tou gegonìtoc ìti h mp�labrÐsketai mèsa se èna apo ta M kouti�. Enac parathrht c, o opoÐoc dengnwrÐzei se poiì koutÐ topojet jhke h mp�la, kuriarqeÐtai apì �gnoia. HempeirÐa mac epib�llei ìti e�n anoiqjeÐ èna apo ta kouti� tuqaÐa, h pijanìthtana brejeÐ th mp�la mèsa eÐnai 1/M . Ac prospaj soume na kwdikopoi soumethn profan aut empeirik diapÐstwsh.

• Pr¸th proteraiìthta opoiasd pote statistik c an�lush-c eÐnai h tautopoÐhsh twn prospel�simwn ( dunat¸n)katast�sewn tou upì exètash sust matoc. Autì prokÔpteiapì touc perisrismoÔc pou orÐzoun to sÔsthma.

• DeÔtero kai kuri¸tero b ma miac statistik c an�lushceÐnai h tautopoÐhsh thc katanom c pijanot twn an�mesastic prospel�simec katast�seic. Ja doÔme sth sunèqeia ìtime b�sh thn empeirÐa mac, aut h katanom ja prèpei na èqei thn idiìth-ta na megistopoieÐ thn entropÐa. Epomènwc, isodÔnama, to b ma autìègkeitai sthn prosp�jeia tautopoÐhshc thc katanom c pijanot twn poumegistopoieÐ thn entropÐa me b�sh to axÐwma Boltzmann - Gibbs pouanaparist� aut thn empeirik mac apaÐthsh.

Sthn perÐptwsh tou sust matoc mp�la mèsa se èna apì ta M kouti� kai tadÔo b mata eÐnai profan . Efìson h mp�la eÐnai mÐa, oi prospel�simeckatast�seic tou sust matoc eÐnai h mp�la na brÐsketai sto kajèna apo taM kouti�. Epomènwc, to sÔnolo twn prospel�simwn katast�sewnapoteleÐtai apì M stoiqeÐa,to kajèna antistoiqeÐ se mia diaforetik jèsh thc mp�lac an�mesa sta M kouti�.Oson afor� sto deÔtero b ma, thn tautopoÐhsh thc katanom c twn pijan-ot twn, eÐnai shmantikì na shmei¸soume akìmh kai sto aplì par�deigma poumac apasqoleÐ to gegonìc ìti sto tèloc kaloÔmaste na epilèxoume thn katanom me b�sh thn empeirik mac logik . Enac shmantikìc periorismìc eÐnai ìtito �jroisma twn pijanot twn ìlwn twn prospel�simwn katast�sewn prèpei


na isoÔtai me th mon�da. Autìc eÐnai o periorismìc thc legìmenhc kanon-ikopoÐhshc twn pijanot twn kai isqÔei gia ìla ta sust mata. Dedomè-nou ìti den èqoume kamÐa �llh plhroforÐa statistikoÔ qarakt ra (p.q. k�poiakouti� eÐnai piì pijan� klp.), o mìnoc statistikìc periorismìc pou èqoume eÐnaih kanonikopoÐhsh twn pijanot twn.

Apomènei t¸ra na tautopoi soume thn katanom twn pijanot twn an�mesastic prospel�simec katast�seic h opoÐa upakoÔei touc statistikoÔc perior-ismoÔc pou èqoume, dhlad sthn perÐptws mac mìnon sthn kanonikopoÐhshtwn pijanot twn. EÐnai profanèc ìti up�rqoun �peirec sumbatèc katanomècpijanot twn an�mesa stic prospel�simec katast�seic oi opoÐec eÐnai tètoiec¸sste to �jroisma twn pijanot twn eÐnai h mon�da. Sto shmeÐo autì den è-qoume k�poio formalistikì periorismì o opoÐoc na mac epitrèpei na dialèxoumean�messa stic �peirec autèc katanomèc pijanot twn. EÐmaste upoqrewmènoina dialèxoume mia sugkekrimènh katanom pijanot twn diìti dÔo diaforetikècepilogèc katanom¸n pijanot twn antistoiqoÔn se dÔo diaforetikèc eikìnectou probl matoc oi opoÐec ja odhg soun kai se diaforetikèc problèyeic. Giapar�deigma, e�n epilèxoume ìti up�rqei ekatì toic ekatì pijanìthta na broÔmethn mp�la sto pr¸to koutÐ kai mhdèn pijanìthta se ìla ta upìloipa èqoumek�nei mia epilog katanom c pijanot twn h opoÐa eÐnai sumbat me ton perior-ismì thc kanonikopoÐhshc. An ìmwc epilèxoume pen nta toic ekatì pijanìthtana brÐsketai h mp�la mèsa sto pr¸to koutÐ, pen nta toic ekatì pijanìthtana brÐsketai sto teleutaÐo koutÐ kai mhdèn pijanìthta se ìla ta upìloipa è-qoume epÐshc k�nei mia sumbat epilog katanom c pijanot twn. Oi dÔo autèceikìnec eÐnai entel¸c diaforetikèc perièqontac h pr¸th mhdenik �gnoia en¸h deÔterh peperasmènh �gnoia.

Gia na bgoÔme apì to adièxodo thc Ôparxhc �peirou arijmoÔ sumbat¸n katanom¸n,eÐmaste upoqrewmènoi na epikalesjoÔme èna prìsjeto empeirikì krit rio. Hempeirik mac logik epib�llei me b�sh ta dedomèna na jew-r soume ìti ìlec oi prospel�simec katast�seic èqoun thn Ðdi-a pijanìthta na sumboÔn. ParathroÔme ìti aut h katanom pijanot twn antistoiqeÐ sth mègisth �gnoia. H epilog thc isopÐ-janhc katanom c eÐnai h mình pou den prosjètei teqnhèntwc plhroforÐa stosÔsthma, h opoÐa den up�rqei ston orismì tou probl matoc. Opoiad pote�llh katanom e�n epilegìtan ja isodunamoÔse sthn pragmatikìthta me thnparamìrfwsh tou arqikoÔ mac probl matoc.

Dedomènou ìti to �jroisma ìlwn twn pijanot twn isoÔtai me th mon�da èqoumetelik�:

M∑i

Pi =M∑i

1

M= 1 (1.1.1)

1.1. Par�deigma Statistik c An�lushc: Se poiì koutÐ eÐnai h mp�la ? 7

Par' ìlo pou sthn apl perÐptws mac h epilog thc isopijan c katanom ceÐnai profan c, sthn pio genik perÐptwsh h epilog thc swst c katanom -c pijanot twn mporeÐ na eÐnai exairetik� polÔploko prìblhma. Odhgìc giathn epilog aut ja eÐnai h parat rhsh pou mìlic k�name: Den ja prèpeih epilog thc katanom c pijanot twn pou k�noume na prosjètei plhroforÐateqnhèntwc sto sÔsthma. Prèpei loipìn h epilog thc katanom c na anti-stoiqeÐ sth mègisth dunat �gnoia. Gia na eÐmaste se jèsh na epilèxoumesusthmatik� thn katanom aut , eÐnai aparaÐthto na èqoume mia susthmatik katagraf thc �gnoiac, kai bèbaia na èqoume mia èkfrash thc entropÐac(thc sun�rthshc pou katametr� thn �gnoia) san sun�rthshthc katanom c pijanot twn. Mìnon ètsi eÐnai dunatìn na brejeÐ susth-matik� h katanom pijanot twn h opoÐa megistopoieÐ thn elleÐpousa plhrhfhrÐ-a. Autì anadeiknÔei th shmasÐa thc entropÐac se k�je statistik an�lush.Gia to aplì par�deigma thc mp�lac me ta M kouti�, ja doÔme sth sunèqeiap¸c mporeÐ na orisjeÐ mia sun�rthsh I(M) h opoÐa na posotikopoieÐ thnelleÐpousa plhroforÐa. EÐnai profanèc ìti h elleÐpousa plhroforÐa prèpeina eÐnai sun�rthsh tou arijmoÔ twn kouti¸n M . Gia mia tètoia sun�rthshapaitoÔme na isqÔoun ta akìlouja:

• An up�rqei mìnon èna koutÐ, tìte I(1) = 0. Autì eÐnai pro-fanèc, afoÔ h mp�la ja brÐsketai upoqreawtik� s' autì to koutÐ opìteden up�rqei elleÐpousa plhroforÐa.

• I(M1) > I(M2) efìson isqÔei M1 >M2. 'Oso aux�netai o ari-jmìc twn kouti¸n, tìso aux�netai h elleÐpousa plhrhfhrÐa.

• H plhroforÐa pou apokt�tai stadiak� prèpei na prostÐjetai akìmh kaiìtan èqoume na antimetwpÐsoume sÔnjeta (me th statistik ènnoia) gegonì-ta. MporeÐ na apodeiqjeÐ ìti h apaÐthsh aut epib�llei sthn ousÐa thnlogarijmik ex�rthsh

Oi pio p�nw periorismoÐ epib�lloun thn akìloujh sunarthsi-ak dom I(M) = k ln(M).

1.1.2 Genik perÐptwsh: Anomoiìmorfh katanom pijan-ot twn

Upojètoume t¸ra ìti up�rqei prìsjeth statistik plhroforÐa. DeneÐnai plèon isopijanèc oi prospel�simec katast�seic ìpwc prohgoumènwc al-l� up�rqei mia (anomoiìmorfh) katanom pijanot twn. Sto par�deigm� thcmp�lac me ta M kouti�, mporoÔme na upojèsoume ìti h pijanìthta na broÔme


th mp�la se k�je èna apo ta kouti� exart�tai kai apì th jèsh tou koutioÔ.Epomènwc, se k�je mÐa apì ticM prospel�simec katast�seic (pou onom�zoumekai gegonìta) antistoiqeÐ kai mia pijanìthta Pi na brÐsketai to sÔsthma sthnkat�stash aut . Dedomènou ìti e�n h mp�la brÐsketai mèsa se èna apì takouti�, den mporeÐ sugqrìnwc na brÐsketai kai se k�poio �llo, oÔte ephre�zeithn pijanìthta na brÐsketai se k�poio �llo koutÐ mia �llh for�, lème ìti oiprospel�simec katast�seic eÐnai anex�rthtec.

'Estw M anex�rthta gegonìta, me pijanìthtec P1, P2, . . ., PM1, gia tic

opoÐec isqÔei∑M

i Pi = 1. AnazhtoÔme mÐa kat�llhlh sun�rthsh IM ({Pi})h opoÐa na katametr� thn elleÐpousa plhroforÐa2. EÐnai profanèc ìti jaeÐnai sun�rthsh thc katanom c twn pijanot twn {Pi} ≡ P1, P2, ..., PM de-domènou ìti diaforetikèc katanomèc antistoiqoÔn se diaforetik elleÐpousaplhroforÐa. Apì thn IM ({Pi}) apaitoÔme na ikanopoieÐ ta akìlouja:

• Prèpei h IM({Pi}), na eÐnai suneq c sun�rthsh twn Pi. Denupp�rqei lìgoc na anamènoume asunèqeiec thc elleÐpousac plhroforÐacwc proe thn katanom twn pijanot twn. Epiplèon, dedomènou ìti japrèpei na brejeÐ h katanom pou megistopoieÐ th sun�rthsh aut , japrèpei na eÐnai kai paragwgÐsimh.

• Prèpei h IM({Pi}) na eÐnai summetrik kat� thn antallag twn Pi. Autì shmaÐnei ìti opoiad pote kai an eÐnai h seir� twn kouti¸n,h elleÐpousa plhroforÐa pou antistoiqeÐ se mia sugkekrimènh katanom pijanot twn eÐnai Ðdia. Gia par�deigma, h katanom pijanot twn kat�thn opoÐa up�rqei pen nta toic ekatì pijanìthta h mp�la na brÐske-tai sto pr¸to koutÐ, pen nta toic ekatì pijanìthta na brÐsketai stodeÔtero koutÐ kai mhdèn pijanìthta na brÐsketai se opoiod pote apì taupìloipa kouti� emperièqei thn Ðdia ellèipousa plhroforÐa me opoiad -pote katanom pou antistoiqeÐ se pen nta toic ekatì pijanìthta se dÔoopoiad pote kouti� (par�deigma to trÐto kai to teleutaÐo) kai mhdèn seìla ta upìloipa.

• 'Otan èqoume isopÐjana gegonìta, me pijanìthtec Pi =1M ,

prèpei h sun�rthsh f(M) = IM( 1M , 1

M , . . . , 1M), na eÐnai aÔx-

ousa sun�rthsh tou M . Profan¸c ìtan ta gegonìta eÐnai isopÐ-jana ìpwc sthn prohgoÔmenh upoenìthta, ìso perissìtera eÐnai ta k-outi�, tìso perissìterh eÐnai kai h elleÐpousa plhroforÐa.

1oi pijanìthtec Pi den eÐnai en gènnei Ðsec metaxÔ touc2Prìkeitai gia mia sun�rthsh poll¸n metablht¸n, me tìsec metablhtèc Pi ìsec kai oi

prospel�simec katast�seic: IM ({Pi}) ≡ IM (P1, P2, ..., PM )

1.2. Statistik EntropÐa 9

• Prèpei h morf thc sun�rthshc I(n) na eÐnai tètoia, ¸-ste na exasfalÐzei thn prosjetikìthta thc plhroforÐac.Dhlad , k�je �kainoÔria� plhroforÐa pou lamb�noume gia to sÔsthma,ja prèpei na prostÐjetai sthn dh up�rqousa3. H ap�ithsh aut mazÐkai me tic prohgoÔmenec epib�llei th logarijmik dom .

EÐnai dunatì na apodeiqjeÐ ìti h sun�rthsh IM({Pi}), pou ikanopoieÐtic parap�nw apait seic, èqei th morf :

IM({Pi}) ≡ IM(P1,P2, . . . ,PM) = −k

M∑i=1

Pi ln(Pi). (1.1.2)

1.2 Statistik EntropÐa

H sun�rthsh IM ({Pi}) h opoÐa katametr� thn elleÐpousa plhroforÐa, onom�ze-tai Statistik EntropÐa tou sust matoc. Lìgw aut c thc onomasÐac,ja metonom�soume th sun�rthsh IM se SM , ¸ste na gÐnei �mesa antilhpt hsÔndesh me th Statistik Fusik kai th jermodunamik .

SM({Pi}) = −kM∑i=1

Pi ln(Pi). (1.2.1)

Sto trÐto kef�laio ja doÔme ìti eÐnai dunat h tautopoÐhsh thc statis-tik c entropÐac fusik¸n susthm�twn me th jermodunamik entropÐa, opìteh k prèpei ja tautopoihjeÐ me th stajer� Boltzmann pou isoÔtai me k =1.38 · 10−23JK−1. H entropÐa sth statistik fusik ja èqei ticfusikèc diast�seic pou thc prosdÐdei h stajer� Boltzmann.Lìgw thc tautopoÐhshc thc sun�rthshc pou metr� thn elleÐpousa plhroforÐafusik¸n susthm�twn me thn fusik posìthta pou tan dh gnwst wc en-tropÐa sth jermodunamik , epikr�thse h onomasÐa entropÐa gia th sun�rthshaut akìmh kai gia mh fusik� sust mata sta plaÐsia thc jewrÐac thc plhro-forÐac.

1.2.1 Idiìthtec thc Statistik c EntropÐac

H statistik entropÐa enìc sust matoc, eÐte prìkeitai gia fusikì sÔsthmaeÐte prìkeitai gia k�poio �llo statistikì sÔsthma, upakoÔei tic parak�twidiìthtec:

3Na shmeiwjeÐ sto shmeÐo autì ìti e�n dÔo gegonìta èqoun to kajèna pijanìthta P1

kai P2 antÐstoiqa, tìte h pijanìthta na sumboÔn tautìqrona kai ta dÔo isoÔtai me P1×P2.


• H entropÐa eÐnai summetrik kat� thn antallag twn pi-janot twn Pi.

S(P1, . . . ,Pi, . . . ,Pj, . . . , PM ) = S(P1, . . . ,Pj, . . . ,Pi, . . . , PM ).

• S ≥ 0, gia opoiesd pote timèc twn pijanot twn Pi. Dhlad h elleÐpousa plhroforÐa eÐna mhdèn (e�n mÐa apì tic prospel�simeckatast�seic èqei ekatì toic ekatì pijanìthta) jetik pr�gma to opoÐoeÐnai profanèc.

• H entropÐa parousi�zei el�qisto ìtan isqÔoun: Pimin= 1

kai tautìqrona Pi =imin= 0. Se aut n thn perÐptwsh isqÔei:

S(0,0, . . . ,1, . . . ,0) = 0.

Mìnon sthn perÐptwsh aut h elleÐpousa plhroforÐa eÐnai mhdenik .

• H entropÐa èqei èna mègisto pou antistoiqeÐ sthn isopi-jan katanom . Gia na broÔme to mègisto, ja qrhsimopoi soume thmèjodo twn pollaplassiast¸n Lagrange. Jèloume na broÔme to mègis-to thc entropÐac, S, upì ton periorismì

∑Mi=1 Pi = 1 ⇒

∑Mi=1 Pi−1 = 0.

OrÐzoume mÐa sun�rthsh Υ, h opoÐa enswmat¸nei ton prohgoÔmeno pe-riorismì, wc akoloÔjwc:

Υ({Pi}) = S({Pi})− λ

(M∑i=1

Pi − 1

).

ìpou λ, o pollaplassiast c Lagrange. H elaqistopoÐhsh thc sun�rthsh-c Υ, ja mac d¸sei to mègisto thc entropÐac. 'Eqoume loipìn:

∂Υ

∂Pi= −k (lnPi + 1)− λ = 0 ⇒

lnPi = −1− λ

k= Stajer�.

Sunep¸c Pi = C. Apì th sqèsh∑M

i=1 Pi = 1, prokÔptei Pi = 1/M .Ara, to mègisto thc entropÐac lamb�netai gia isopijan�gegonìta kai isoÔtai me Smax = k lnM.

• H sun�rthsh SM({Pi = 1/M}) = Smax, eÐnai aÔxousa sun�rthshtou M. To eÐdame kai sthn prohgoÔmenh upoenìthta ìti sthn isopi-jan katanom h elleÐpousa plhroforÐa eÐnai megalÔterh ìtan aux�neio arijmìc twn isopijan¸n katast�sewn.


• H entropÐa qarakthrÐzetai apì thn idiìthta thc pros-jetikìthtac. 'Estw ìti èqoume dÔo diaforetik� anex�rthta sÔnolagegonìtwn, me pijanìthtec Pi kai P

′i . Se aut n thn perÐptwsh, èqoume

sÔnjeta gegonìta, twn opoÐwn h pijanìthta dÐnetai apì to ginìmenoPi × P

′i . H entropÐa tou sust matoc, dÐnetai apì th sqèsh:

S({Pi, P′j }) = −k

M∑i=1

N∑j=1

Pi · P′j ln(Pi · P

′j )

= −kM∑i=1

N∑j=1

Pi · P′j [ln(Pi) + ln(P

′j )]

= −k

N∑j=1

P′j

(M∑i=1

Pi lnPi

)+

M∑i=1

Pi

N∑j=1

P′j lnP

′j

.Sthn perÐptwsh pou ta dÔo sÔnola apoteloÔntai apì isopÐjana gegonì-ta, h entropÐa lamb�nei thn apl prosjetik morf :

S({Pi, P′j }) = S({Pi}) + S({P ′

j }). (1.2.2)

Sthn perÐptwsh pou ta dÔo sÔnola, den eÐnai anex�rthta,up�rqei susqetismìc. H Ôparxh susqetismoÔ, prokaleÐthn upo-prosjetikìthta. Dhlad , den isqÔei h sqèsh(1.2.2), all� h anisìthta:

S({Pi, P′j }) ≤ S({Pi}) + S({P ′

j }). (1.2.3)

O susqetismìc, aux�nei to posostì thc plhroforÐac,mei¸nontac thn entropÐa.

• Gia thn entropÐa isqÔei hAnisìthta Kurtìthtac (concavity). Jew-roÔme ìti èqoume dÔo sÔnola pijanot twn, pou perigr�foun to Ðdio sÔno-lo twn dunat¸n gegonìtwn. Onom�zoume to èna sÔnolo pijanot twn wcP1, P2, . . ., PM kai to �llo wc P

′1 , P

′2 , . . ., P

′M . Katìpin, jewroÔme

èna pragmatikì arijmì λ gai ton opoÐo isqÔei 0 < λ < 1. OrÐzoume mÐanèa katanom (sÔnolo pijanot twn), me ton akìloujo trìpo:

P′′i = λPi + (1− λ)P

′i .

MporeÐ na deiqjeÐ, ìti gia th nèa katanom isqÔei h sqèsh:


S({P ′′i }) > λS({Pi}) + (1− λ)S({P ′

j }). (1.2.4)

1.2.2 SuneqeÐc Katanomèc Pijanìthtac

Wc ed¸, asqolhj kame me diakritèc katanomèc pijanìthtac. Sthn perÐptwshpou èqoume suneqeÐc katanomèc pijanìthtac, h entropÐa èqei mÐa diaforetik morf . Gia na ex�goume thn exÐswsh gia thn entropÐa, sth suneq perÐptwsh,jewroÔme èna di�sthma (α, β), sto opoÐo orÐzetai h suneq c katanom pi-janìthtacD(x). 'Epeita, qwrÐzoume to di�sthma seM tm mata∆x. OrÐzoumewc xi, ta mèsa twn parap�nw tmhm�twn, ìpou i = 1 . . .M . To eÔroc twn tmh-m�twn, den eÐnai en gènei to Ðdio. H katanom tou eÔrouc twn tmhm�twn mporeÐna perigrafeÐ apì mÐa sun�rthsh f(xi). Eidikìtera, to eÔroc k�je tm matoc,dÐnetai apì th sqèsh:

∆xi = εf(xi), (1.2.5)

ìpou ε ènac pragmatikìc arijmìc, pou apoteleÐ to eÔroc twn upodiasthm�twn,ìtan aut� isapèqoun. O arijmìc ε teÐnei sto mhdèn kai h tim tou exart�taiapì thn akrÐbeia twn metr sewn pou ja ektelèsoume sto sÔsthma.EpÐshc, jewroÔme ìti h puknìthta pijanìthtac se k�je upodi�sthma, eÐnaistajer , kai Ðsh me thn tim thc katanom c sto mèso tou diast matoc xi.Dhlad , gia tic pijanìthtec Pi, isqÔei:

Pi = D(xi)∆xi = P (xi)ε.

H diakritopoÐhsh tou diast matoc (α, β), mac epitrèpei na upologÐsoume thzhtoÔmenh sqèsh gia thn entropÐa, qrhsimopoi¸ntac thn exÐswsh

SM (P1, P2, . . . , PM ) = −kM∑i=1

Pi ln(Pi). (1.2.6)

Sugkekrimèna èqoume:

S = −kM∑i=1

D(xi)εf(xi) ln[D(xi)εf(xi)]

= −kM∑i=1

D(xi)εf(xi) ln[D(xi)f(xi)]− kM∑i=1

ρ(xi)εf(xi) ln ε.(1.2.7)


Lìgw tou ìti h katanom thc pijanìthtac eÐnai suneq c, den mporoÔme naorÐsoume thn entropÐa kat� apìluth tim . Autì ofeÐletai, sto ìti h aÔxhshthc akrÐbeiac thc mètrhshc aux�nei thn entropÐa. MporoÔme na orÐsoumemìno th sqetik entropÐa. H sqetik entropÐa S, orÐzetai apì th sqèsh:

S = −kM∑i=1

D(xi)εf(xi) ln[D(xi)f(xi)] = −kM∑i=1

D(xi)∆(xi) ln[D(xi)f(xi)]

(1.2.8)Sto ìrio lim∆xi → 0, to k�je di�sthma gÐnetai apeirostì. Telik� prokÔptei:

S = −k∫dxD(x) ln[D(x)f(x)]. (1.2.9)

An jewr soume ìti ìla ta upodiast mata èqoun to Ðdio eÔroc, dhlad f(x) =1, katal goume sth sqèsh:

S = −k∫dxD(x) ln[D(x)]. (1.2.10)


Kef�laio 2

Majhmatikìc FormalismìcKbantomhqanik c

Mia diakrit om�da apì kets, {| ui⟩} eÐnai orjokanonik e�n ta kets upakoÔounsth sqèsh:

⟨ui | uj⟩ = δij (2.0.1)

Mia suneq c om�da apì kets, {| wα⟩} eÐnai orjokanonik e�n ta kets upakoÔ-oun sth sqèsh:

⟨wα | wα′⟩ = δ(α−α′) (2.0.2)

MÐa diakrit om�da {| ui⟩} , mÐa suneq c om�da {| wα⟩} apoteleÐ mÐab�sh, e�n k�je ket, | ψ⟩ ∈ E èqei èna monadikì an�ptugma p�nw sta {| ui⟩} ta {| wα⟩}, antÐstoiqa.

| ψ⟩ =∑i

ci | ui⟩ ⇒ ⟨uj | ψ⟩ = cj (2.0.3)

| ψ⟩ =∫

dα c(α) | wα⟩ ⇒ ⟨wα′ | ψ⟩ = c(α′) (2.0.4)

An ta {| ui⟩} ta {| wα⟩}, apoteloÔn mÐa orjokanonik b�sh, tìte isqÔei:

P{ui} =∑

i | ui ⟩⟨ui | = 1 (2.0.5)

P{wα} =∫dα | wα⟩⟨wα | = 1 (2.0.6)

An λ eÐnai ènac migadikìc arijmìc (λ ∈ C), isqÔoun ta akìlouja:

15

16 Kef�laio 2. Majhmatikìc Formalismìc Kbantomhqanik c

1. | ψ⟩λ = λ | ψ⟩ (2.0.7)

2. ⟨ψ | λ = λ⟨ψ | (2.0.8)

3. A | ψ⟩λ = λA | ψ⟩ (2.0.9)

4. ⟨ϕ | λ | ψ⟩ = λ⟨ϕ | ψ⟩ (2.0.10)

Telestèc

| ϕ⟩ A→| ψ⟩ | ψ⟩ = A | ϕ⟩

Metajèthc dÔo telest¸n A kai B:

[A, B] = A · B − B · A (2.0.11)

Je¸rhma 1:

An [A, B] = 0 kai A | ψ⟩ = α | ψ⟩, tìte | ϕ⟩ = B | ψ⟩ eÐnai epÐshc idiodi�nusmatou A me thn Ðdia idiotim α.

Je¸rhma 2:

An [A, B] = 0 kai {A | ψ1⟩ = α1 | ψ1⟩A | ψ2⟩ = α2 | ψ2⟩

(2.0.12)

ìpou α1 = α2, tìte ⟨ψ1 | B | ψ2⟩.

Je¸rhma3:An [A, B] = 0, tìte mporoÔme na broÔme mÐa orjokanonik b�shtou q¸rou twn katast�sewn, pou na apoteleÐtai apì koin� id-iodianÔsmata twn A kai B.

'Otan [A, B] = 0, ta megèjh A kai B onom�zontai sumbat� kai m-poroÔn na metrhjoÔn tautìqrona, me jewrhtik� �peirh akrÐbeia.

'Otan [A, B] = 0, ta megèjh A kai B onom�zontai asumbat� kaiisqÔei h arq Abebaiìthtac tou Heisenberg:

17

∆A ·∆B ≥ 1

2| ⟨[A, B]⟩ | (2.0.13)

ìpou

∆A =

√⟨A2⟩ − ⟨A⟩2 kai ⟨A⟩ = ⟨ψ | A | ψ⟩ (2.0.14)

Idiìthtec Metajet¸n

1. [A, B] = −[B, A] (2.0.15)

2. [A, (B + C)] = [A, B] + [A, C] (2.0.16)

3. [A, BC] = [A, B]C + B[A, C] (2.0.17)

4. [A, B]† = [B†, A†] (2.0.18)

5. [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0 (2.0.19)

ErmitianoÐ TelestècO Ermitianìc Suzug c, enìc telest A, orÐzetai apì th sqèsh:

⟨ϕ | A | ψ⟩ = ⟨ψ | A† | ϕ⟩∗ (2.0.20)

Sto ket A | ψ⟩ antistoiqeÐ to bra ⟨ψ | A†.'Enac telest c eÐnai ermitianìc, ìtan isqÔei:

A = A† (2.0.21)

⇒ ⟨A⟩ = ⟨A⟩∗.⇒ Oi idiotimèc enìc ermitianoÔ telest eÐnai pragmatikèc.⇒ Oi idiosunart seic pou antistoiqoÔn se diaforetikèc idiotimèc, eÐnai or-jog¸niec metaxÔ touc.

Gia ìlouc touc telestèc, isqÔoun:

(A†)†

= A (2.0.22)(λA)†

= λ∗A† (λ ∈ C) (2.0.23)

(A+ B)† = A† + B† (2.0.24)(AB)†

= B†A† (2.0.25)


MonadiaÐoi Telestèc'Enac telest c onom�zetai monadiaÐoc (Unitary ), ìtan isqÔei:

U−1 = U † (2.0.26)

O parap�nw orismìc, sunep�getai:

U †U = U U † = 1 (2.0.27)

E�n ènac telest c eÐnai monadiaÐoc, isqÔoun:{| ψ′

1⟩ = U | ψ1⟩| ψ′

2⟩ = U | ψ2⟩⇒ ⟨ψ′

1 | ψ′2⟩ = ⟨ψ1 | ψ2⟩ (2.0.28)

O monadiaÐoc metasqhmatismìc pou antistoiqeÐ se ènan mona-diaÐo telest U DiathreÐ to Eswterikì Ginìmeno (kai �ra thnkanonikopoÐhsh) sto q¸ro twn katast�sewn E.E�n o telest c A eÐnai ermitianìc, tìte o telest c T = eiA, eÐnai monadiaÐoc.Autì mporeÐ na deiqjeÐ wc akoloÔjwc:

T † = e−iA†= e−iA (2.0.29)

(2.0.30)

T †T = e−iAeiA = 1 (2.0.31)

T T † = eiAe−iA = 1 (2.0.32)

Anapar�stash Telest¸n me M trec

Aij = ⟨ui | A | uj⟩ =

A11 A12 · · · A1j · · ·A21 A22 · · · A2j · · ·...

.... . .

......

Ai1 Ai2 · · · Aij · · ·...

.... . .

.... . .

(2.0.33)

A(α, α′) = ⟨wα | A | wα′⟩ =

− − −

α′

− →

|...

|...

| α · · · · · · Aα,α′

↓

(2.0.34)

19

(A†)ij= A ∗

ij (2.0.35)

(A†)(α,α′)

= A ∗(α,α′) (2.0.36)

Pern�me apì ton A ston A† antall�ssontac grammèc me st lec kai èpeitapaÐrnontac touc migadikoÔc suzugeÐc twn stoiqeÐwn.

Ta diag¸nia stoiqeÐa enìc ermitianoÔ telest eÐnai p�nta pragmatikoÐ arijmoÐ.Gia èna monadiaÐo telest , to �jroisma twn ginomènwn twn stoiqeÐwn mÐacst lhc kai twn suzug¸n migadik¸n twn stoiqeÐwn mÐac �llhc st lhc, eÐnai:

• Mhdèn e�n oi st lec eÐnai diaforetikèc.

• 'Ena e�n den eÐnai.

'Iqnoc Telest

'Iqnoc, eÐnai to �jroisma twn diag¸niwn stoiqeÐwn thc m trac pou anaparist�ènan telest :

Tr{A}=∑i

⟨ui | A | ui⟩ (2.0.37)

Tr{A}=

∫dα⟨wα | A | wα⟩ (2.0.38)

To Ðqnoc enìc telest A den exart�tai apì th b�sh sthn opoÐa anaparist�taio telest c.

Idiìthtec tou Ðqnouc:

Tr(AB)= Tr

(BA)

(2.0.39)

Tr(ABC

)= Tr

(BCA

)= Tr

(CAB

)(2.0.40)

Anapar�stash Jèshc

'Otan endiaferìmaste gia th jèsh enìc swmatidÐou, profan¸c dialègoume thb�sh pou apoteloÔn oi idiosunart seic tou telest jèshc. EÐnai mÐa suneq cb�sh, tètoia ¸ste:

X | wx⟩ = x | wx⟩ (2.0.41)


h opoÐa sun jwc gr�fetai san:

XΨ(x) = xΨ(x) (2.0.42)

Me ton idiìmorfo sumbolismì thc anapar�stashc jèshc, èqoume:

⟨ϕ | ψ⟩ =∫ +∞

−∞dxϕ∗(x)ψ(x) (2.0.43)

⟨ψ | ψ⟩ =∫ +∞

−∞dx | ψ(x) |2= 1 kanonikopoÐhsh thcψ (2.0.44)

⟨A⟩ = ⟨ψ | A | ψ⟩ =∫ +∞

−∞dxψ∗(x)Aψ(x) (2.0.45)

apì ìpou prokÔptei:

⟨X⟩ =∫ +∞−∞ dxψ∗(x)Xψ(x) (2.0.46)

(2.0.47)

=∫ +∞−∞ dxψ∗(x)xψ(x) (2.0.48)

GnwstoÐ Telestèc sthn Anapar�stash Jèshc

Telest c Jèshc : XΨ(x) = xΨ(x) (2.0.49)

Telest c Orm c : PΨ(x) = −i~ ∂∂xΨ(x) (2.0.50)

Qamiltonian (1-D) : HΨ(x) ={− ~2

2m∂2

∂x2 + V (x)}Ψ(x)(2.0.51)

Qamiltonian (3-D) : HΨ(r) ={− ~2

2m∇2 + V (r)}Ψ(r)(2.0.52)

Troqiak Stroform : L = ˆr × ˆp (2.0.53)

me sunist¸sec : (2.0.54)

Lx = ypz − zpy ⇒ LxΨ(r) = −i~(y ∂∂z − z ∂

∂y

)Ψ(r)(2.0.55)

Ly = zpx − xpz ⇒ LyΨ(r) = −i~(z ∂∂x − x ∂

∂z

)Ψ(r) (2.0.56)

Lz = xpy − ypx ⇒ LzΨ(r) = −i~(x ∂∂y − y ∂

∂x

)Ψ(r)(2.0.57)

Kef�laio 3

Axi¸mata thcKbantomhqanik c

AxÐwma 1

Thn stigm t0 h kat�stash tou sust matoc orÐzetai apì èna ket | Ψ(t0)⟩ pouan kei sto q¸ro twn katast�sewn E.

AxÐwma 2

K�je metr simh posìthta perigr�fetai apì ènan telest A pou dra mèsa stoq¸ro E. Autìc o telest c mÐa parathr simh posìthta (observable).

AxÐwma 3

To mìno dunatì apotèlesma thc mètrhshc miac fusik c posìthtac eÐnai mÐaapì ti idiotimèc tou telest pou antistoiqeÐ sth fusik aut posìthta.

AxÐwma 4

a. PerÐptwsh diakritoÔ mh ekfulismènou f�smatoc:

'Otan metr�me mÐa fusik posìthta p�nw se mia kanonikopoihmènh kat�stash| Ψ⟩, h pijanìthta P (αn) na p�roume th mh ekfulismènh idiotim αn toutelest A pou antistoiqeÐ, eÐnai:

P (αn) =| ⟨un | Ψ⟩ |2 (3.0.1)

21

22 Kef�laio 3. Axi¸mata thc Kbantomhqanik c

ìpou un eÐnai to kanonikopoihmèno idiodi�nusma tou A pou antistoiqeÐ sthnidiotim αn.

b. PerÐptwsh diakritoÔ ekfulismènou f�smatoc:

'Otan h fusik posìthta metr�tai p�nw se mÐa kanonikopoihmènh kat�stash| Ψ⟩, h pijanìthta P (αn) na p�roume (san apotèlesma mètrhshc) thn idiotim αn tou telèst A pou antistoiqeÐ eÐnai:

P (αn) =

gn∑i=1

| ⟨ui | Ψ⟩ |2 (3.0.2)

ìpou gn eÐnai o bajmìc ekfulismoÔ thc idiotim c αn kai {| u in ⟩} (i = 1, 2, . . . , gn)

eÐnai èna orjokanonikì sÔnolo dianusm�twn to opoÐo apoteleÐ b�sh ston upì-qwro E pou antistoiqeÐ sthn idiotim A.

g. PerÐptwsh suneqoÔc mh ekfulismènou f�smatoc:

'Otan h fusik posìthta metr�tai p�nw se èna sÔsthma to opoÐo brÐsketaisthn kanonikopoihmènh kat�stash | Ψ⟩, h pijanìthta P (α) na p�roume ènaapotèlesma thc mètrhshc metaxÔ α kai α+ dα, isoÔtai me:

dP (α) =| ⟨wα | Ψ⟩ |2 dα (3.0.3)

ìpou | wα⟩ eÐnai to idiodi�nusma pou antistoiqeÐ sthn idiotim α tou parathr -simou telest A pou antistoiqeÐ sth fusik posìthta pou metr�me.AxÐwma 5

E�n h mètrhsh mÐac fusik c posìthtac sthn kat�stash | Ψ⟩ dÐnei to apotè-lesma αn, h kat�stash tou sust matoc amèswc met� th mètrhsh eÐnai hkanonikopoihmènh probol :

Pn | Ψ⟩√⟨Ψ | Pn | Ψ⟩

(3.0.4)

thc | Ψ⟩ p�nw ston upìqwro E pou antistoiqeÐ sthn idiotim αn tou telest A o opoÐoc antistoiqeÐ sth metr simh fusik posìthta.

AxÐwma 6

23

H exèlixh sto qrìno tou dianÔsmatoc kat�stashc | Ψ(t)⟩ kubern�tai apì thnexÐswsh Schrodinger:

i~d

dt| Ψ(t)⟩ = H(t) | Ψ(t)⟩ (3.0.5)

ìpou H(t) eÐnai o telest c pou antistoiqeÐ sthn olik enèrgeia tou sust matoc(Qamiltonian ).

24 Kef�laio 3. Axi¸mata thc Kbantomhqanik c

Kef�laio 4

Qr sh tou exwterikoÔginomènou sthnKbantomhqanik

Exwterikì ginìmeno Tanustikì ginìmeno ginìmenoKronecker

IsqÔei:

E = E1 ⊗ E2 (4.0.1)

e�n isqÔoun ta akìlouja:

•

| ϕ(1)⟩ ∈ E1 (4.0.2)

| χ(2)⟩ ∈ E2 (4.0.3)

•| ϕ(1)⟩ ⊗ | χ(2)⟩ =| χ(2)⟩ ⊗ | ϕ(1)⟩ ∈ E (4.0.4)

• E�n λ ∈ C kai µ ∈ C tìte:

[λ | ϕ(1)⟩]⊗ | χ(2)⟩ = λ [| ϕ(1)⟩ ⊗ | χ(2)⟩] (4.0.5)

| ϕ(1)⟩ ⊗ [µ | χ(2)⟩] = µ [| ϕ(1)⟩ ⊗ | χ(2)⟩] (4.0.6)

25

26 Kef�laio 4. Qr sh tou exwterikoÔ ginomènou sthn Kbantomhqanik

• Epimeristikìthta wc proc thn prìsjesh:

| ϕ(1)⟩ ⊗[| χ1(2)⟩+ | χ2(2)⟩] = | ϕ(1)⟩ ⊗ | χ1(2)⟩+ | ϕ(1)⟩ ⊗ | χ2(2)⟩(4.0.7)

[| ϕ1(1)⟩+ | ϕ2(1)⟩] ⊗ | χ(2)⟩ =| ϕ1(1)⟩ ⊗ | χ(2)⟩+ | ϕ2(1)⟩ ⊗ | χ(2)⟩(4.0.8)

E�n :

{| ui(1)⟩} eÐnai b�sh tou E1 (4.0.9)

{| vi(2)⟩} eÐnai b�sh tou E2 (4.0.10)

tìte ta dianÔsmata | ui(1)⟩⊗ | vj(2)⟩ eÐnai b�sh tou E .

E�n N1 eÐnai h di�stash tou E1 kai N2 h di�stash tou E2, h di�stash tou EeÐnai N1 ·N2.

'Ena tuqaÐo di�nusma tou E mporeÐ p�nta na analujeÐ se grammikì sunduasmìdianusm�twn pou eÐnai exwterikì ginìmeno enìc dianÔsmatoc b�shc tou E1 meèna di�nusma b�shc tou E2.

| Ψ⟩ =∑i,j

ci,j | ui(1)⟩ ⊗ | vj(2)⟩ (4.0.11)

'Estw:

| ϕ(1)χ(2)⟩ = | ϕ(1)⟩ ⊗ | χ(2)⟩ (4.0.12)

| ϕ ′(1)χ

′(2)⟩ = | ϕ ′

(1)⟩ ⊗ | χ ′(2)⟩ (4.0.13)

tìte:

⟨ϕ ′(1)χ

′(2) | ϕ(1)χ(2)⟩ = ⟨ϕ ′

(1) | ϕ(1)⟩⟨χ ′(2) | χ(2)⟩ (4.0.14)

⟨ui′(1)vj′(2) | ui(1)vj(2)⟩ = ⟨ui′(1) | ui(1)⟩⟨vj′(2) | vj(2)⟩ = δii′δjj′ (4.0.15)

Exwterikì ginìmeno telest¸n:

27

'Estw ènac grammikìc telest c A(1) pou dra ston E1 kai B(2) ènac grammikìctelest spou dra sto q¸ro E2. O A(1)⊗ B(2) eÐnai ènac grammikìc telest cpou dra sto q¸ro E , me ton akìloujo trìpo:

[A(1)⊗ B(2)][| ϕ(1)⟩ ⊗ | χ(2)⟩] = [A(1) | ϕ(1)⟩]⊗ [B | χ(2)⟩] (4.0.16)


Exis¸seic idiotim¸n ston E

'Estw ènac telest c A(1) pou dra sto q¸ro E1, èqontac, gia par�deigma,diakritì f�sma.

A(1) | ϕ in (1)⟩ = αn | ϕ i

n (1)⟩ i = 1, 2, . . . , gn ⇒ (4.0.17)

A(1) | ϕ in (1)⟩ ⊗ | χ(2)⟩ = αn | ϕ i

n (1)⟩ ⊗ | χ(2)⟩ (4.0.18)

• E�n o A(1) eÐnai parathr simoc ston E1, eÐnai epÐshc parathr simoc stonE = E1 ⊗ E2.

• To f�sma tou A(1) sto q¸ro E , paramènei Ðdio me to f�sma sto q¸roE1.

• MÐa idiotim αn pou eÐnai gn forèc ekfulismènh sto q¸ro E1, èqei mèsasto q¸ro E bajmì ekfulismoÔ N2 × gn.

'Estw A(1) kai B(2) tètoioi ¸ste:

A(1) | ϕn(1)⟩ = αn | ϕn(1)⟩ (4.0.19)

B(2) | χp(2)⟩ = βp | χp(2)⟩ (4.0.20)

(par�panw jewr same gia aplopoÐhsh, ìti to f�sma twn telest¸n eÐnai di-akritì kai mh ekfulismèno.)

'Estw o telest c C = A(1) + B(2), o opoÐoc dra sto q¸ro E . Oi idiotimèctou C prokÔptoun apì to �jroisma mÐac idiotim c tou A(1) kai mÐac idiotim ctou B(2). Dhlad :

C[| ϕ(1)⟩ ⊗ | χ(2)⟩] = (αn + βp)[| ϕ(1)⟩ ⊗ | χ(2)⟩] (4.0.21)

MporoÔme na broÔme mÐa b�sh apì idiodianÔsmata tou C pou eÐnai exwterik�ginìmena enìc idiodianÔsmatoc tou A(1) kai enìc idiodianÔsmatoc tou B(2).

29

Efarmog

Sqèsh metaxÔ twn 1−D kai 3−D katast�sewn enìc swmatidÐou

Exyz = Ex ⊗ Ey ⊗ Ez (4.0.22)

'Estw {| x⟩} b�sh tou Ex, {| y⟩} b�sh tou Ey kai {| z⟩} b�sh tou Ez.

{| x, y, z⟩} eÐnai h b�sh tou q¸rou E .

| r⟩ =| x, y, z⟩ =| x⟩ | y⟩ | z⟩ =| x⟩ ⊗ | y⟩ ⊗ | z⟩ (4.0.23)

X | x, y, z⟩ = x | x, y, z⟩ (4.0.24)

Y | x, y, z⟩ = y | x, y, z⟩ (4.0.25)

Z | x, y, z⟩ = z | x, y, z⟩ (4.0.26)

'Estw | ϕ, χ, ω⟩ =| ϕ⟩ ⊗ | χ⟩ ⊗ | ω⟩, tìte isqÔei:

⟨r | ϕ, χ, ω⟩ = ⟨x | ϕ⟩⟨y | χ⟩⟨z | ω⟩ = ϕ(x) · χ(y) · ω(z) (4.0.27)

⟨r | r0⟩ = δ(r − r0) = δ(x− x0)δ(y − y0)δ(z − z0) (4.0.28)

H pio genik kat�stash tou q¸rou E , gr�fetai:

| Ψ⟩ =∫

dxdydz Ψ(x, y, z) | x, y, z⟩ (4.0.29)

SthnΨ(x, y, z) = ⟨x, y, z | Ψ⟩, oi x, y, x exart seic den eÐnai en gènei paragontopoi -simec (factorizable)

Efarmog

H = Hx + Hy + Hz (4.0.30)

Hx | ϕn⟩ = E nx | ϕn⟩ (4.0.31)

Hy | χp⟩ = E py | χp⟩ (4.0.32)

Hz | ωr⟩ = E rz | ωr⟩ (4.0.33)


Oi idiotimèc tou H, èqoun th morf :

En,p,r = E nx + E p

y + E rz (4.0.34)

Oi antÐstoiqec idiosunart seic èqoun th morf :

| ϕn⟩ ⊗ | χp⟩ ⊗ | ωr⟩ ≡| ϕn⟩ | χp⟩ | ωr⟩ (4.0.35)

Oi antÐstoiqec idiosunart seic:

ϕn(x)χp(y)ωr(z) = ⟨x | ϕn⟩⟨y | χp⟩⟨z | ωr⟩ (4.0.36)

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