ANTWNIOU A. STREKLAMAJHMATIKOU

H DIATAXIS TWN TELESTWNEIS THN KBANTOMHQANIKH

DIATRIBH EPI DIDAKTORIAEgkrijeÐsa upì thc Fusikomajhmatik c Sqol c tou PanepisthmÐou Patr¸n

PATRAI 1980

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H ègkrish thc Didaktorik c Diatrib c upì thc Fusikomajhmatik c Sqol c touPanepisthmÐou Patr¸n den upodhloÐ apodoq n twn gnwm¸n tou suggrafèwc.

(N. 5343/1932, rjro. 202)

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PROLOGOS

H paroÔsa diatrib exepon jh eic thn 'EdranJewrhtik c Fusik c tou PanepisthmÐou Patr¸nth upodeÐxei tou KajhghtoÔ k. AsterÐou Gian-noÔsh, proc ton opoÐon ekfrzw thn eugnwmosÔ-nhn mou dia thn kajod ghsin kai thn amèristonsumparstashn tou kaj' ìlhn thn dirkeian thcekpon sewc thc.

Euqarist¸ epÐshc touc bohjoÔc thc 'EdracJewrhtik c Fusik c k. K. Blqon kai k. B.Papajèou dia thn en gènei sunergasÐan twn.

Tèloc euqarist¸ ìlouc ìsouc kaj' oiond -pote trìpon me ebo jhsan.

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4

PINAX PERIEQOMENWNSelÐc

EISAGWGH 2

KEFALAION I. KbantomhqanikaÐ parastseic.

1. Shrodinger parstasic. 62. Wigner parstasic. 83. Feynman parstasic. 12

KEFALAION II. Mèjodoi upologismoÔ twn propagator.

4. Stsimai katastseic. 175. Mh stsimai katastseic. 186. MÐa mesoc lÔsic. 22

KAFALAION III. H Ditaxic twn telest¸n kai efarmogaÐ thcjewrÐac diatxewc eic thn mh sqetikistik nkbantomhqanik n.

7. Ditaxic twn Boson telest¸n. 288. Kanonik ditaxic tou telestoÔ

expk1a

2 + k2a+2 + k3 (aa+ + a+a)

. 31

9. Mh stsima sust mata tetragwnik c morf c. 3510. Mh stsima sust mata tetragwnik c morf c.

me grammikoÔc ìrouc 41

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KAFALAION IV. EfarmogaÐ thc jewrÐac diatxewc eic thnsqetikistik n kbantomhqanik n.

11. Wigner parstasic. 4412. Sqetikistikìc propagator. 48

SUMPERASMATA 54

SUMMARY 56

BIBLIOGRAFIA 58

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1 EISAGWGH

Eic thn kbantomhqanik n lambnetai wc arqik ènnoia h katstasic enìc sust -matoc, h opoÐa perigrfetai eic mian dedomènhn stigm apì mÐa kumatosunrthshΨ(q). H kumatosunrthsic Ψ(q) eÐnai stoiqeÐon tou q¸rou Hilbert toi eÐnai tetragw-nik¸c oloklhr¸simoc, kai toÔto diìti to mègejoc |Ψ(q)|2 lambnetai wc h antÐs-toiqoc puknìthc pijanìthtoc. EpÐ plèon aut eÐnai suneq c kai èqei pargwgonsuneq kat tm mata wc proc q. H dunamik tou sust matoc perigrfetai th bo-hjeÐa miac qronik¸c exartwmènhc kumatosunart sewc Ψ(q, t), h opoÐa epÐ plèoneÐnai suneq c wc proc t kai ikanopoieÐ thn exÐswshn kin sewc tou Shrodinger. Taparathr sima megèjh parÐstantai amfimonoshmntwc apì ermhtianoÔc telestc kaih mèsh tim twn dÐdetai ek thc sqèsewc.

< F >=∫

Ψ∗(q, t)F(

q,−ih∂

∂q, t

)Ψ(q, t)dq (1)

To 1943 mia llh parstasic aneptÔqjh upì tou Feynman. Eic thn parstashnaut n h exÐswsic kin sewc tou Shrodinger antikajÐstatai apì mÐan oloklhrwtik nexÐswshn, o pur n thc opoÐac eÐnai h Green sunrthsic thc exis¸sewc tou Shrodinger,kai onomzetai propagator tou sust matoc. O Feynman èdeixe epÐshc ìti o en lìgwpur n dÐnetai ex enìc path integral. H mèjodoc aut en toÔtoic, den dÔnatai namelet sh sust mata ta opoÐa perièqoun spin llouc parìmoiouc telestèc me aplìntrìpon. EurÐskei perissìteron efarmog n eic sust mata dia ta opoÐa ai suntetag-mènai kai ai suzugeÐc ormaÐ twn eÐnai eparkeÐc dia na ta perigryoun. Poll apìta sumpersmata twn path integrals dÔnantai na epanekfrasjoÔn kai arket nèana prokÔyoun, apì mÐan llhn majhmatik n mèjodon, èna eÐdoc logismoÔ diatxewctwn telest¸n. H ditaxic tou telestoÔ qronik c exelÐxewc eic kpoian morf n eÐnaiidiaitèrwc qr simoc, diìti o telest c autìc dÐdei thn qronik n exèlixhn tou sust -matoc. Eidik¸teron h epÐdrasic tou en lìgw telestoÔ epÐ thc δ(q2−q1) sunart sewcdÐdei ton propagator tou sust matoc.

Prosftwc gÐnetai prospjeia dia thn perigraf n twn kbantomhqanik¸n susth-mtwn kat' eujeÐan eic ton q¸ron twn fsewn. H arq egèneto upì tou Wigner to1932 o opoÐoc susqètise thn katstashn enìc sust matoc me mian sun jhn migadik nsunrthshn gnwst wc Wigner sunrthsic katanom c. 'Edeixe epÐshc ìti me thn bo -jeian thc katanom c aut c, ai kbantomhqanikaÐ mèsai timaÐ dÔnantai na parastajoÔnme thn Ðdian majhmatik n morf n wc oi anamenìmenai timaÐ thc klassik c statistik c

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mhqanik c. Argìteron ègine safèc ìti h susqètisic aut , sundèetai me mian antis-toiqÐan metaxÔ sunart sewn telest¸n kai twn sun jwn migadik¸n sunart sewn, hopoÐa eis qjh upì tou Weyl. H katanom tou Wigner antistoiqeÐ proc ton telest nthc m trac puknìthtoc kai h exÐswsic kin sewc tou Heisenberg autoÔ antistoiqeÐproc thn exÐswshn kin sewc tou Wigner. 'Alloi kanìnec susqetismoÔ èqoun epÐshcmelethj kai orismènoi ex aut¸n èdwsan thn bshn dia thn eisagwg n diafìrwnllwn sunart sewn katanom c. Oi en lìgw kanìnec susqetismoÔ basÐzontai epÐshceic thn jewrÐan diatxewc twn telest¸n.

H ditaxic enìc telestoÔ prosfèrei dÔo wfèleiac eic thn kbantomhqanik n.Di twn antistoiqi¸n twn diatetagmènwn telest¸n proc touc migadikoÔc arijmoÔc,dunmeja na antikatast swmen touc antimetajètac kai tac troqic thc kbantomhqa-nik c, me sun jeic diaforikc exis¸seic kai sun jh oloklhr¸mata eic ton q¸ron twnfsewn. H dunatìthc aut prosfèrei perissìteron eic thn katanìhshn twn kban-tomhqanik¸n ekfrsewn par eic thn aplopoÐhshn twn upologism¸n.

Kat deÔteron lìgon h anptuxic twn telest¸n eic kpoian katllhlon morf ndieukolÔnei kai aplopoieÐ thn epÐdrashn twn telest¸n epÐ twn diafìrwn sunart -sewn. Atuq¸c h ditaxic twn telest¸n an kai eÐnai èna apì ta pr¸ta probl ma-ta thc kbantomhqanik c, den eÐnai mia apl diadikasÐa. En toÔtoic dia ekjetikoÔctelestc tetragwnik c morf c wc proc tac kanonikc suzugeÐc metablhtc h ditaxicepitugqnetai dia thc mejìdouc thc parametrik c paragwgÐsewc.

HmeÐc ed¸ ja asqolhj¸men me tac dÔo autc efarmogc thc jewrÐac diatxewctwn telest¸n. Idiaitèrwc ja eÔrwmen touc akribeÐc propagators mh sqetikistik¸nsusthmtwn, stasÐmwn mh, tetragwnik c morf c, angontec to prìblhma eic thnlÔshn klassik¸n exis¸sewn thc kin sewc. EpÐshc ja epekteÐnwmen thn mèjodon kaieic thn sqetikistik n kbantomhqanik n, ènja ja eÔrwmen ton advance propagator touhlektromagnhtikoÔ pedÐou. Ja epekteÐnomen tèloc thn Wigner parstashn eic thnsqetikistik n kbantomhqanik n, ènja ja orÐswmen kai ja lÔswmen thn sqetikistik nexÐswshn tou Wigner.

H paroÔsa diatrib perilambnei tèssara keflaia. Eic to pr¸to keflaionperigrfontai ai treic parastseic thc kbantomhqanik c. Eidikìteron anafèretai hklassik kai h kbantomhqanik ènnoia thc katastsewc enìc sust matoc kaj¸c kaiai antÐstoiqai exis¸seic thc kin sewc. Perigrfetai h parstasic tou Wigner kaiapodeiknÔetai h metatrop thc exis¸sewc kin sewc tou Heisenberg eic thn exÐsw-shc tou Wigner, th bohjeÐa twn telest¸n tou Bopp. AkoloÔjwc anaptÔssetaih parstasic tou Feynman, dÐdwmen thn exÐswshn thn opoÐan ikanopoieÐ o prop-agator, kaj¸c kai wrismènac idiìthtac autoÔ. Anafèromen epÐshc thn qrhsimìth-ta thc parastsewc aut c kai dÐdomen tèloc thn ènnoian tou path integral. Eicto deÔtero keflaion perigrfomen wrismènac mejìdouc upologismoÔ twn propa-gators dia stsima kai mh stsima sust mata. Melettai h proseggistik mè-jodoc, kaj¸c kai h mèjodoc upologismoÔ dia twn oloklhrwmtwn thc kin sewc.

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Diexodik¸teron melettai mia mesoc lÔsic dia ton propagator, thn opoÐan kaiqrhsimopoioÔmen dia touc upologismoÔc mac. Eic to trÐton keflaion anaptÔs-sontai orismèna jewr mata diatxewc twn telest¸n. AnaptÔssetai o telest cexp

k1a

2 + k2a+2 + k3 (aa+ + a+a)

eic kanonik n morf n kai dÐdomen mia mmeson

efarmog n eic thn statistik n. AkoloÔjwc eurÐskomen touc propagators wrismènwnmh stsimwn susthmtwn. Mèroc tou kefalaÐou autoÔ apetèlese thn dhmosÐeushneic to periodikìn Physica Scripta vol. 18 13 – 17 en¸ to upìloipon einai mia epèktasicthc en lìgw dhmosieÔsewc dia thn perÐptwsin qronik¸c exartwmènwn telest¸n touHamilton. Eic to tètarton keflaion orÐzetai o sqetikistikìc Wigner telest c touopoÐou eurÐskomen tìson tac idiosunart seic ìson kai tac idiotimc, ai opoÐai eÐnaih diafor twn idiotim¸n dÔo exis¸sewn tou Dirac. H pargrafoc aut apetèlesenthn dhmosÐeushn eic to periodikìn Lettere al Nuovo Cimento vol. 18, No 11, 1997.Tèloc dia thc anaptuqjeÐshc mejìdou, upologÐzomen ton sqetikistikìn propagatortou hlektromagnhtikoÔ pedÐou.

2 KbantomhqanikaÐ Parastseic

2.1 Shrodinger parstasic

Ac jewr swmen èna sÔsthma to opoÐon perigrfetai apì thn klassik sunrthshH(q, p, t) tou Hamilton. Ta p kai q eÐnai to sÔnolon twn kanonik¸n suntetagmènwnkai orm¸n kai H(q, p, t) eÐnai h enèrgeia tou sust matoc h opoÐa dÐdetai wc sunrth-sic aut¸n kai pijan¸c kai tou qrìnou. H sunrthsic aut wc gnwstìn lambnetaimèsw thc sunart sewc L(q, q, t) tou Lagrange. Eic thn klassik n mhqanik n touHamilton to prìblhma to opoÐon melettai èqei wc ex c:

Dedomènhc thc jèsewc q0 kai thc orm c p0 tou sust matoc dia kpoian arqik nqronik n stigm n t0 na eurejeÐ h katstasic autoÔ di ìlouc tou metèpeita qrìnouc.ToÔto shmaÐnei, na eurej h troqi q(t) tou sust matoc h opoÐa ikanopoieÐ tacarqikc sunj kac q(t0) = q0, p(t0) = p0.

To prìblhma sunÐstatai kat' ousÐan eic thn lusin twn kanonikwn diaforik¸nexis¸sewn tou Hamilton [1].

p = −∂H

∂q, q =

∂H

∂p(2)

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ApeikonÐzoume tèloc tac sunart seic q(t) kai p(t) eic ton q¸ron twn p kai q, oopoÐoc onomzetai q¸roc twn fsewn kai oÔtw eÐmeja eic jèsin na gnwrÐzomen anpsan stigm n thn jèsin kai thn orm n tou sust matoc.

Eic thn kbantomhqanik n to anlogon prìblhma tÐjetai wc akoloÔjwc.Dedomènhc thc arqik c katastsewc ψ(q) dia kpoian qronik n stigm n t0 zhteÐ-

tai h katstasic ψ(q, t) di ìlouc touc metèpeita qrìnouc, h opoÐa na ikanopoi thnarqik n sunj khn ψ(q, t0) = ψ0(q). Eic thn perÐptwsin aut n h dunamik tou prob-l matoc perigrfetai me thn bo jeian thc exis¸sewc tou Shrodinger.

ih∂

∂tψ(q, t) = Hψ(q, t) (3)

O telest c H sqetÐzetai me thn klassik n sunrthsin tou Hamilton mèsw thcsqèsewc

H = H

(−ih

∂

∂q, q, t

)(4)

En o telest c tou Hamilton den exarttai analutik¸c ek tou qrìnou, h lÔsicthc exis¸sewc tou Shrodinger h opoÐa ikanopoieÐ kai thn arqik n sunj khn eÐnai thcmorf c:

ψ(q, t) = exp−(i/h)Ht

ψ0(q) (5)

O ekjetikìc telest c thc sqèsewc aut c onomzetai telest c exelÐxewc kai dÐdeithn qronik exèlixin thc kumatosunart sewc. DÔnatai eukìlwc na deiqj ìti hsqèsic aut grfetai kai upì thn akìloujon oloklhrwtik n morf n

ψ(q, t) =∫

K(q, t; q ′, 0)ψ0(q ′)dq ′ (6)

'Enjak(q, t; q ′, 0) = exp

−(i/h)Ht

δ(q − q ′) (7)

O pur n K onomzetai propagator tou sust matoc kai ja exetasj leptomerèsteroneic llhn pargrafon.

Wc gnwstìn h kbantomhqanik qreizetai to misu tou arijmoÔ twn anexart twnmetablht¸n ènanti thc klassik c mhqanik c. ToÔto èqei wc sunèpeian h kumato-sunrthsic na eÐnai sunrthsic mìnon twn suntetagmènwn.

Eic mian llhn parstasin to fusikì sÔsthma perigrfetai apì mian kumato-sunrthsin Φ(p, t) twn orm¸n. Ai dÔo kumatosunart seic sundèontai metaxÔ twnmèsw metasqhmatismoÔ Fourier. Eic thn parstasin aut n o telest c tou Hamiltonthc exis¸sewc tou Shrodinger sundèetai me thn klassik n sunrthshn tou Hamiltonmèsw thc akoloÔjou sqèsewc:

H = H

(p, ih

∂

∂p, t

)(8)

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2.2 Wigner parstasic

H ènnoia thc paragwgÐsewc enìc fusikoÔ megèjouc wc proc ton qrìnon, den dÔ-natai na orisjeÐ kat ton sun jh trìpon thc klassik c dunamik c. Eic thn kban-tomhqanik n orÐzomen wc pargwgon ˆF tou megèjouc F to mègejoc tou opoÐou hmèsh tim isoÔtai proc thn pargwgon thc mèshc tim c < F > toi ex orismoÔ èqoume

< ˆF >=(< F >

)˙ (9)

Lambnontec up' ìyin thn exÐswsin Shrodinger h anwtèrw tautìthc th bohjeÐa thcsqèsewc (8), dÐdei thn akìloujon exÐswsin thc kin sewc

dF

dt=

∂F

∂t=

1ih

[F , H] (10)

O antimetajèthc dÔo telest¸n orÐzetai wc ex c:

[A, B] = AB − BA (11)

Wc klassikì pardeigma anarèromen ton antimetajèthn twn telest¸n p kai qthc orm c kai thc jèsewc

[qi, qj ] = [pi, pj ] = 0, [qi, pj ] = ihδij (12)

Dexi thc teleutaÐac sqèsewc uponnoeÐtai o monadiaÐoc telest c. EpÐ plèon hsqèsic aut sundèetai amèswc me thn arq abebaiìthtoc tou Heisenberg.

Eic thn perÐptwsin kat thn opoÐan o telest c tou Hamilton den exarttaianalutik¸c ek tou qrìnou, h lÔsic thc exis¸sewc kin sewc èqei wc ex c:

F (t) = exp(i/h)Ht

F

(q,−ih

∂

∂q

)exp

−(i/h)Ht

(13)

H exÐswsic kin sewc tou telestoÔ F dÔnatai na eurej ekìlwc paragwgÐzontec thnsqèsin (13)

ih∂F

∂t= [F , H] (14)

Kat sunèpeian h mèsh tim tou megèjouc F dÐdetai ek thc sqèsewc

< F >=∫

ψ∗0(q)F (t)ψ0(q)dq (15)

Eic thn kbantomhqanik n diakrÐnomen dÔo eÐdh susthmtwn. Ta kajar sust ma-ta ta opoÐa dÔnantai na parastajoÔn apì kumatosunart seic kai ta meikt sust ma-ta ta opoÐa perigrfontai apì ton telest n thc m trac puknìthtoc. H parstasic

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aut thc kbantomhqanik , mèsw tou telestoÔ thc m trac puknìthtoc eÐnai idiaitèrwcqr simoc dia sust mata poll¸n swmatidÐwn dia sust mata dia ta opoÐa èqomenolÐgac plhroforÐac. H qronik exèlixic tou en lìgw telestoÔ, eic thn perÐptwsintwn stasÐmwn katastsewn, dÐdetai ek thc sqèsewc (14) kai h statistik mèsh tim enìc parathrhsÐmou eÐnai to Ðqnoc

< A >= Tr(ρA) (16)

H basik exÐswsic kin sewc (14) thc en lìgw parastsewc, dunmei basik¸n sqè-sewn thc jewrÐac twn telest¸n [2] dÔnatai na graf

ihd

dtρ(p, q, t) =

H

(p + ih

∂

∂q, q − ih

∂

∂p, t

)− H(p, q, t)

ρ(p, q, t) (17)

H exÐswsic aut eÐnai dÔskolon na luj genik¸c, dedomènou ìti ai sunart seicρ(p, q, t) eÐnai sunart seic telest¸n. Shmei¸nomen ìti h pargwgoc sunart sewctelest¸n [3] wc proc kpoion telest n kpoian parmetron dÐdetai ek thc sqèsewc:

d

dtF (A(t)) =

∫ 1

0dλDF (Aλ)

dA

dt(18)

'Enja to D dhl¸nei parag¸gisin wc proc to ìrisma kai o telest c Aλ epidrwc akoloÔjwc

AλB = A− λ[A, B] (19)

Sumf¸nwc proc tou Bopp kai Kubo[[4], [5]], h telestik diaforik exÐswsic (17)dÔnatai na metatrap eic mÐan sun jh diaforik n exÐswsin me thn antikatstasintwn telest¸n p kai q me touc ktwji telestc:

P = p− ih

2∂

∂q, Q = q +

ih

2∂

∂p(20)

Epidr¸ntec telik¸c ek dexi¸n me ton monadiaÐon telest n, lambnomen thn exÐsw-shn

ihd

dtf(p, q, t) =

H(P , Q, t)−H(P ∗, Q∗, t)

f(p, q, t) (21)

H anwtèrw exÐswsic apoteleÐ thn exÐswsin Wigner. H lÔsic thc exis¸sewc eÐnai hkatanom Wigner [6] h opoÐa sundèetai me tac lÔseic thc exis¸sewc tou Shrodingermèsw thc sqèsewc:

f(p, q, t) = h−3∫ ∞

−∞e

ipτh ψ∗

(q +

τ

2

)ψ

(q − τ

2

)dτ (22)

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H statistik mèsh tim enìc parathrhsÐmou megèjouc eic thn parstasin aut n dÐde-tai upì morf n oloklhr¸matoc paromoÐou me ekeÐnou thc klassik c pijanojewrÐaceic ton q¸ron twn fsewn

< A >= TrρA =∫

dpdqA(p, q, t)f(p, q, t) (23)

'Enja A(p, q, t) eÐnai o Weyl [7] metasqhmatismìc tou telestoÔ A(p, q, t) toi:

A(p, q, t) = (2π)−2∫

dτdθe−i(θq+τp)Tr[A(p, q, t) exp i(θq + τ p)

](24)

kai f(p, q, t) h katanom Wigner.DÔnatai na apodeiqj ìti h katanom Wigner eÐnai o Weyl metasqhmatismìc tou

telestoÔ thc m trac puknìthtoc.H sunrthsic Wigner eÐnai pragmatik kai kanonikopoihmènh eic thn monda, al-

l den eÐnai jetik¸c orismènh dia orismèna kbantomhqanik sust mata. Ta oloklh-r¸mata thc sunart sewc aut c wc proc tac suntetagmènac tac ormc eÐnai jetik¸corismèna kai wc ek toÔtou dÔnantai na jewrhjoÔn wc puknìthtec pijanìthtoc. DÔ-natai na deiqj eukìlwc, efarmìzontec thn anisìthta tou Shwarts eic thn èkfrasin(22) ìti h sunrthsic tou Wigner eÐnai apolÔtwc fragmènh ek twn nw

|f(p, q, t)| ≤ (2/h)3 (25)

Epeid h sunrthsic f(p, q, t) eÐnai kanonikopoihmènh èpetai apì thn anisìthta aut ìti h f(p, q, t) eÐnai diforoc tou mhdenìc eic ènan ìgkon tou q¸rou twn fsewnmegalÔteron tou (h/2)3. To gegonìc autì shmaÐnei ìti h sunrthsic tou Wigner dendÔnatai na entopisj wc proc p kai q kai toÔto eÐnai mia mesoc sunèpeia thc arq cthc abebaiìthtoc. An kai h sunrthsic katanom c tou Wigner den eÐnai jetik¸corismènh, en toÔtoic h parstasic aut eÐnai idiaitèrwc qr simoc diìti dÐdei miansusthmatik n mèjodon analÔsewc twn fusik¸n megej¸n eic dunmeic tou h. Kat'autìn ton trìpon dunmeja na lbwmen tac kbantikc diorj¸seic twn klassik¸nsqèsewn. En lbwmen to ìrion thc exis¸sewc (21) tou h teÐnontoc proc to mhdènkal gomen eic thn gnwst n exÐswsin thc klassik c dunamik c tou Liouville, opìte oantimetajèthc metapÐptei eic thn agkÔlhn tou Poison. Mia susthmatik melèth thckatanom c aut c èqei dojeÐ eic thn bibliografÐan [8] ìpou apedeÐqjh ìti ai idiotimaÐthc exis¸sewc tou Wigner, dia thn perÐptwsin twn stasÐmwn katastsewn eÐnai hdiafor twn idiotim¸n thc antistoÐqou exis¸sewc tou Shrodinger.

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2.3 Feynman parstasic

To 1932 o Dirac me thn ergasÐan tou epÐ tou rìlou twc mhqanik c tou Lagrangeeic thn kbantik n jewrÐan, èjese tac bseic dia mian nèan parstasin thc kban-tomhqanik c h opoÐa aneptÔqjh argìteron to 1943 upì tou Feynman [10]. H kurÐaidèa thc jewrÐac aut c sunÐstatai eic thn melèthn twn lÔsewn twn exis¸sewn thckbantomhqanik c antÐ twn idÐwn twn exis¸sewn. En h kumatosunrthsic ψ(q1, t1)eÐnai gnwst dia mÐan dedomènhn qronik stigm t1 aÔth dÔnatai na eurej di ìlouctouc metèpeita qrìnouc. JewroÔmen ìti kat thn qronik n stigm n t1 kje shmeÐontou q¸rou eÐnai mia phg sfairik¸n kumtwn ta opoÐa diadÐdontai ek tou shmeÐouq1. To kÔma to opoÐon fjnei eic to shmeÐon q2 kat thn qronik n stigm n t2 prèpeina eÐnai anlogon proc to arqikìn pltoc ψ(q1, t1), h dè stajer analogÐac eÐnai hK(q1, t1; q2, t2). Sumf¸nwc proc thn arq n thc epiprosjèsewc to sunolikìn kÔmadÐdetai ek thc sqèsewc

ψ(2) =∫

K(2, 1)ψ(1)dq1 (26)

'Enja dia suntomÐan qrhsimopoioÔmen ton sumbolismìn j = 1, 2 dia to shmeÐon (qj , tj).O pur n K paristnei mian didosin tou sunìlou twn suntetagmènwn ek tou

shmeÐou (q1, t1) proc to shmeÐon (q2, t2) kai onomzetai propagator tou sust matoc.En h sunrthsic K eÐnai gnwst dunmeja na eÔrwmen thn exèlixin miac oiasd potearqik c katastsewc wc proc ton qrìnon. Wc ek toÔtou eÐnai isodÔnamoc procthn pl rh lÔsin thc exis¸sewc tou Shrodinger. 'Ena kbantomhqanikìn sÔsthmaperigrfetai ex Ðsou kal¸c eÐte gnwrÐzontec thn sunrthsin K eÐte ton telest ntou Hamilton ek tou opoÐou eÐnai dunatìn na exaqj aut . Sunep¸c h K mac parèqeiìlac tac plhroforÐac tac opoÐac apaiteÐ h kbantik je¸rhsic thc fÔsewc.

H pijanìthc na eurej to sÔsthma eic thn katstasin X(2) thn qronik n stigm nt2 ìtan kat thn qronik n stigm n t1 toÔto eurÐsketo eic thn katstasin ψ(1) dÐdetaiex tou tetrag¸nou tou oloklhr¸matoc

∫X∗(2)k(2, 1)ψ(1)dq1dq2 (27)

Ta probl mata ta opoÐa antimetwpizomen me thn parstasn aut n tou FeynmaneÐnai sun jwc probl mata skedsewc. To fusikì prìblhma tÐjetai wc ex c:

'Ena eleÔjeron swmtion skedzetai eic kpoion shmeÐon apì èna dunamikìn.ZhteÐtai h kumatosunrthsic tou met thn diadikasÐan thc skedsewc. To en lìgwswmtion, eic to ap¸teron pareljìn tou, perigrfetai apì mÐan kumatosunrthsintÔpou Gauss thc morf c:

ψ(q, 0) =1

(πδ2)3/4exp

ipq

h− q2

2δ2

(28)

14

H sqèsic aut apoteleÐ thn ψ(1) thc (26) dia t1 = 0. En upojèswmen ìti opropagator tou sust matoc èqei upologisj tìte h katstasic ψ(2) tou swmatÐoueÐc to ap¸teron mèllon tou dÐdetai ek thc sqèsewc (26).

Kumatosunart seic thn morf c tou kumatodèmatoc parousizoun thn elqisth-n diasporn toi h sqèsic abebaiìthtoc tou Heisenberg isqÔei me to Ðson. Katsunèpeian ai katastseic autaÐ eÐnai plhsièsteron proc tac klassikc katastseic.Autìc eÐnai o lìgoc dia ton opoÐon protim¸ntai wc arqikaÐ kumatosunart seic.

Ed¸ to endiafèron mac entopÐzetai eic kumatosunart seic ai opoÐai exelÐssontaiwc proc ton qrìnon, kai ìqi dia stasÐmouc energeiakc idiosunart seic toi stsimakÔmata (parstasic tou Shrodinger).

Dia na antilhfj¸men perissìteron thn fusik n shmasÐan thc sunart sewc K,ja anazht swmen thn exÐswsin thn opoÐan ikanopoieÐ. ParagwgÐzomen thn sqèsin(26) wc proc t2 kai th bohjeÐa thc exis¸sewc tou Shrodinger thn opoÐan ikanopoieÐh kumatosunrthsic lambnomen telik¸c thn ktwji exÐswsin.

ih

∂

∂t2− H(2)

k(2, 1) = ihδ(2, 1) (29)

ToÔto shmaÐnei ìti o propagator eÐnai h Green sunrthsic thc exis¸sewc touShrodinger kai wc ek toÔtou paristnei thn antapìkrisin tou sust matoc eic mÐanexwterik c allag n.

Eic thn mh sqetikistik n kbantomhqanik n o propagator lambnetai Ðsoc proc tìmhdèn dia qrìnouc t2 < t1. EpÐ plèon dia t2 → t1 metapÐptei eic thn dèlta sunrthsinδ(q2 − q1).

Shmei¸nomen tèloc ìti o en lìgw pur n diadìsewc prèpei na ikanopoi thnsqèsin:

k(2, 1) =∫

K∗(2, 3)K(3, 1)dq3 (30)

dit1 ≤ t ≤ t2 (31)

Kaj¸c epÐshc kai thn akìloujon sqèsin thc unitarity∫ ∞

−∞K∗(x, t; x′1, t1)K(x1, t1;x, t)dx = δ(x′1 − x1) (32)

EÐnai profanèc ek thc sqèsewc (30) ìti o K(2, 1) apì thn qronik n stigm n t1proc thn t2 dÔnatai na eurejeÐ ek twn K(2, 3) kai K(3, 1) apì tac qronikc stigmct3 proc thn t2 kai apì thn t1 proc thn t3 antistoÐqwc, prosjètontec epnw eicìlac tac dunatc endiamèsouc jèseic q3 eic ton qrìnon t3. EpÐshc exarmìzontecepanalhptik¸c thn sqèsin aut dunmeja na upologÐswmen ton pur na diadìsewcdia peperasmèna diast mata t2− t1, ek twn pur nwn dia apeirostìc qrìnouc t2− t1.H parat rhsic aut odhgeÐ telik¸c eic to path integral.

15

O pur n K dÐdetai kat ton Feynman ek tou akoloÔjou path integral.

k(2, 1) =∫

exp (i/h)S (33)

H olokl rwsic gÐnetai epnw eic ìlouc tou dunatoÔc drìmouc ek tou shmeÐou q1 =q1(t) proc to shmeÐon q2 = q2(t). H sunrthsic S eÐnai to olokl rwma thc klassik cdrsewc kai dÔnatai na upologisj eukìlwc th bohjeÐa thc sunart sewc Lagrangetou sust matoc.

Kat' autìn ton trìpon h klassik mhqanik tou Lagrange genikeÔetai upì thnmorf n aut n twn path integrals eic thn kbantomhqanik n. En toÔtoic h efarmog thc sqèsewc (33) dia ton upologismìn propagators èqei epitÔqei eic olÐgac peript¸-seic [11] lìgw twn analutik¸n duskoli¸n tac opoÐac parousizoun ta oloklhr¸mataaut. EpÐ plèon sust mata me analutik n exrthsin twn sunart sewn Lagrange ektou qrìnou èqoun elqista melethj [12].

3 Mèjodoi UpologismoÔ twn Propagators

3.1 Stsimec katastseic

'Ena sÔsthma eurÐsketai eic stsimon katstasin, ìtan o telest c tou HamiltoneÐnai anexrthtoc tou qrìnou. Eic thn perÐptwsin aut n o pur n eÐnai sunrthsicthc diaforc t2− t1 kai h kumatosunrthsic talantoÔtai me mia orismènh suqnìthtah opoÐa eÐnai anlogoc thc enèrgeiac tou sust matoc. H pijanìthc na eurej toswmtion eic kpoian perioq n eÐnai anexrthtoc tou qrìnou.

Genikìteron stsimai eÐnai ai katastseic dia ta opoÐac h puknìthc pijanìthtocψ∗ψ kai h puknìthc reÔmatoc pijanìthtoc

~J =ih

2m

(ψ~∇ψ∗ − ψ∗~∇ψ

)

eÐnai anexrthtoi tou qrìnou.Ai stsimai katastseic perigrfontai apì kumatosunart seic thc morf c

ψ(q, t) = exp −(i/h)EtU(q) (34)

Eisgontec thn kumatosunrthsin aut n eic thn exÐswsin tou Shrodinger eurÐskomenthn akìloujon exÐswsin idiotim¸n ek thc opoÐac dunmeja na upologÐswmen tac

16

sunart seic U(q) kai thn stajern E.

H

(q,−ih

∂

∂q

)U(q) = EU(q) (35)

En o telest c H dèqetai èna pl rec kai orjokanonikìn sÔnolon idiosunart -sewn Un(q) me antistoÐqouc idiotimc En toi:

∫U∗

n(q)Um(q)dq = δnm,∑n

U∗n(q ′′)Uu(q ′) = δ(q ′′ − q ′) (36)

Tìte o propagator tou sust matoc dÔnatai na grafeÐ upì thn akìloujon analutik nmorf n.

k(q, t; q ′, 0) =∑n

U∗n(q)Uu(q ′)e−iEnt/h t > 0

K(q, t; q ′, 0) = 0 t < 0 (37)

H sqèsic aut prokÔptei eukìlwc en antikatast swmen thn èkfrasin (36) thcδ sunart sewc eic thn sqèsin (7).

ParathroÔme ìti dia tou aploÔ tupikoÔ metasqhmatismoÔ it/h → b h sqèsic(refsx44) dÐdei thn statistik n m tran puknìthtoc dia thn kanonik olothta. Toktwji olokl rwma,

z(b) =∫

K(q,−ihb ; q, 0)dq =∑n

e−bEn (38)

eÐnai to jroisma katastsewn enìc kbantikoÔ statistikoÔ sust matoc [13].

3.2 Mh stsimai katastseic

'Ena sÔsthma eurÐsketai eic mh stsimon katstasin ìtan o telest c HamiltonautoÔ perièqei analutik¸c ton qrìnon.

H plèon gnwst mèjodoc dia thn melèthn aut¸n twn susthmtwn eÐnai h qronik¸cexartwmènh jewrÐa diataraq¸n [14].

Upojètomen ìti o telest c Hamilton tou sust matoc dÔnatai na analuj wcex c:

H = H0 + V (q, t) (39)

H diataraq V (q, t) jewreÐtai mikr sugkrinìmenh me to H0.

17

Upojètomen epiplèon ìti o pur n K0(2, 1) thc kin sewc dia ton H0 eÐnai gnwstìc(gia pardeigma to H0 dÔnatai na eÐnai tetragwnik c morf c kai anexrthton touqrìnou). Tìte o propagator dèqetai thn anlusin

K(2, 1) = K0(2, 1) + K ′(2, 1) (40)

AntikajistoÔmen ton pur na (40) eÐc thn exÐswsin (29) kai lambnomen

ih∂K ′

∂t2−H0K

′ = V K (41)

H Green sunrthsic tou telestoÔ tou pr¸tou mèlouc thc (41) eÐnai o gnwstìcpropagator K0(2, 1), èpomènwc èqomen:

K ′(2, 1) = − i

h

∫ t2

t1

∫

VK0(2, 3)V (3)k(3, 1)d3 (42)

ènja d3 = dq3dt3.Ta ìria tou oloklhr¸matoc wc proc t prokÔptoun ek thc asuneqeÐac tou K(2, 1)

dia t2 − t1 → 0.En prosjèswmen kai to K0(2, 1) eurÐskomen thn oloklhrwtik n exÐswsin

K(2, 1) = K0(2, 1)−∫

K0(2, 3)V (3)K(3, 1)d3 (43)

h opoÐa dÔnatai na luj dia thc epanalhptik c mejìdou.Mia analutik èkfasic wc h sqèsic (37) dÔnatai na eurej kai eic thn perÐptwsin

twn mh stasÐmwn katastsewn me thn bo jeian twn oloklhrwmtwn thc kin sewc.'Enac ermhtianìc telest c onomzetai olokl rwma stajer thc kin sewc en

h mèsh tim tou eÐnai anexrthtoc tou qrìnou [12]. H idiìthc aut eÐnai isodÔnamocme thn tautìthta

dI(t)dt

=∂I(t)∂t

+1ih

[I(t), H] = 0 (44)

En o telest c tou Hamilton exarttai analutik¸c ek tou qrìnou tìte oÔtoc deneÐnai plèon mia stajer thc kin sewc. DÔnameja apl¸c na eÐpwmen ìti apoteleÐ ènangenn tora thc kin sewc.

Kje olokl rwma thc kin sewc dÔnatai na ekfrasj me thn bo jeia tou teles-toÔ qronik c exelÐxewc U(t) wc akoloÔjwc

I(t) = U(t)I0U−1(t) (45)

I0 eÐnai ènac telest c anexrthtoc tou qrìnou. Kat sunèpeian h exÐswsic (44)èqei lÔsin. Dia kje sÔsthma me N bajmoÔc eleujerÐac uprqoun 2N anexrthta

18

oloklhr¸mata thc kin sewc.

qj(t) = U(t)qjU−1(t)

pj(t) = U(t)

(−ih

∂

∂qj

)U−1(t) j = 1, 2, · · · , N (46)

Ta oloklhr¸mata thc kin sewc èqoun thn ex c idiìthta:En Ψ eÐnai mia lÔsic thc exis¸sewc Shrodinger tìte dia kje telest I o

opoÐoc ikanopoieÐ thn exÐswsin (44) h sunrthsic Ψ = IΨ eÐnai epÐshc lÔsic thcidÐac exis¸sewc. Oi idiosunart seic twn oloklhrwmtwn thc kin sewc dÔnantai naeklegoÔn eic trìpon ¸ste na ikanopoioÔn thn exÐswsin Shrodinger.

'Estw ìti Ψn(q, t) eÐnai èna pl rec kai orjokanonikìn sÔnolon idiosunart -sewn tou I(t) me antistoÐqouc idiotimc λn

I(t)Ψn(q, t) = λnΨn(q, t) (47)

ApodeiknÔetai [16] tìte ìti o propagator dèqetai mia anlusin thc morf c

K(2, 1) =∑n

exp i [an(t2)− an(t1)]Ψ∗(q1, t1)Ψn(q2, t2) (48)

Oi fseic an(t) orÐzontai ek thc sqèsewc

ihd

dtan =

∫Ψ∗

n(q, t)[ih

∂

∂t− H(t)

]Ψn(q, t)dq (49)

DÔnatai epÐshc na deiqj ìti h lÔsic Ψn(q, t) thc exis¸sewc Shrodinger dÐdetaiek thc sqèsewc

Ψn(q, t) = eian(t)Ψn(q, t) (50)

kai kat sunèpeian o propagator grfetai

K(2, 1) =∞∑

n=0

Ψ∗n(q1, t1)Ψn(q2, t2) (51)

Dia thn eidik n perÐptwsin twn stasÐmwn katastsewn o analloÐwtoc I(t) sum-pÐptei me ton telest n tou Hamilton kai ta Ψn eÐnai anexrthta tou qrìnou. AiidiotimaÐ λn eÐnai ai energeiakaÐ idiotimaÐ En kai oi fseic an(t) dÐdontai ek thcsqèsewc an = −Ent/h. Kat sunèpeia oi sqèseic (43) kai (46) metapÐptoun eic thngnwst n èkfrasin (37) tou propagator twn stasÐmwn katastsewn.

H mèjodoc aut upologismoÔ twn propagators mèsw twn oloklhrwmtwn thckin sewc èqei epitÔqei eic wrismèna probl mata, wc p.q. h kbantik perigraf thctrib c [[17] [23]].

19

En toÔtoic h mèjodoc qreizetai tac idiosunart seic Ψn(q, t) tac Ψn(q, t), oupologismìc twn opoÐwn dia telestc Hamilton exarthmènouc ek tou qrìnou eÐnaigenik¸c dÔskoloc. Eic thn epomènhn pargrafon ja prospaj swmen na lÔswmenap' eujeÐac thn exÐswsin (29).

3.3 Mia mesoc lÔsic

Mia mesoc tupik lÔsic thc exis¸sewc (29) prokÔptei eukìlwc en pollaplasi-swmen thn exÐswsin aut n me ton antÐstrofon telest n tou dexioÔ mèlouc

K(2, 1) = ih

(ih

∂

∂t2− H(2)

)−1

δ(2, 1) (52)

Prosjètomen eic ton paranomast n thc sqèsewc aut c mian peirost n fantastik posìthta iε kai lambnomen to ìrion tou ε → 0+

K(2, 1) = ih limε→0+

(ih

∂

∂t2− H(2)± iε

)−1

δ(2, 1) (53)

Kat ton Feynman [24] antikajistoÔmen tac δ sunart seic me ton gnwstìnFourier metasqhmatismìn twn kai met thn epÐdrasin twn telest¸n epÐ twn ekjetik¸nsunart sewn lambnomen

K(2, 1) = ih limε→0+

∫ ∞

−∞e−ik(q2−q1)−iω(t2−t1)

hω −H(k)± iεdkdω (54)

H olokl rwsic gÐnetai epnw eic ton pragmatikìn èxona kai h prìsjesic tou ìrou±iε èqei wc sunèpeia na metafèrh touc pìlouc, en uprqoun, tou oloklhr¸matocelafr¸c epnw ktw tou pragmatikoÔ xonoc. Met thn oloklhrwsin lambnomento ìrion tou ε → 0+.

O propagator me to shmeÐon + onomzetai advanced epeid paristnei mia di-dosin proc touc metèpeita qrìnouc en¸ ekeÐnoc me ton shmeÐon− onomzetai retarded,epeid paristnei mian didosin pÐsw wc proc ton qrìnon.

Ed¸ grfomen ton telest n thc sqèsewc (53) upì thn akìloujon oloklhrwtik nmorf n, lambnontec kat' arqc to shmeÐon meÐon.

K(2, 1) =∫ ∞

0dt exp

− i

ht

(ih

∂

∂t2− H(t2)− iε

)δ(2.1) (55)

Prgmati h olokl rwsic dÐdei eic to 0 thn sqèsin (53), en¸ eic to peiron dÐdeito mhdèn lìgw tou ìrou exp −tε/h o opoÐoc dia t → +∞ dÐdei to 0 akìmh kai

20

dia apeirost ε/h. H prìsjesic tou ìrou −iε èqei wc sunèpeian thn sÔgklisin touoloklhr¸matoc.

AkoloÔjwc upojètomen ìti o ekjetikìc telest c dèqetai thn anptuxin

exp

t

(∂

∂t2+

i

hH(2)− ε

h

)= U(t)et ∂

∂t2 e−tε/h (56)

O telest c U(t) eÐnai mia sunrthsic twn telestwn p, q kai den perièqei parag¸gisinwc proc ton qrìnon. Eisgwmen thn anptuxin (56) eic thn sqèsin (55) kai lam-bnomen

K(2, 1) =∫ ∞

0dte

t ∂∂t2 δ(t2 − t1)U(t)δ(q2 − q1)e−tε/h

=∫ ∞

0dtδ(t2 − t1 + t)U(t)δ(q2 − q1)e−tε/h (57)

Eic thn perÐptwsin ènja t2 − t1 < 0 to ìrisma thc δ(t2 − t1 + t) sunart sewcmhdenÐzetai entìc twn orÐwn thc oloklhr¸sewc prgmati t = t1 − t2 > 0. Toolokl rwma lambnontec telik¸c kai to ìrion ε → 0+ gÐnetai

K(2, 1) = U(t2, t1)δ(q2 − q1), t2 − t1 < 0 (58)

En t2 − t1 > 0 to ìrisma thc δ(t2 − t1 + t) den mhdenÐzetai entìc twn orÐwn touoloklhr¸matoc kai to olokl rwma gÐnetai mhdèn.

K(2, 1) = 0, t2 − t1 > 0 (59)

Kat sunèpeian o propagator autìc eÐnai o retarded propagator.Lambnontec to shmeÐon + thc sqèsewc (53) aut dÔnatai na graf upì thn

akìloujon oloklhrwtik morf n

K(2, 1) =∫ 0

−∞exp

ii

h

(ih

∂

∂t2− H(2) + iε

)δ(2, 1) (60)

Prgmati h olokl rwsic dÐdei eic to mhdèn thn sqèsin (53) en¸ eic to meÐonpeiron dÐdei to mhdèn lìgw tou ìrou exp tε/h o opoÐoc dia t → −∞ gÐnetai mhdèn.O ekjetikìc telest c dèqetai mian anlusin paromoÐan thc (56) o de propagator katton Ðdion trìpon grfetai

K(2, 1) =∫ 0

−∞dtδ(t2 − t1 + t)U(t)δ(q2 − q1)etε/h (61)

Eic thn perÐptwsin t2− t1 < 0 to ìrisma thc δ(t2− t1 + t) mhdenÐzetai entìc twnorÐwn tou oloklhr¸matoc kai kat sunèpeian to olokl rwma mhdenÐzetai

K(2, 1) = 0 t2 − t1 < 0 (62)

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Eic thn perÐptwsin t2 − t1 > 0 uprqei mia tim tou t entìc twn orÐwn touoloklhr¸matoc dia thn opoÐan isqÔei h sqèsic t2− t1 + t = 0. Sunep¸c lambnontecto ìrion ε → 0+ to olokl rwma grfetai

K(2, 1) = U(t1, t1)δ(q2 − q1) t2 − t1 > 0 (63)

kai kat sunèpeian o propagator autìc eÐnai o advanced propagator.Eic thn klassik n kbantik n mhqanik n kanèna sÔsthma kai kanèna s ma den

diadÐdetai pÐsw wc proc ton qrìnon kai kat sunpeian o retarded propagator denèqei ènnoian.

Antijètwc eic thn sqetikistik n kbantik n mhqanik n h ènnoia tou redardedpropagator paramènei kai perigrfei katastseic arnhtik c energeÐac.

Dia na katano swmen perissìteron thn fusik n shmasÐan tou telestoÔ U(t) jaanazht swmen thn exÐswsin thn opoÐan ikanopoieÐ.

H pargwgoc thc sqèsewc (56) wc proc t dÐdei:

− ∂

∂tU(t) +

∂

∂t2U(t) = − i

hH(2)U(t) (64)

jewroÔmen akoloÔjwc ton metasqhmatismon

t = t1 − t2 t2 = t1 (65)

kai h exÐswsic (64) gÐnetai

∂

∂t2U(t1, t2) =

i

hH(2)U(t1, t2) (66)

EpÐshc ek thc sqèsewc (56) faÐnetai eukìlwc ìti o telest c U(t1, t2) ikanopoieÐ thnakìloujon oriak n sunj khn

U(t, t) = 1 (67)

toi o U(t1, t2) eÐnai o telest c qronik c exelÐxewc.Eic thn perÐptwsin twn stasÐmwn katastsewn ènja o telest c H(2) den perièqei

analutik¸c ton qrìnon, h lÔsic thc exis¸sewc aut c, h opoÐa ikanopoeÐ kai thnarqik n sunj khn lambnei thn akìloujon gnwst n morf n:

U(t1, t2) = e−ih(t2−t1)H(2) (68)

Dia tac mh stasÐmouc katastseic dunmeja na lbwmen mia proseggistik èk-frasin tou telestoÔ autoÔ metatrèpontec thn diaforik n exÐswsin me thn oriak nsunj khn (66) eic thn oloklhrwtik n exÐswsin

U(t1, t2) = 1− i

h

∫ t1

t2H(t)U(t, t2)dt (69)

22

H lÔsic thc exis¸sewc aut c me thn bo jeian thc epanalhptik c mejìdou dÐdetai ektou anaptÔgmatoc [25].

U(t1, t2) = 1− i

h

∫ t2

t1dtH(t) +

(−i)2

h2

∫ t2

t1dt3

∫ t2

t1dt4H(t3)H(t4) · · · (70)

apl¸c dia tou ktwji Dyson’s time ordered oloklhr¸matoc

U(t1, t2) = T exp− i

h

∫ t2

t1H(t)dt

(71)

'Enja prèpei na apokleÐswmen idizousec (singular) sunart seic H kai ja jewr -swmen monon sust mata gia ta opoÐa o telest c (71) uprqei pnta.

To sÔmbolo T paristnei ton qronologikìn telest n

TH(t2)H(t2) =

H(t1)H(t2) δια t1 < t2

H(t2)H(t1) δια t1 > t2

(72)

H anptuxic (71) tou telestoÔ qronik c exelÐxewc èqei qrhsimopoiejeÐ epituq¸ceic thn kbantik n jewrÐan, idiaitèrwc eic thn kbantik n hlektrodunamik n, dia nalbwmen proseggistikc ekfrseic gia ta plth pijanìthtoc.

Wc gnwstìn h exÐswsic (66) lÔetai akrib¸c eic olÐgac monon peript¸seic. EntoÔtoic dia mian omdan susthmtwn, stasÐmwn mh, eÐnai dunatìn na lbwmenakribeÐc lÔseic mllon na ekfrswmen tac kbantomhqanikc lÔseic th bohjeÐatwn antistoÐqwn klassik¸n. H txic aut sunÐstatai apì sust mata twn opoÐwnoi telestaÐ tou Hamilton eÐnai polu¸numa deutèrac txewc wc proc tac kanonikcsuzugeÐc metablhtc. Eic to epìmenon keflaion melet¸men ta anwtèrw sust matame thn bo jeian thc jewrÐac diatxewc twn telest¸n. Sugkekrimèna diatssomenton telest n qronik c exelÐxewc U(t) eic mian katllhlon morf n ikan na d¸shton propagator tou sust matoc kai ta oloklhr¸mata thc kin sewc me ènan aplìnkai eujÔn trìpon.

23

4 H Ditaxic twn Telest¸n kai EfarmogaÐthc JewrÐac Diatxewc eic thn mh Sqetik-istik n Kbantomhqanik n

4.1 Ditaxic twn Boson telest¸n

Oi Boson telestaÐ genèsewc kai exafanÐsewc a+ kai a dÔnantai na orisjoÔn diakje sÔsthma to opoÐon perigrfetai apì to zeÔgoc twn ermhtian¸n telest¸n p, qdia tou grammikoÔ sunduasmoÔ

a = (2h)−12 (λq + iλp) (73)

a = (2h)−12 (λq − iλp) (74)

'Enja to λ eÐnai mia pragmatik parmetroc.Oi en lìgw telestaÐ ikanopoioÔn thn sqèsin:

[a, a+] = 1 (75)

Onomzomen coherent katastseic ta idiodianÔsmata tou telestoÔ a kai ta sum-bolÐzoume me |α >.

a|α >= α|α >, < α|a+ =< α|α∗ (76)

Oi coherent katastseic eÐnai katastseic elaqÐsthc diasporc.Oi Boson telestaÐ a kai a+ den eÐnai ermhtianoÐ kai ai idiotimaÐ twn den eÐnai

en gènei pragmatikoÐ arijmoÐ. EpÐshc ai coherent katastseic den eÐnai orjog¸nioi.En toÔtoic sqhmatÐzoun èna pl rec sÔnolon katastsewn kai h sqèsic plhrìthtocekfrzetai dia thc sqèsewc:

∫|α >< α|d

2α

π= 1 (77)

enja 1 eÐnai o tautotikìc telest c kai h olokl rwsic gÐnetai eic to migadikìnepÐpedon.

Eic ta epìmena kje sunrthsic twn telest¸n a kai a+ orÐzetai apì mÐan anp-tuxin thc morf c:

f(a, a+) =∑

i,j,···,ka+i

aja+r · · · · ak (78)

24

ènja i, j, · · · , k eÐnai jetikoÐ akèraioi kai mhdèn.Dunmeja na qrhsimopoi swmen thn sqèsin antimetajèsewc (75) dia na diatxw-

men kje sunrthsin f(a, a+) eic oiand pote epijumit n morf . Ja lègomen ìti hsunrthsic f(a, a+) èqei graf upì kanonik n (mh kanonik n) morf n, en ìlai oidunmeic tou telestoÔ a eurÐskontai eic ta dexi (arister) ìlwn twn dunmewn toua+.

f(a, a+) = f (n)(a, a+) =∑

i,j

f(n)i,j a+i

aj (79)

f(a, a+) = f (a)(a, a+) =∑

i,j

f(a)i,j aja+i (80)

an kai f(a, a+) = f (n)(a, a+) = f (a)(a, a+) ai stajeraÐ anaptÔxewc f(n)i,j kai f

(a)i,j

eÐnai en gènei diforai. H sunrthsic f(a, a+) dÔnatai epÐshc na anaptuqj eicmian morf n pl rwc summetrik (Weyl morf ) wc proc a kai a+, h qrhsimìthc thcopoÐac èqei anaptuqj eic thn pargrafo 2. Tèloc wrismènai llai diatxeic èqounanaptuqj eic thn bibliografÐa. [[26] [29]]

Epeid h kanonik kai mh kanonik morf twn telest¸n eÐnai monadikaÐ, dunme-ja na orÐswmen mÐan amfimonos manton antistoiqÐan metaxÔ twn sunart sewn twntelest¸n f (n)(a, a+) kai f (a)(a, a+) kai twn sun jwn sunart sewn f (n)(α, α∗),f (a)(α, α∗) miac migadik c metablht c α. Di na lbwmen thn sunrthsin f (n)(α, α∗) thn f (a)(α, α∗) grfomen ton telest n f(a, a+) upì thn kanonik mh kanonik morf kai antikajist¸men touc telestc a kai a+ me touc migadikoÔc arijmoÔc α kaiα∗ antistoÐqwc. En sumbolÐswmen me N−1 kai A−1 tac antistoiqÐac autc, èqomen:

N−1 f(a, a+)

= N−1

∑

i,j

f(n)i,j a+i

aj = f (n)(α, α∗) (81)

A−1 f(a, a+)

= A−1

∑

i,j

f(a)i,j a+i

aj = f (a)(α, α∗) (82)

EpÐshc h kanonik kai mh kanonik morf thc sunart sewc f(a, a+) dÔnatai naeurej ek twn sqèsewn

f (n)(a, a+) = N

f

(α +

∂

∂α∗, α∗

)1

(83)

f (a)(a, a+) = A

f(α, α∗ − ∂

∂α)1

(84)

Oi sqèseic autaÐ metatrèpoun touc metajètac eic diaforikc exis¸seic [30]. EpÐshcai antistoiqÐai N−1 kai A−1 metatrèpoun ta Ðqnh twn telest¸n eic sun jh oloklhr¸-

25

mata.

Trρ(a, a+) =∫

ρ(a)(α, α∗)f (n)(α, α∗)d2α

π=

∫ρ(n)(α, α∗)f (a)(α, α∗)

d2α

π(85)

Eic ta dÔo autc sqèseic ègkeitai kai h qrhsimìthc thc jewrÐac thc diatxewctwn telest¸n. Ai antÐstoiqai sqèseic dia thn summetrik n antistoiqÐan eÐnai ai (21)kai (23). Ai treic aÔtai amfimonos mantoi antistoiqÐai sundèontai me tac coherentkatastseic (76).

4.2 Kanonik ditaxic tou telestoÔexp

k1a

2 + k2a+2 + k3(aa+ + a+a)

H ditaxic twn sunart sewn f(a, a+) eic kanonik n mh morf n eÐnai dunat di ìlactac sunart seic ai opoÐai dÔnantai na anaptuqjoÔn eic seirc dunmewn twn a kai a+.En toÔtoic, ektìc orismènwn peript¸sewn, wc p.q. ai tetragwnikaÐ morfaÐ, h ditaxictwn telest¸n eÐnai arket dÔskoloc. Ai mèjodoi ai opoÐai qrhsimopoi jhsan katkairoÔc [[31] [34]] faÐnetai ìti èqoun periorismènhn efarmog n. Di thn anptuxintou ekjetikoÔ telestoÔ f(a, a+) qrhsimopoioÔmen ed¸ thn mèjodon thc parametrik cparagwgÐsewc [35].

Prosjètomen ston telest n mÐan bohjhtik n parmetron b wc akoloÔjwc

f(a, a+, b) = expb[k1a

2 + k2a+2 + k3

(aa+ + a+a

)](86)

Upojètwmen ìti o telest c f(a, a+, b) dÔnatai na graf upì thn morf n

f(a, a+, b) = expf0(b) + f1(b)a+2

exp

f2(b)a+a

exp

f3(b)a2

(87)

Lìgw thc ktwji kanonik c anaptÔxewc

exp−f2a

+a

=∞∑

l=0

1l!

(e−f2 − 1

)la+l

al (88)

o telest c touc deutèrou mèlouc thc (87) eÐnai grammènoc upì thn kanonik n toumorf n. Sunep¸c apomènei na prosdiorÐsoume tac agn¸stouc sunart seic fi(b) aiopoÐai profan¸c ikanopoihjoÔn tac arqikc sunj kac

fi(0) = 0 (89)

26

Ek thc sqèsewc (86) apodeiknÔetai eukìlwc ìti h sunrthsic f(a, a+, b) ikanopoieÐthn diaforik n exÐswsin

f−1(a, a+, b)∂

∂bf(a, a+, b) = k1a

2 + k2a+2 + k3

(aa+ + a+a

)(90)

UpologÐzomen akoloÔjwc thn èkfrasin tou pr¸tou mèlouc thc (90) me thn bo jeianthc kanonik c anaptÔxewc (87)

f−1(a, a+, b)∂

∂bf(a, a+, b) = exp

−f3a

2

exp−f2a

+a [

∂f0

∂b+

∂f1

∂ba+2

]

expf2a

+a

expf3a

2

+ exp−f3a

2 [

∂f2

∂ba+a

]exp

f3a

2

+∂f3

∂ba2 (91)

To deÔteron mèloc thc (90) dÔnatai na aplopoihj th bohjeÐa thc gnwst c tautìth-toc.

eξAf(B)e−ξA = f

(B + ξ[A,B] +

ξ2

2![A, [A,B]] + · · ·

)(92)

h opoÐa isqÔei dia kje sunrthsin thc morf c f(B) =∑

n cnBn. Met apì oris-mènouc upologismoÔc lambnomen

f−1(a, a+, b)∂

∂bf(a, a+, b) =

∂f0

∂b+

∂f1

∂be−2f2

(a+ − 2f3a

)2 +

∂f2

∂b

(a+ − 2f2a

)a +

∂f3

∂ba2 (93)

Ek twn ekfrsewn (93) kai (90) lambnomen to ktwji diaforikìn sÔsthma.

∂f3

∂b= k1 + 4k3f3 + 4k2f

23

∂f1

∂b= k2e

2f2

∂f2

∂b= 2k3 + 4k2f3 f0 =

12f2 (94)

fi(0) = 0 (95)

H pr¸th twn exis¸sewn aut¸n eÐnai tÔpou Riccati, en¸ ai upìloipai eÐnai grammikaÐ.H lÔsic tou sust matoc autoÔ dÐdei tac zhtoumènac sunart seic

f0 = −12

ln

cosh (2λb)− k3

λsinh (2λb)

f1 =

(k2/2λ) sinh (2λb)cosh (2λb)− (k3/λ) sinh (2λb)

f2 = − ln

cosh (2λb)− k3

λsinh (2λb)

f3 =

(k1/2λ) sinh (2λb)cosh (2λb)− (k3/λ) sinh (2λb)

(96)

27

λ =(k2

3 − k1k2

)1/2(97)

jètomen b = 1 kai lambnomen telik¸c thn ktwji kanonik n anptuxin.

expk1a

2 + k2a+2 + k3

(aa+ + a+a

)=

(cosh (2λ)− k3

λsinh (2λ)

)−1/2

exp

(k2/2λ) sinh (2λ)cosh (2λ)− (k3/λ) sinh (2λ)

a+2

exp − ln cosh (2λ)−

(k3/λ) sinh (2λ) a+a

exp

(k1/2λ) sinh (2λ)cosh (2λ)− (k3/λ) sinh (2λ)

a2

(98)

H antÐstoiqoc kanonik sunrthsic f (n)(α, α∗) tou telestoÔ f(a, a+) dÔnataina eurej eukìlwc, lambnontec up' ìyin kai thn kanonik n ditaxin (88). Antika-jistoÔme apl¸c touc telestc a kai a+ me tic metablhtèc α kai α∗.

N−1exp

[k1a

2 + k2a+2 + k3

(aa+ + a+a

)]=

(cosh (2λ)− k3

λsinh (2λ)

)−1/2

exp

k1α

2 + k2α∗2 + 2 (k3 − λ tanhλ) αα∗

2 (λ coth 2λ− k3)

(99)

H parmetroc λ dÔnatai na lbei ìlac tac migadikc timc kai to mhdèn, ektìc apìthn idizousa perÐptwsin λ coth 2λ = k3.

Exetzomen akoloÔjwc mian meson efarmog n thc kanonik c diatxewc eic thnstatistik n

Ek thc sqèsewc (68) kai thc parathr sewc thc paragrfou 4 prokÔptei ìti hm tra puknìthtoc enìc sust matoc to opoÐon perigrfetai apì ton anexrthton touqrìnou telest n tou Hamilton H dÐdetai ek thc sqèsewc.

ρ(~r, ~r ′, b) = e−bHδ(~r − ~r ′) (100)

Eic thn perÐptwsin tou armonikoÔ talantwtoÔ eic mÐan distasin, h sqèsic (8.13)grfetai.

ρ(~r, ~r ′, b) = exp

b

[h2

2m

∂2

∂x2− 1

2mω2x2

]δ(x− x ′) (101)

O telest c thc anwtèrw sqèsewc anaptÔssetai eic kanonik n morf n th bohjeÐathc sqèsewc (98). Jètontec

a =∂

∂x, a+ = x =⇒ [a, a+] = [

∂

∂x, x] = 1 και

k1 =h2b

2m, k2 =

12mω2b, k3 = 0, λ = 2f = hωb (102)

28

H sqèsic (101) grfetai:

ρ(x, x ′, b) = (cosh (2f))−1/2 exp

mω

2htanh (2f)x2

exp

ln(cosh (2f))−1

x

∂

∂x

exp

h

2mωtanh (2f)

∂2

∂x2

δ(x− x ′) (103)

Oi telestaÐ thc sqèsewc (103) epidroÔn epÐ thc δ(x− x ′) sunart sewc wc akoloÔ-jwc.

eb ∂2

∂x2 δ(x− x ′) =12π

∫ ∞

−∞eb ∂2

∂x2 e−ik(x−x ′)dk =12π

∫ ∞

−∞ebk2−ik(x−x ′)dk =

1√4πb

e−14b

(x−x ′)2dk (104)

ebx ∂∂x g(x) = g

(ebx

)(105)

Kat sunèpeian èqomen:

ρ(x, x ′, b) =mω

2πh sinh (2f)exp

− mω

2h sinh (2f)

[(x2 + x ′2

)cosh (2f)− 2xx ′

]

(106)Eic thn bibliografÐan [26], èqei upologisj h kanonik anptuxic telest¸n

tetragwnik c morf c wc proc p kai q kai me grammikoÔc ìrouc. Th bohjeÐa twnanaptÔxewn aut¸n èqei upologisj h m tra puknìthtoc tou hlektromagnhtikoÔpedÐou [37]. EpÐshc èqoun upologisj kai ai summetrikaÐ Weyl morfaÐ twn telest¸naut¸n ai opoÐai qrhsimopoi jhsan eic ton upologismìn thc Wigner sunart sewckatanom c. Eic tac epomènac paragrfouc, exetzomen ta anwtèrw probl mata diathn perÐptwsin qronik¸c exartwmènwn telest¸n Hamilton.

4.3 Mh stsima sust mata tetragwnik c morf c

Exetzoume eic thn pargrafon aut n sust mata ta opoÐa dÔnantai na perigrafoÔnapì telestc tou Hamilton tetragwnikoÔc wc proc tac jèseic kai tac ormc.

Eic mÐan distasin oi en lìgw telestaÐ grfontai

H

(−ih

∂

∂x, x, t

)= −h2a(t)

∂2

∂x2+ b(t)x2 (107)

29

O telest c (107) perigrfei ènan talantwt n metablht c suqnìthtoc ω(t) =4a(t)b(t) parousÐa trib c γ(t) = a

a h opoÐa eÐnai epÐshc sunrthsh tou qrìnou.O telest c qronik c exelÐxewc tou en lìgw sust matoc, wc prokÔptei ek thc

(6.5) eÐnai:

U(t) = exp

t

[∂

∂t2− iha(t2)

∂2

∂x22

+i

hb(t2)x2

2

]exp

−t

∂

∂t2

(108)

Upojètomen oti o anwtèrw telest c dèqetai thn kanonik n anptuxin

U(t) = expf0(t) + f1(t)x2

2

exp

f2(t)x2

∂

∂x2

exp

f3(t)

∂2

∂x22

(109)

AkoloÔjwc ja upologÐswmen tac agn¸stouc sunart seic fi(t) ai opoÐai profan¸cikanopoioÔn tac arqikc sunj kac.

fi(0) = 0 (110)

UpologÐzomen thn èkfrasin U−1 ∂∂t U eic to shmeÐon t = t1 − t2. Ek twn sqèsewn

(108) kai (109) lambnomen.

U−1 ∂

∂tU

∣∣∣∣t=t1−t2

= −iha(t1)∂2

∂x22

+i

hb(t1)x2

2 =

∂f0

∂t1+

∂f1

∂t1e−2f2

(x2 − 2f3

∂

∂x2

)2

+∂f2

∂t1

(x2 − 2f3

∂

∂x2

)∂

∂x2+

∂f3

∂t1

∂2

∂x22

(111)

Ek thc sqèsewc aut c, met apì wrismènac prxeic, lambnomen telik¸c to di-aforikìn sÔsthma.

f3 = −iha(t1) + 4i

hb(t1)f2

3 f1 =i

hb(t1)e2f2

f2 = 4i

hb(t1)f3 f0 =

12f0 (112)

'Enja ai teleÐai shmaÐnoun parag¸gisin wc proc t1. Ta fi eÐnai sunart seic twn t1kai t2 kai ikanopoioÔn tac arqikc sunj kac:

fi(t1, t2) = 0 δια t1 = t2 (113)

To sÔsthma (112) diafèrei apì to antÐstoiqon (94) twn stasÐmwn katastsewn, eicto ìti ed¸ ta a kai b eÐnai sunart seic tou qrìnou t1, en¸ o qrìnoc t2 eisèrqetai eictac arqikc sunj kac.

30

H pr¸th twn exis¸sewn aut¸n eÐnai tÔpou Riccati kai wc gnwstìn exis¸seic toutÔpou autoÔ dÔnantai na metatrapoÔn eic grammikc diaforikc exis¸seic deutèractxewc. Jètomen wc lÔsin thc en lìgw exis¸sewc thn ktwji èkfrasin

f3 = iha(t1)Y

Y(114)

kai lambnomen

Y + 4abY − a

aY = 0 (115)

H sunrthsic Y (t1, t2), lìgw twn sqèsewn (113), ikanopoieÐ tac arqikc sunj kac.

Y (t1, t2) = 0, Y (t1, t2) = a(t1) δια t1 = t2 (116)

En X(t1, t2) eÐnai h llh anexrthtoc lÔsic thc (115) me arqikc sunj kac

X(t1, t2) = 1, X(t1, t2) = 0 δια t1 = t2 (117)

tìte oi dÔo lÔseic sundèontai dia twn sqèsewn.

XY − XY = a(t1), X = −Y

∫a

Y 2dt (118)

H klassik exÐswsic (115) dèqetai kai mÐan llhn lÔsin [39] thc morf c:

Y = S(t1, t2) sin∫ t2

t1Ω(t)dt (119)

'Enja h suqnìthc Ω(t) kai to pltoc S(t1, t2) thc talant¸sewc dÐdontai ek twnsqèsewn.

Ω =a

S2(120)

S +(ω2 − Ω2

)S − a

aS = 0 (121)

Ai zhtoÔmenai sunart seic fi dÔnantai na ekfrasjoÔn sunart sei twn dÔo anexa-rt twn lÔsewn X kai Y .

f3 = −ihaY

Yf1 = − i

4h

X

Y

f2 = ln

a

Y

f0 =

12

ln

a

Y

(122)

31

O telest c qronik c exelÐxewc (109) tou sust matoc lambnei thn ktwji kanonik nmorf c.

U(t1, t2) =√

a

Yexp

− i

4h

X

Yx2

2

exp

ln

a

Y

x2

∂

∂x2

exp

−iha

Y

Y

∂2

∂x2

(123)H epÐdrasic tou telestoÔ autoÔ epÐ thc δ(x2−x1) sunart sewc dÐdei ton propagatortou sust matoc. Me thn bo jeia twn sqèsewn (104) kai (105) lambnomen [40]

K(2, 1) =1√

4ihπYexp

− i

4hY

[Xx2

2 +Y

ax2

1 − 2x2x1

](124)

H anptuxic tou telestoÔ qronik c exelÐxewc eic kanonik n morf n, eÐnai idi-aitèrwc qr simoc eic ton upologismìn twn dÔo anexart twn oloklhrwmtwn thckin sewc.

x2(t)

−ih ∂∂x2

(t)

= U(0, t)

x2

−ih ∂∂x2

U(0, t)−1 =

X 2Y

12

Xa

Ya

x2

−ih ∂∂x2

(125)H anwtèrw mèjodoc dÔnatai na epektaj kai eic perissotèrac diastseic, è-

nac talantwt c p.q. eic dÔo diastseic perigrfetai ek tou ktwji telestoÔ touHamilton.

H(−ih~∇, ~r, t

)= H1

(−ih

∂

∂x, x, t

)+ H2

(−ih

∂

∂y, y, t

)(126)

ènja Hi, i = 1, 2 eÐnai o telest c (107)Epeid oi telestaÐ H1 kai H2 antimetatÐjentai, o propagator tou sust matoc

dÐdetai ek thc sqèsewc:

K(2, 1) =1

4ihπYexp

− i

4hY

[Xr2

2 +Y

ar21 − 2~r2 · ~r1

](127)

Exetzomen akoloÔjwc to anwtèrw sÔsthma entìc hlektromagnhtikoÔ pedÐou,dianusmatikoÔ dunamikoÔ ~A(~r, t) = 1

2~H (t) × ~r, ènja to H (t) èqei thn z dieÔjunsin.

O telest c tou Hamilton dÐdetai ek thc sqèsewc.

H(−ih~∇, ~r, t

)= a(t)

(−ih~∇− e

2cH(t)× ~r

)2

+ δ(t)r2 (128)

O telest c qronik c exelÐxewc eic thn perÐptwsin aut n grfetai

U(t) = exp

t

[∂

∂t2− iha(t2)

(∂2

∂x22

+∂2

∂y22

)− ω(t2)

(x2

∂

∂y2− y2

∂

∂x2

)+

i

hb(t2)

(x2

2 + y22

)](129)

32

'Enjaω(t2) =

eHc

a(t2) b(t2) =1

4a(t2)

(ω2(t2) + 4δ(t2)a(t2)

)(130)

Epeid o telest c thc stroform c x2∂

∂y2− y2

∂∂x2

antimetatÐjetai me touc telestcthc kinhtik c enèrgeiac ∂2

∂x22+ ∂2

∂y22thc dunamik c enèrgeiac x2

2 + y22 kaj¸c kai me ton

telest n thc trib c, x2∂

∂x2−y2

∂∂y2

, o telest c (129) dÔnatai na anaptuqj wc ex c

U(t) = exp

t

[∂

∂t2− ω(t2)

(x2

∂

∂y2− y2

∂

∂x2

)]exp

−t2

∂

∂t2

Uτ (t)

= exp−

∫ t

0ω(t2 + t ′)dt ′

(x2

∂

∂y2− y2

∂

∂y2

)Uτ (t) (131)

'Enja Uτ (t) eÐnai o telest c qronik c exelÐxewc tou talantwtoÔ eic dÔo diastseic.Epidr¸men tèloc ton telest n autìn epÐ thc δ(~r2−~r1) sunart sewc kai jètontec

t = t1 − t2 lambnomen ton zhtoÔmenon propagator

K(2, 1) = exp∫ t2

t1ω(t)dt

(x2

∂

∂y2− y2

∂

∂x2

)Kτ (2, 1)

(132)

O pur n Kτ (2, 1) dÐdetai ek thc sqèsewc (127).O ekjetikìc telest c thc sqèsewc (132) paristnei mia peristrof eic to x, y

epÐpedon upì gwnÐan

f =∫ t2

t1ω(t)dt (133)

kai epidr epÐ mÐan tuqoÔsan analutik n sunrthsin g(x2, y2) wc ex c:

expf

(x2

∂

∂y2− y2

∂

∂x2

)g(x2, y2)=g(x2 cos f−y2 sin f, y2 cos f+x2 sin f) (134)

Kat sunèpeian h sqèsic (132) dÔnatai na graf analutikìteron wc ex c:

K(2, 1) =1

4ihπYexp

− i

4hY

[X(x2

2 + y22) + (Y /a)(x2

1 + y21)−

2(x2x1 + y2y1) cos f − 2(x2y1 − x1y2) sin f ] (135)

EpÐshc ta tèssera anexrthta oloklhr¸mata thc kin sewc dÔnantai na eurejoÔneukìlwc th bohjeÐa thc anaptÔxewc (131).

x2(t) y2(t)

−ih ∂∂x2

(t) −ih ∂∂y2

(t)

=

X 2Y

12

Xa

Ya

x2 y2

−ih ∂∂x2

−ih ∂∂y2

cos f sin f

− sin f cos f

(136)

33

ènjaf =

∫ t

0ω(t ′)dt ′ (137)

4.4 Mh stsima sust mata tetragwnik c morf c megrammikoÔc ìrouc

Eic thn pargrafon aut n melet¸men sust mata ta opoÐa dÔnantai na perigrafoÔnapì telestc tou Hamilton thc morf c.

H(−ih~∇, ~r, t

)= a(t)

(−ih~∇− e

c~A(~r, t)

)2

+ c(t)r2 − ih~d(t) · ~∇− ~E(t) · ~r (138)

O telest c qronik c exelÐxewc eic thn perÐptwsin aut n grfetai:

U(t) = exp

t

[∂

∂t2+

i

hHn(2) + ~d(t) · ~∇2 − i

h~E(t) · ~r2

]exp

−t

∂

∂t2

(139)

kai dÔnatai na anaptuqj wc akoloÔjwc,

U(t) = exp

i

hk0 +

i

hk1x2 +

i

hk2y2

exp

λ1

∂

∂x2+

∂

∂y2

(140)

Oi telestaÐ Hm(2) kai Um(2) dÐdontai ek twn sqèsewn (128) kai (131) antistoÐqwc.Ek twn sqèsewn (139) kai (140) lambnomen thn isìthta:

U−1 ∂

∂tU(t)

∣∣∣∣t=t1−t2

=i

hHm(t1) + ~d(t) · ~∇2 − i

h~E(t) · ~r2 = k0 − k1λ1 − k2λ2 +

U−1m

(∂

∂t+

i

hk1x2 +

∂

∂t+

i

hk2y2 + λ1

∂

∂x2+ λ2

∂

∂y2

)Um(t)

∣∣∣∣t=t1−t2

(141)

O teleutaÐoc ìroc thc tautìthtoc (141) upologÐzetai th bohjeÐa thc sqèsewc (9.28).Exis¸nontec touc suntelestc twn omoÐwn ìrwn lambnomen to ktwji diaforikìnsÔsthma.

E1 E2

d1 d2

=

− Y

aX2a

−2Y X

k1 k2

λ1 λ2

cos f sin f

− sin f cos f

(142)

k0 = d1λ1 + k2λ2 (143)

34

To opoÐon lÔetai eukìlwc wc proc tac agn¸stouc sunart seic kai dÐdei tac akoloÔ-jouc klassikc (den perièqoun to h) diaforikc exis¸seic

k1 k2

λ1 λ2

=

−X X

2a

−2Y Ya

E1 E2

d1 d2

cos f − sin f

sin f cos f

(144)

k0 = d1λ1 + k2λ2 (145)

me arqikc sunj kac

ki(t1, t2) = λi(t1, t2) = 0 δια t1 = t2 (146)

EpidroÔmen katìpin ton telest n qronik c exelÐxewc eic thn δ(~r2 − ~r1) sunrthsin

K(2, 1) =

14ihπY

exp− i

4hY

[X(x2

2 + x21) + (Y /a)(y2

2 + y21) + 2(Xλ1 − 2Y k1)x2+

2(Xλ2 − 2Y k2)y2 − 2(λ1 cos f − λ2 sin f)x1 − 2(λ2 cos f + λ1 sin f)y1+

X(λ21 + λ2

2)− 4Y k0 − 2(x2x1 + y2y1) cos f − 2(y2x1 − y1x2) sin f]

(147)

EpÐshc ek thc anaptÔxewc (140) ta tèssera anexrthta oloklhr¸mata thc kin sewcupologÐzontai eukìlwc.

x2(t) y2(t)

−ih ∂∂x2

(t) −ih ∂∂y2

(t)

=

X 2Y

X2a

Ya

x2 y2

−ih ∂∂x2

−ih ∂∂y2

cos f sin f

− sin f cos f

+

X 2Y

X2a

Ya

λ1 λ2

−k1 −k2

cos f sin f

− sin f cos f

(148)

Oi telestaÐ (148) kaj¸c kai oi (134) kai (125) eÐnai lÔseic thc exis¸sewc kin sewc(44) tou Heisenberg h opoÐa eic thn prokeimènhn perÐptwsin grfetai.

q(t) =1ih

[q,H] =∂H(t)∂p(t)

, p(t) =1ih

[p, H] = −∂H(t)∂q(t)

(149)

35

5 EfarmogaÐ thc JewrÐac Diatxewc eic thnSqetikistik n Kbantomhqanik n

5.1 Wigner Parstasic

Eic to kefalaion autì genikeÔomen thn parstasin Wigner kaj¸c kai thn mèjodonupologismoÔ twn propagators dia thc diatxewc twn telest¸n eic thn sqetikistik nkbantomhqanik n. Ja periorisj¸men eic stsima sust mata me spin 1/2.

H kumatik exÐswsic enìc swmatÐou entìc dedomènou exwterikoÔ pedÐou grfetaiwc kai eic thn m sqetikistik n kbantomhqanik c [41] wc akoloÔjwc.

ih∂

∂tΨ = HΨ (150)

SumbolÐzomen me Ψ thn tetradistaton kumatosunrthsin h opoÐa eic thn spinorparstasin eÐnai èna bispinor. En Aµ = (V, ~A) eÐnai to tetradistaton dunamikìnenìc exwterikoÔ stajeroÔ hlektromagnhtikoÔ pedÐou, o telest c H tou Dirac eÐnai

H(p, q) = c~a ·(

~p− e

c~A

)+ mc2 + eV (151)

Ai m trai ai, b ikanopoioÔn tac sqèseic:

aib + bai = 0, aiaj + ajai = 2δij , b2 = 1 (152)

Mia llh parstasic [42] thc exis¸sewc Dirac (150) prokÔptei pollaplasi-zontec thn (150) me thn m tran b kai jètontec

γi = b ai, γ0 = b, x0 = ct (153)

lambnomen [c~γ ·

(~p− e

c~A

)−mc2

]Ψ = 0 (154)

Ai m trai γ orÐzontai mèsw twn mhtr¸n σ tou Pauli

γ0 =[

0 1−1 0

]~γ =

[0 ~σ−~σ 0

](155)

Ai pr¸thc txewc exis¸seic (154) dÔnatai na metasqhmatisjoÔn eic deutèractxewc exis¸seic pollaplasizontec me ton telest n c~γ · (~p − e

c~A)2 + mc2. To

apotèlesma eÐnai:[c2

(~p− e

c~A

)2

−m2c4 − 12iec2Fµνσ

µν

]Ψ = 0 (156)

36

ènjaFµν = ∂µAν − ∂νAµ (157)

eÐnai o hlektromagnhtikìc pediakìc tanust c kai

σµν =12

(γµγν − γνγµ) = (~a, i~Σ) (158)

H exÐswsic (156) eic sun jeic mondac grfetai[(

ih∂

∂t− eV

)2

− c2(

ih~∇+e

c~A

)2

−m2c4 + ehc ~H · ~Σ− iehc ~E · ~a]

Ψ = 0

(159)OrÐzomen ton sqetikistikìn Wigner telest n W dia thc ekfrsewc [43]

W (F ) = H(P , Q)F (P , Q)− F (P , Q)H(P ∗, Q∗) (160)

Ta P , Q eÐnai oi telestaÐ (20) tou Bopp.H exÐswsic Wigner lambnei thn morf n

ih∂

∂tF (p, q, t) = H

(p− ih

2∂

∂q, q +

ih

2∂

∂p

)F (p, q)−

F (p, q)H(

p +ih

2∂

∂q, q − ih

2∂

∂p

)(161)

H diafor thc exis¸sewc aut c apì thn antÐstoiqon mh sqetikistik n (21) ofeÐletaieic to gegonìc ìti ta H kai F eÐnai m trai kai en gènei den antimetatÐjentai.

En jèswmen wc lÔsin thc (161) thn

F (p, q, t) = e−ih

Etf(p, q) (162)

lambnomen thn exÐswsin idiotim¸n

WF (p, q) = Ef(p, q) (163)

Thn exÐswsin aut n ja melet swmen katwtèrw, toi ja eÔrwmen tac idiosunart seickai idiotimc thc. Qrin aplìthtoc periorizìmeja eic ton telest n

H(p, q) = c~a · ~p + bmc2 + eV (q) (164)

H exÐswsic idiotim¸n lambnei thn morf n

c

(pi − ih

2∂

∂qi

)aiF − c

(pi +

ih

2∂

∂qi

)Fai + mc2(bF − Fb) =

E − eV

(q +

ih

2

)∂

∂q+ eV

(q − ih

2

)∂

∂q

F (165)

37

Ekfrzomen tac idiosunart seic f(p, q) th bohjeÐa tou ktwji Fourier metasqhma-tismoÔ

f(p, q) =∫ ∞

−∞exp

2(i/h)pq ′

Ψ(q, q ′)dq ′ (166)

ApodeiknÔetai eukìlwc, met apì mÐan merik n olokl rwsin ìti h sunrthsic Ψ(q, q ′)epalhjeÔei thn exÐswsin.

ihc

2

(∂

∂qi− ∂

∂q ′i

)aiΨ−

(∂

∂qi+

∂

∂q ′i

)Ψai + mc2(bΨ−Ψb) =

E − eV (qi − q ′i) + eV (qi + q ′i)

ψ (167)

H exÐswsic (167) dia tou metasqhmatismoÔ

q + q ′ = x, q − q ′ = y (168)

lambnei thn morf n

−ihc∂

∂xiaiΨ(x, y) +

∂

∂yiΨ(x, y)ai + mc2bΨ(x, y)−Ψ(x, y)b =

E − eV (x) + eV (y)Ψ(x, y) (169)

Epeid h idiosunrthsic thc exis¸sewc tou Dirac èqei 4 diastseic h sunrthsicΨ(x, y) prèpei na eÐnai mia m tra 4× 4. Prgmati dia tou metasqhmatismoÔ

Ψij(x, y) = Ψi(x)Ψ∗(y), E = Ex − Ey (170)

H exÐswsic (169) diaqwrÐzetai eic ta exis¸seic−ihcai

∂

∂xi+ mc2b

Ψ(x) = Ex + eV (x)Ψ(x) (171)

ihc

∂

∂yiΨ∗(y)ai + mc2

Ψ∗(y)b = Ψ∗(y) Ey + eV (y) (172)

Kat sunèpeian h idiosunrthsic thc exis¸sewc idiotim¸n tou Wigner eÐnai mia m tra4× 4 me stoiqeÐa

fij(p, q) =∫ ∞

−∞exp

2(i/h)pq ′

Ψi(q + q ′)Ψ∗

j (q − q ′)dq ′ (173)

ènja ta Ψi(x) eÐnai ta spinor tou Dirac [44]. Ai de idiotimaÐ thc eÐnai h diafor twnidiotim¸n dÔo isodunmwn exis¸sewn tou Dirac.

38

Dia thn perÐptwsin twn eleujèrwn swmatÐwn oi idiosunart seic kai idiotimaÐ thcsqetikistik c exis¸sewc tou Wigner dÐdontai ek twn sqèsewn.

Fij(p, q, t) = e−ih

EtAi(k)Aj(k ′)ei(k−k ′)δ

(p

h− k + k ′

2

)

E = ±c

√m2c2 + h2k2 ± c

√m2c2 + h2k2 (174)

H sqetikistik sunrthsic Wigner paÐzei kai ed¸ ton rìlon katanom c kai sundèetaiepÐshc me ton telest n thc m trac puknìthtoc [45].

5.2 Sqetikistikìc propagator

OrÐzomen wc electron propagator èna bispinor txewc dÔo wc akoloÔjwc

Kij(2, 1) = −i < 0|TΨi(x1)Ψj(x2)|0 > (175)

'Ekastoc twn telest¸n ψ eÐnai èna jroisma thc morf c

Ψi =∑p

apΨpi + b+p Ψ−pj

Ψi =∑p

a+p Ψpi + bpΨ−pj (176)

O deÔteroc ìroc twn ekfrsewn aut¸n perièqei touc positron ìrouc. Mia sqèsicparomoÐa me thn mh sqetikistik n (37) dÔnatai na eurej kai ed¸. Eisgontec touctelestc (176) eic thn (175) kai lambnontec up' ìyin ta sqèseic:

< 0|apa+p |0 >=< 0|bpb

+p |0 >= 1 (177)

eurÐskomen

Kij(2, 1) = −i∑p

e−iEp(t2−t1)Ψpi(~r2)Ψpk(~r1), δια t2 > t1

Kij(2, 1) = i∑p

eiEp(t2−t1)Ψ−pi(~r2)Ψ−pk(~r1), δια t2 < t1 (178)

H epèktasic aut twn mh sqetikistik¸n sqèsewn dia qrìnouc t2 < t1 eÐnai anagkaÐadia na sumperilbwmen kai tac sqetikistikc katastseic arnhtik c energeÐac.

39

DÔnatai na deiqj ìti o propagator (175) ikanopoieÐ thn sqèsin.

c~γ ·(

~p− e

c~A

)−mc2

K(2, 1) = iδ(2, 1) (179)

EÐnai dhlad h Green sunrthsic thc sqetikistik c Dirac exis¸sewc.Katwtèrw dia thc anaptuqjeÐshc mejìdou ja upologÐswmen ton advanced prop-

agator dia tac peript¸seic tou eleujèrou swmatÐou kai dia to hlektromagnhtikìnpedÐon. Eic touc upologismoÔc mac ja paraleÐywmen thn bohjhtik n stajern ε,deqìmenoi ìti h mza perièqei mÐan apeirost n fantastik n posìthta m → m− iε.

O sqetikistikìc propagator dia èn swmtion entìc hlektromagnhtikoÔ pedÐoudÐdetai ek thc sqèsewc.

K+(2, 1) =∫ 0

−∞dt exp

− it

h

[c~γ ·

(~p− e

c~A

)−mc2

]δ(2, 1) (180)

H sqèsic (180) grfetai

K+(2, 1) =

c~γ ·(

~p− e

c~A

)+ mc2

∫ 0

−∞dt exp

− it

h2

[c~γ ·

(~p− e

c~A

)+ mc2

] [c~γ ·

(~p− e

c~A

)−mc2

]δ(2, 1) (181)

Ek thc sqèsewc (159) lambnomen

K+(2, 1) =

c~γ · (~p− e

c~A) + mc2

∆(2, 1) (182)

ènja o spinor ∆(2, 1) eÐnai eic sun jeic mondac o ex c:

∆(2, 1) =∫ 0

−∞dt exp

− it

h2

[(ih

∂

∂t2− eV (~r2)

)2

− c2(

ih~∇2 +e

c~A(~r2)

)2

−

m2c4 + ehc ~H · ~Σ− iehc ~E · ~a]

δ(~r2 − ~r1)δ(t2 − t1)(183)

Dia thn perÐptwsin twn eleujèrwn swmatÐwn toi ~E = ~H h sqèsic (183) grfetai

∆(2, 1) =∫ 0

−∞dt exp

it

∂2

∂t22− ic2t~∇2

2 + im2c4

h2 t

δ(~r2 − ~r1)δ(t2 − t1) =

∫ 0

−∞dt exp

it

m2c4

h2t

exp

it

∂2

∂t2

δ(t2 − t1) exp

ic2t~∇2

2

δ(~r2 − ~r1) (184)

Eic thn perÐptwsin aut n to ∆(2, 1) eÐnai mia bajmwt sunrthsic.

40

Ek thc sqèsewc (104) lambnomen

∆(2, 1) = − 1(4π)2c3

∫ 0

−∞dt

t2exp

i

4t

[(t2 − t1)2 − 1

c2(~r2 − ~r1)2

]+ i

m2c4

h2 t

(185)

To olokl rwma autì eÐnai stoiqei¸dec [46]. Jètontec

s =[(t2 − t1)2 − 1

c2(~r2 − ~r1)2

]1/2

∆t2 > ∆r2

s = i

[1c2

(~r2 − ~r1)2 − (t2 − t1)2]1/2

∆t2 < ∆r2 (186)

lambnomen

∆(2, 1) =m

8πsH

(2)1

(msc2

h

)+

12π

δ(s2

)(187)

ènja H(2)1 eÐnai h sunrthsic Hankel.

O spinor ∆(2, 1) dÔnatai na upologisj akrib¸c kai eic thn pl rhn morf n tou.Eklègomen ta dunamik thc morf c.

~A(~r) = (0,Hx, 0) και V (~r) = −Ex (188)

Htoi stajerìn magnhtikìn pedÐon parllhlon proc ton z xona kai stajerìnhlektrikìn pedÐon parllhlon proc ton x axona

O spinor ∆(2, 1) grfetai:

∆(2, 1) =∫ 0

−∞dt exp

t

[i∂2

∂t22− ic2~∇2

2 +iω2

c2x2

2 + 2x2

(ω2

c

∂

∂t2− ω1

∂

∂y2

)

+im2c4

h2 − iω2Σ3 + ω1a1

]δ(~r2 − ~r1)δ(t2 − t1) (189)

ènja èqomen jèseiω1 =

ec

hH , ω2 =

ec

hE (190)

kai

ω =ec

h

√H 2 − E2 δια 0 ≤ E < H

ω = iec

h

√E2 −H 2 δια 0 ≤ H < E (191)

41

AkoloÔjwc jewroÔmen tou metasqhmatismoÔc.

y ′2 − y ′1 =ω1

ω(y2 − y1)− ω2c

ω(t2 − t1)

t ′2 − t ′1 =ω2

ωc(y2 − y1)− ω1

ω(t2 − t1) (192)

Oi anwtèrw metasqhmatismoÐ wc gnwstìn af noun analloi¸touc tac ekfrseicc2∆t2 − ∆r2, ∂2/∂t22 − c2~∇2

2, kaj¸c kai to ginìmenon twn δ(2, 1) sunart sewn.Antijètwc alloi¸noun ton pargonta ω2

c∂

∂t2− ω1

∂∂y2

, o opoÐoc kai lambnei thnktwji morf n.

ω2

c

∂

∂t2− ω1

∂

∂y2= −ω

∂

∂y ′2(193)

Ekfrzontec thn sqèsin (189) eic to tonoÔmenon sÔsthma kai diatssontec katal-l lwc touc telestc èqomen:

∆(2, 1) =∫ 0

−∞dt exp

t

[i

∂2

∂t ′2− ic2 ∂2

∂z22

]δ(z2 − z1)δ(t′2 − t′1)

exp

t

[−ic2 ∂2

∂x22

+ iω2

c2(x2 − i

c2

ω

∂

∂y ′2)2

]δ(x2 − x1)δ(y′2 − y′1)

exp

t

[im2c4

h2 − iω2Σ3 + ω1a1

](194)

Oi telestaÐ oi opoÐoi perÐeqoun touc spin ìrouc upologÐzontai eukìlwc th bohjeÐatwn sqèsewn.

a21 = Σ2

2 = 1, a2Σ3 + Σ3a1 = 0 (195)

AnalÔontec to ekjetikìn kat taylor lambnomen:

exp −iω1tΣ3 + ω2ta1 = cos (ωt) + (ω2ta1 − iω1tΣ3)sinωt

ωt(196)

EpÐshc ek thc sqèsewc (104) lambnomen:

exp

t

[i

∂2

∂t ′2− ic2 ∂2

∂z22

]δ(z2 − z1)δ(t′2 − t′1) =

14πct

exp

i

4t

[(t ′2 − t ′1)

2 − 1c2

(z2 − z1)2]

(197)

42

AkoloÔjwc analÔomen ton trÐton ekjetikìn telest n thc (194) eic kanonik n mor-f n. Jètontec

a =∂

∂x2, a+ = x2 − ic2

ω

∂

∂y ′2=⇒ [a, a+] = 1

k1 = −ic2t, k2 = itω2

c2, k3 = 0, λ = iωt (198)

lambnomen, dunmei thc sqèsewc (98)

exp

−ic2 ∂2

∂x22

+ iω2

c2

(x2 − i

c2

ω

∂

∂y2′

)2

δ(x2 − x1)δ(y2

′ − y1′) =

(cos (ωt))1/2 exp

i

2c2tan (2ωt)

(x2 − ic2

ω

∂

∂y2′

)2

exp

− ln cos (2ωt)

(x2 − ic2

ω

∂

∂y2′

)∂

∂x2

exp

− ic2

ωtan (2ωt)

∂2

∂x22

δ(x2 − x1)δ(y2

′ − y1′) =

(−2ic2

ωsin (2ωt)

)−1/2

exp

−

i

2c2cot 2ωt

(x2 − ic2

ω

∂

∂y2′

)2

+

(x1 − ic2

ω

∂

∂y2′

)2

− 2cos (2ωt)

(x2 − ic2

ω

∂

∂y2′

) (x1 − ic2

ω

∂

∂y2′

)]δ(y2

′ − y1′) =

ω

4πc2 sinωt

exp− iω

4c2cotωt

[(y2

′ − y1′)2 + (x2 − x1)2

]− iω

2c2(y2

′ − y1′)(x2 + x1)

(199)

Kat sunèpeian ek twn sqèsewn (196), (197) kai (199) lambnomen telik¸c tonzhtoÔmenon propagator.

∆(2, 1) =−1

(4π)2c3exp

− iω

2c2(y ′2 − y ′1)(x2 + x1)

∫ 0

−∞dt

[cos (ωt) +

(ω2ta1 − iω1tΣ3)sin (ωt)

ωt

]1t2

ωt

sin (ωt)exp

it

m2c4

h2 +

i

4t

[(t2 − t1)2 − 1

c2(z2 − z1)2 − 1

ω2ωt cos (ωt)((x2 − x1)2 + (y ′2 − y ′1)

2)]

(200)

43

H sqèsic aut isqÔei di ìlac tac oriakc metabseic toi E = 0 H = 0. EpÐshcen E = H = 0, sumpÐptei me thn sqèsin (185) tou eleujèrou swmatÐou. Thn Ðdiansqèsin lambnomen, ektìc apì tac spin m trac kai dia tou orÐou e = H toi ω = 0.Oloklhr¸mata thc morf c (200) anafèrontai eic thn bibliografÐan [47].

6 SUMPERASMATA

Eic thn paroÔsan ergasÐan, melet¸men tac efarmogc thc jewrÐac diatxewc twntelest¸n eic thn kbantomhqanik n

Dia twn antistoiqi¸n twn diatetagmènwn telest¸n proc tac sun jeic migadikcsunart seic, ai kbantomhqanikaÐ sqèseic lambnoun thn sun jhn klassik n twnmorf n. To sÔsthma perigrfetai katèujeÐan eic ton q¸ron twn fsewn apì mÐansunrthsin katanom c kai ta parathr sima megèjh, parÐstantai amfimonoshmntwcapì sun jeic migadikc sunart seic. H summetrik Weyl antistoiqÐa èqei melethj perissìteron, diìti h antÐstoiqoc sunrthsic tou telestoÔ thc m trac puknìthtocsumpÐptei me thn gnwst n katanom n tou Wigner. Melet¸men thn perigraf n aut thc kbantomhqanik c dia thn mh sqetikistik n perÐptwsin kai thn epeiteÐnomen kaieic thn sqetikistik n toiaÔthn. OrÐzomen ton sqetikistikìn Wigner telest kaieurÐskomen tìson tac idisunart seic ìson kai tac idiotimc tou. ApodeiknÔomen ìtih idiosunrthsic eÐnai mia m tra me stoiqeÐa ton f metasqhmatismon tou ginomènoudÔo spinor tou Dirac, en¸ ai idiotimaÐ eÐnai h diafor twn idiotim¸n duo exis¸sewntou Wigner.

Af etèrou h ditaxic twn telest¸n dieukolÔnei kai apolopoieÐ thn epÐdrasin twntelest¸n epÐ twn diafìrwn sunart sewn. ApodeiknÔetai ìti dunmeja na grywmenton telest n qronik c exelÐxewc enìc sust matoc upì ekjetik n morf n. AnaptÔs-somen akoloÔjwc ton telest n autìn eic mÐan katllhlon diatetagmènhn morf n,toiaÔth ¸ste h epÐdrasic tou en logw telestoÔ epÐ thn dèlta sunrthsin upologÐze-tai me ènan aplìn kai eujÔn trìpon. H prokÔptousa sunrthsic eÐnai o propagatortou sust matoc kai paÐzei ousi¸dh rìlon eic thn parstasin Feynman thc kban-tomhqanik c h opoÐa anaptÔssetai epÐshc. Di qronik¸c exartwmènouc anexrth-touc telestc tou t, oi opoÐoi eÐnai polu¸numa deutèrac txewc wc proc tac kanon-ikc suzugeÐc metablhtc, h ditaxic epitugqnetai dia thc mejìdou thc parametrik cparagwgÐsewc. DÐdwmen peraitèrw thn qronik exèlixin twn telest¸n kai toi eu-rÐskomen ta anexrthta oloklhr¸mata thc kin sewc. Tèloc h mèjodoc epekteÐnetaikai eic thn sqetikistik n kbantomhqanik n ènja upologÐzomen ton propagator touhlektromagnhtikoÔ pedÐou.

44

7 SUMMARY

In this thesis we study the applications of the theory of the ordering of the operatorsin quantum mechanics.

With the help of some correspondences between ordered operators and theusual complex functions, the quantum mechanical relations become similar withthose of classical mechanics. The system is now described directly in the phasespace by a distribution function and the physical observables are represented bythe usual complex functions. The symmetric or Weyl correspondence has beenstudied extensively since the corresponding function of the density matrix operator,coincides with the known Wigner distribution function, We study this Wignerformulation of non relativistic quantum theory and we extend this in the relativisticone. We define the relativistic Wigner operator and we find its eigenfunctions andeigenvalues. We prove that the eigenfunctions are matrices whose elements areFourier transforms of the product of two Dirac spinors, while the eigenvalues aredeferences of the eigenvalues of two Dirac equations.

On the other hand the ordering of the operators simplify the calculation ofthe action of the operators on the various functions. We write the time evolutionoperator in an exponential form and we expand this operator in an appropriateform so that its action on the function, follows in a simple and straightforwardmanner. The resulting function is the propagator of the system which plays acentral role in the Feynman formulation of quantum mechanics, which is alsodeveloped. For Hamiltonians, time dependent or time independent, which arepolynomials of second degree in the canonical conjugate variables, the ordering canbe achieved with the method of parametric differentiation. We also evaluate thetime evolution of the operators p and q which constitute the independent integralsof motion. Finally we extend the method to the case of relativistic mechanicswhere we evaluate the propagator of the electromagnetic field.

45

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48