Ralstc modl fo adaton-m ntacton Rchad A. Pakula Scnc Applcatons Intnatonal Copoaton (SAIC) 41 Noth Fafax D. Sut 45 Alngton VA 3 ABSTRACT Ths pap psnts a alstc modl that dscbs adaton-m ntactons. Ths s achvd by a gnalzaton of fst quantzaton wh th Maxwll quatons a ntptd as th lctomagntc componnt of th Schödng quaton. Ths pctu s complmntd by th consdaton of lctons and otons as al patcls n thdmnsonal spac followng gudng condtons dvd fom th patcl-wavfunctons to whch thy a assocatd. Th gudng condton fo th lcton s takn fom Bohman mchancs whl th oton vlocty s dfnd as th ato btwn th Poyntng vcto and th lctomagntc ngy dnsty. Th cas of many patcls s consdd takng nto account th statstcal popts. Th fomalsm s appld to a two lvl systm povdng a satsfactoy dscpton fo spontanous msson Lamb shft scng absopton dspson sonanc fluoscnc and vacuum flds. Ths modl adquatly dscbs quantum umps by th ntanglmnt btwn th oton and th atomc systm and t wll pov to b vy usful n th smulaton of quantum dvcs fo quantum computs and quantum nfomaton systms. A possbl latvstc gnalzaton s psntd togth wth ts latonshp to QED. 3.65.-w 14.7.Bh 4.5.Ct 4.5.-p - 1 -
Contnts ABSTRACT...1 I. INTRODUCTION... 3 II. REVIEW OF THE CONCEPTS LEADING TO THIS MODEL... 5 A. Vsualzaton of th quantum lctomagntc fld.... 5 B. Fst quantzaton schms fo th lctomagntc fld... 6 C. Quantum umps and th oton patcl.... 9 III. PHOTON-WAVE FUNCTIONS... 11 A. Maxwll s quatons as th sngl oton Schödng quaton... 11 B. Spaaton nto achd and f flds... 1 IV. THE SCHRŐDINGER-MAXWELL EQUATION... 16 V. GENERAL CASE MANY PARTICLES... 4 VI. ELECTRON AND PHOTON IN REAL SPACE... 6 A. Souc cunts n th gnalzd wav quaton... 6 B Total flds. Gudanc condtons... 7 VII. APPLICATION TO A TWO LEVEL SYSTEM... 9 A. Lamb shft and spontanous msson.... 3 Smclasscal modls... 3 Ou modl... 31 B. Scng absopton dspson and sonanc fluoscnc... 35 1. On oton... 35. N-otons... 38 VIII. RELATIVISTIC GENERALIZATION... 39 IX. ANALYSIS OF PREVIOUS CRITICS... 41 X. CONCLUSIONS... 43 Acknowldgmnts... 44 - -
I. INTRODUCTION A mo ntutv undstandng of th natual laws dscbd by quantum mchancs s qud to buld mo fnd vsualzaton and smulaton tools fo th dvlopmnt of quantum dvcs. Such an ntutv alstc and non-local thoy alady xsts fo lctons and oth matal patcls as ognally concvd by d Bogl [1 ] and dfnd by Bohm [3]. Howv to th bst of ou knowldg a smla modl dos not xst fo otons. Ths pap poposs a fst appoach towads th constucton of such a thoy. W notc that th fst stp towads th dvlopmnt of an ntutvly fndly thoy fo otons s th constucton of a fst quantzd vson of quantum lctodynamcs n spac. Th scond stp s to vfy that on can ndd solv th masumnt poblm and dscb th quantum umps n tms of al oton patcls. Th coct fomulaton of ths two tasks wll pov to b of pcous valu at th tm of dsgn and smulaton of quantum computs and communcaton systms. On can stat by notng that th oots of quantum mchancs can b tacd to an mpt fo unfcaton of th classcal dscpton of dspson of lght wth th non-classcal dscpton povdd by Boh s atomc thoy. In fact both thos had bn xtnsvly vfd xpmntally but smd compltly ncompatbl wth on anoth. Dspson thoy mpld a dynamcal contnuous atomc polazaton whl Boh s modl mpld th psnc of statonay stats spaatd by quantum umps assocatd wth th absopton and msson of lctomagntc adaton [4 5]. Th lnk was achvd by placng th dtmnstc dynamcal coffcnts appang n th polazaton fom th classcal dspson fomula wth th pobablstc Enstn coffcnts fom Boh s atomc thoy. That mpt bult xclusvly on th consdaton of th atomc dynamcs lad to th Kams-Hsnbg dspson fomula [6] and vntually to th caton of th hghly countntutv matcal fom of quantum mchancs and to th ntoducton of noncommutatv opatos placng dynamcal vaabls n th classcal quatons of moton [7]. A alstc modl s poposd h whch can concl classcal dspson thoy and Boh s atomc modl. Ths modl s basd on th ognal poposal of Slat [5 8] nspd n th d Bogl-Enstn das about th lctomagntc flds and oton patcls. In Slat s modl both lctomagntc wavs and oton patcls w assgnd yscal xstnc n spac th wavs o oton-wav-functons svng as gudng flds fo th otons. On can vfy that such a modl s abl to conclat wav and patcl aspcts of th ntacton of adaton and m whn th pobablty fo absopton fo an ncdnt oton s mad qual to th latv absopton of ngy povdd by th classcal dspson thoy. Ths wll b achvd as soon as otons a mad to follow th Poyntng vcto and ts pobablty dnsty mad popotonal to th lctomagntc ngy dnsty of th ncdnt fld. In that cas th sam coss scton obtand fom th classcal dspson thoy fo th absopton of ngy would apply qually wll to th pobablty fo th absopton of a oton n Boh s modl. As a coollay th tanston ampltuds appang n th Boh modl can b consdd as - 3 -
numcally qual to th tms appang n th classcal dspson coffcnts as poposd by Ladnbug and followd by Kams and Hsnbg [4 5] n th stps pvous to th caton of quantum mchancs. Whl th ntptaton of absopton as a vtual ntmdat stat n th pocss of dspson n tadtonal quantum mchancs nducs on to consd both as two aspcts of th sam nomnon thy cospond n ou modl to two dffnt yscal pocsss. Dspson occus vy tm an lctomagntc wav o oton-wavfuncton mpngs on an atomc systm ndpndntly fom th poston of th al oton patcl. It s sponsbl among oths fo th gnaton of th ndx of facton of tanspant mda. Absopton on th oth hand taks plac only whn on ncdnt oton patcl occus to com to a dstanc clos than th absopton coss scton and gts tappd by th atom whch now has bn pomotd to an xctd stat. Ths pap dvlops a fomal bass fo ths das towads a wd yscal pctu and s opn fo futu mpovmnt. It psnts n a systmatc and cla way th yscal das mpld by th us of oton-wav-functons and oton patcls. To smplfy th xposton n most of th pap no consdaton s gvn to any ffct assocatd wth th spn of th patcls o wth latvstc dynamcs. In Scton VIII howv a latvstc gnalzaton s psntd and th latonshp of ths modl wth quantum lctodynamcs (QED) s gvn. Th dffnt aspcts of th ntacton btwn lght and m povd th bass fo ou modl. Th fst aspct s th tansfomaton ov tm suffd by th ntutv dscpton of quantum lctomagntc flds. Ths dscpton statd as a mathmatcal tool dpvd fom any al yscal xstnc and volvd to a fom mo mnscnt of classcal lctodynamcs n cnt tms. Ths voluton ncouagd ou vsualzaton of th flds as havng a alty n spac and xpssd by th gudng condtons to whch thy gv s n ths modl. Th scond aspct s th dvlopmnt of a fst quantzaton schm fo lctodynamcs. In ths schm th Maxwll quatons a ntptd as th quvalnt to th Schödng quaton and th lctomagntc flds a consdd th oton-wav-functons. Ths s an mpotant pont bcaus n Bohman mchancs th gudng condtons a povdd by th wav functons assocatd wth th matal patcls n a fst quantzaton thoy. Th thd aspct s stll an opn quston n quantum mchancs: can w xplan quantum umps? Ths pap s oganzd as follows. In Scton II a vw of th concpts ladng to ou modl s povdd. In Scton III th ntptaton of th Maxwll quatons as th sngl oton Schödng quaton th poblms assocatd to th dvgnc of th flds at th soucs and a possbl dfnton of achd and f flds a consdd. In Scton IV th fomalsm assocatd wth th gnalzd Schödng-Maxwll quaton s psntd wh otons a tatd on th sam footng as any oth matal patcl. In Scton V th gnalzaton to many patcls and th slcton of ndpndnt vaabls a analyzd. In Scton VI th gudanc condton fo th - 4 -
lcton and oton n th xtndd confguaton spac s povdd. In Scton VII th nw fomalsm s appld to scng absopton dspson sonanc fluoscnc quantum umps and vacuum flds n a two lvl systm. In Scton VIII a possbl latvstc gnalzaton s psntd togth wth ts latonshp to QED. In Scton IX pvous ctcsm asd agan alstc modls s consdd and n Scton X w psnt ou conclusons. II. REVIEW OF THE CONCEPTS LEADING TO THIS MODEL Ths pap poposs a fomulaton of som of th das and quatons pvalng n th dscpton of quantum optcs today. Thfo a latvly complt vw of th most lvant dvlopmnts ladng to th das xposd n ths wok s ndd n od to gan th mo gnal pspctv qud fo a btt appcaton of ths poposal. A. Vsualzaton of th quantum lctomagntc fld. W stat consdng how th ntutv vsualzaton of th pocss of msson of lctomagntc adaton volvd wth tm snc th fst woks at th bgnnng of quantum mchancs. In th famwok of th old quantum thoy of Boh [9] msson of adaton was accomplshd though a quantum ump fom an atomc xctd stat to a low ngy stat. In th thoy of Dac [1] ths pocss was dscbd by th abstact caton of a on-oton stat of th lctomagntc fld wth wll-dfnd momntum psntd by a st of quantum mchancal opatos. Ths opatos as wll as th lcton-wav-functons had no al xstnc n spac. Thy w sn as a puly mathmatcal and fomal psntaton of th pocss that was bng dscbd. An mpotant popty of that thoy was th fact that t mad no allowanc fo th adaton acton flds sponsbl fo th dcay of th oscllatoy moton n classcal lctodynamcs. By contast quantum lctodynamcs consdd th msson pocss as nducd by th so-calld vacuum flds o vacuum fluctuatons [11 1]. In th 196s a dscpton of spontanous msson n tms of adaton acton was dvlopd fst as a smclasscal thoy [13] and lat makng full us of th quantum mchancal quatons of moton n th Hsnbg s pctu [14]. In addton th van d Waals and Casm focs w calculatd and xpland wthout th nd fo vacuum fluctuatons [15]. In th 197s th cohnc popts of flds mttd by atoms w futh analyzd [16] and t was shown that tadaton ffcts of long known hstoy n classcal lctomagntsm could b nvokd fo th hgh fquncy cut off n th non-latvstc calculaton fo th Lamb shft [17]. Rtadaton ffcts w also consdd fo th ntacton btwn dffnt atoms [18] n th coopatv ffcts dscbd by Dck s modl [19]. - 5 -
In th aly 198s wokng on causalty ffcts n th poxmty of atoms and molculs Bykov [] was abl to show that th na flds opatos follow xactly th sam quatons and hav th sam solutons as th classcal flds. At th sam tm Baut [1] statd a ss of paps showng that many of th ffcts pvously consdd of puly quantum ogn poducd by th vacuum flds could also b xpland by classcal flds allowng fo a mo complcatd mathmatcs. Among ths ffcts w can mnton spontanous msson and Lamb shft van d Waals focs Casm focs acclatng thmal flds tc. In th followng yas th woks by th goups of Psco [] and Pow [3] yldd an ncasd vsualzaton of th lctomagntc flds suoundng atoms. Thy also showd that causalty could b fully -stablshd by qung th tnton of non-sonant tms n th flds commonly though away n th otatng wav appoxmaton [4]. All ths woks povdd a mo complt undstandng of th lctomagntc flds suoundng th atoms and th ntactons. As a sult th ognally abstact and ndtmnstc pctu of msson of adaton nto a stat of wll-dfnd momntum was gvng plac to a mo ntutv and alstc pctu n spac and tm ncasngly mnscnt of classcal lctodynamcs. B. Fst quantzaton schms fo th lctomagntc fld. Classcal lctodynamcs [5] and optcs [6] show that lctomagntc flds dsplay a vy ch gomty dpndng on th shap of th souc cunts and bounday condtons. Th classcal thoy fo th scng by small patcls [7 8] maks us of scal Bssl functons basd on th wok of M [9]. Also classcal antnna thoy s concnd mostly wth th na and fa flds n th dpol appoxmaton [3]. Smla tatmnts can b found n th smclasscal dscptons commonly usd n X- ay dffacton and nucla yscs. X-ay dffacton by cystals can b xpland n tms of th fa fld dpol msson o plan wavs [31 3 33] as ognally shown by Dawn [34] and by Ewald [35]. Nucla adaton [36 37 38 39 4] has always bn dscbd n tms of th scal hamoncs dvlopd by Hansn [41] and Htl [4]. All ths tatmnts contast wth th tadton n quantum optcs usd to dcompos th quantum flds n plan wavs [43 44 45 46 47 48] whl wokng n th Coulomb gaug wth tansvs and longtudnal dlta functons [49 5 51 5]. A supfcal consdaton of quantum optcs may cat th mpsson that a gomtcal dstncton btwn classcal and quantum flds ally xst: whl classcal flds may hav any gomty quantum flds a stctd n most cass to plan wavs. That ths s ally not th cas s dmonstatd by th souc fld xpsson n quantum optcs [53 54 55] whch shows that quantum lctomagntc flds poducd by a pont systm hav a scal dscpton vy smla to th classcal on [56 57 58]. W fnd h agan that on could unfy th ch gomtc fatus of classcal lctodynamcs wth th quantum popts of quantum optcs by consdng a fst quantzd vson of quantum lctodynamcs wh th solutons to th Maxwll - 6 -
quatons w consdd wav functons fo oton patcls. Ths s not th standad way howv n whch quantum lctodynamcs has volvd. In fact n smclasscal appoxmatons [59] and n quantum optcs [6] th systms a dscbd n tms of patcl postons and lctomagntc flds whl n scond (o canoncal) quantzaton [61 6 63 64] th only yscal quantts a th flds thmslvs. So dpndng on th appoach on slcts on has th choc to dscb matal patcls by th poston o by quantzd flds but not such opton s offd fo otons. Th tadtonal thoy povds no plac fo a oton poston o vlocty as a yscal quantty n th dscpton of th quantzd lctomagntc fld. W should mmb howv that n th fst appoach to a quantum dscpton of lght Enstn [65] poposd n 195 that lght was composd of al patcls n spac. Ths stat of facts had ts ogn n Dac s ognal wok [66] on th quantzaton of th lctomagntc fld n 197. In that pap th lctomagntc fld was not consdd th quvalnt to th wav functon fo otons but Dac consdd nstad th ampltud of th lctomagntc fld tslf as th yscal quantty to b quantzd. Each Fou componnt of th fld was consdd quvalnt to an ndpndnt hamonc oscllato coupld to th soucs of adaton though th ntacton Hamltonan. Ths thoy was futh dvlopd by Hsnbg and Paul [67] and captulatd by Fm [68] achng th gnal fom that has bn n us untl today. Th ason fo consdng th fld as th pmay vaabl to b quantzd nstad of th possbl oton coodnats sms to b th gnalzd luctanc to th da that lght could vntually b composd of matal patcls. On th oth hand Landau and Pls [69] publshd n 193 a wok about th gnalzd wav functon of a systm consstng of m patcls and of oton patcls. Ths wav functon consstd of a collcton of functons wth fx numb of patcls multpld by tm dpndnt coffcnts n confguaton spac. Thy assumd that th ol of oton-wav functon was playd xclusvly by th lctc fld. Th th otons w consdd on an qual footng wth th oth patcls. Th man dffnc btwn th matal patcls and th otons was th xstnc of soucs fo th oton fld. Ths fact was not consdd hamful fo th gnal dvlopmnt of th thoy. Th authos w abl to show nstad that th sum of th noms of th functons blongng to dffnt numb of patcls mand constant. Th vaaton of th nom of ths functons had two soucs: th tm vaaton of th spctv coffcnts and th adaton soucs fo th oton flds. Ths was a makd dffnc wth th customay mthods of tatng systms wth vaabl numb of patcls as dscbd n th 193 pap by Fock [7] wh no fncs to oton soucs was mad at all. In that wok Fock dmonstatd th full quvalnc of wokng wth th caton opato fomalsm of scond quantzaton fo bosons and fmons and th us of wav functons wth constant nom fo dffnt numb of patcls but multpld by tm dpndnt coffcnts. - 7 -
Th two ways of consdng th quantzaton of th lctomagntc fld (namly th Dac and th Landau-Pls fomalsm) w compad by Oppnhm [71] n 1931 who concludd that Landau and Pls souc thoy was not ght bcaus of th appaanc of ngatv ngs and oton dnsts. Ths w howv xactly th sam poblms ntally butd to Dac s latvstc lcton quaton. Evn whn thos poblms ddn t pvnt a full succss fo Dac s lcton thoy no mo wok sms to hav bn don n ths dcton fo otons untl 1949 wth th xcpton of a wok by Kmm [7] n 1939 who analyzd a spnoal xpsson fo th Maxwll quatons ncludng th potntals n th Lontz gaug. In 1949 Mol [73] poposd th us of th lctomagntc fld vctos n complx notaton pvously dvlopd by Slbstn n 197 [74] as th oton-wav-functon and to us scal solutons fo th Maxwll s quatons. Slbstn dfnd th complx lctomagntc flds functons as F E+ cb wh Eand B w th lctc and magntc flds as usual. It can b notcd that Oppnhm [71] n hs pvous pap also cognzd th Slbstn functons as th appopatd wav functons fo th oton. Fo a pod of about yas wok about oton-wav-functons focusd manly on th poblm of localzablty of patcls wth zo mass. Th woks by Pyc [75] Nwton and Wgn [76] Whghtman and collaboatos [77] cam nvaably to th concluson that zo mass patcls wth spn qual to on (lk th oton) would not allow fo th xstnc of wav functons abl to localz a f patcl n spac as was possbl wth th oth patcls. As a coollay thy concludd that th oton as an ndpndnt patcl could not xst. Ths was ntmatly latd to th fact that th f lctomagntc fld s dvgnc lss n f spac and that a satsfactoy and latvstcally wll-bhavd poston opato wth commutng componnts could not b dfnd mathmatcally at that tm [78]. W would lk to mak that th poblms psntd n thos paps w latd to th mpossblty fo dfnng oton-wavfunctons abl to confn th oton to a pcsly dfnd gon of spac. On th oth hand thy could not pvnt th possblty of postulatng th xstnc of otons as patcls n spac ndpndntly fom th popts of th wav functons to whch thy w assocatd. Two dffnt solutons appad to b possbl at that tm: 1) to nounc to th full localzaton of th oton and accpt only patal localzaton n small volums o ) popos that th oton mass could b dffnt fom zo and to wok nsd th famwok of th Poca quaton [79 8] fo spn 1 patcls wth nonzo mass. Th fst appoach was poposd by Jauch and Pon [81] and Amn [8] and dvlopd xplctly n th modl by Mandl [83] whl th scond appoach was dvlopd by Blnfant [84] and dscussd by Bass and Schödng [85]. W should mnton h th stong analogy that xsts btwn th Slbstn s functons fo th lctomagntc fld and th solutons of th Dac s quaton fo th latvstc lcton. Th lctomagntc flds a typcally consdd solutons of th Maxwll s scond od dffntal quatons but n th Slbstn s notaton thy a also solutons to a spnoal lna quaton as shown by Good [86] and Moss [87]. Smlaly th lcton Dac quaton whch ognally was psntd n spnoal fom - 8 -
can also b wttn as a st of scond od dffntal quatons fo th lcton functon and fo th tm dvatv of that functon as was shown by Fynman and Gll- Mann [88]. Th ognal fom of th Dac quaton and th spnoal fom of th Maxwll quatons a almost dntcal. Bsds th fnt lcton mass th only dffnc s that th Dac s lcton-wav-functons hav fou componnts whl th lctomagntc flds n complx notaton hav only th componnts namly th th componnts of th lctomagntc fld n spac. Supassng th poblms mntond abov wth th patcl ntptaton fo th oton som woks w publshd n th 197-199 pod. Fo xampl th wok by Moss [87] who showd that by wokng wth scal gnvctos of th cul opato not only ntptatonal advantags could b achvd but also pactcal bnfts such as th goous dtmnaton of tanston matx lmnts. In th aly 198s Cook [89] was abl to show that th oton dynamcs n f spac was Lontz covaant. Aft 199 th man pont was th alzaton that th wav functon psntd th pobablty fo fndng th oton ngy at a gvn poston n spac. Among thm w can mnton th woks by I. Balynck-Bula [991] Inagak [9] and Gstn [93]. Th fst on to consd th xplct ntacton of adaton and m n th contxt of oton-wavfunctons appas to b Sp [94] who also showd that th scond quantzaton fomalsm whn appld to th oton-wav-functons povds th sam sults as th tadtonal mthod n th dscpton of th oton-atom ntacton. Kll has dvlopd a dtald modl cntly [95] wh howv colatons btwn th m flds and oton flds a not consdd and t s not possbl to gnat ntanglmnt btwn th otons and th atomc systm. A al bakthough has bn gvn cntly by Magat Hawton and cowoks [96] who obtand an xplct xpsson fo th poston opato wth commutng componnts fo th oton thfo nvaldatng th wok don n th dcads of th 5 and 6 about th fomal mpossblty of dfnng oton as matal patcls. C. Quantum umps and th oton patcl. Fom th bgnnng of th study of atomc pocsss t smd appant that m undwnt dscontnuous voluton. In th old quantum thoy dvlopd by Nls Boh [9] n th 191s ths was xpssd by th assumpton of th xstnc of quantum umps. In that thoy th atom could b found n th xctd statonay stats fo a pod compaabl to th man lftm of th stat and thn t dcayd spontanously n a quantum ump to th gound stat smla to th adoactv msson of patcls by th atomc nuclus [97]. Wth th full dvlopmnt of quantum mchancs ths dscontnuous tanstons w lat dntfd wth th pocss of masumnt. It was von Numann [98] n 193 who stablshd claly th Copnhagn ntptaton fo th act of masumnt. In ths ntptaton bcaus of vy masumnt th wav functon collapss nto th subspac of th Hlbt spac spannd by th sult of th masumnt. Also as a sult of th masumnt th obsvd systm plus th masumnt appaatus a lft n a colatd stat ndcatng that f th appaatus shows - 9 -
a sult α fo th obsvabl A thn th obsvd systm s lft n th subspac cospondng to th gnvcto of A wth gnvalu α. A followng masumnt of A wll gv nvtably th sam valu α. In ths ntptaton t could b sad that th act of masumnt of th oton by th dtcto collapss th stat of th atom fom th ognal xctd stat to th gound stat. Th xact dfnton of masud systm and masung appaatus s not a cla cut n quantum mchancs but at th sam tm dffnt slctons fo th ln spaatng systm fom masung appaatus cannot modfy th sult of th masumnt. In fact on can agu that f w masu th oton at a gvn tm ths mans that th atom dcayd to th gound stat mttng th oton at a pvous tm allowng th oton to tavl though spac and ach th dtcto. Und ths ntptaton t would b th oton th masung appaatus and th collaps of th atomc wav functon would tak plac bfo th masumnt by th dtcto namly dung th oton msson tslf. A fst xpmntal obsvaton of ths quantum umps was pfomd n th lat 197s by th obsvaton of th antbunchng ffct n th sngl atom msson by Mandl [99] and collaboatos. Ths ffct shows that aft th msson of a oton by an atom on has to wat a fnt amount of tm n od fo th atom to b abl to mt a scond oton vn whn th las pump mans at a constant pow. Ths fact s ntptd by sayng that f th atom collapss (umps) to th gound stat aft th msson of th fst oton t wll nd som tm to b -xctd agan and -mt a scond oton as shown by th quantum gsson thom [1]. Snc thn quantum umps hav bn obsvd n oth xpmnts [11 1] poposd ntally by Dhmlt [13] n 1975 wh th atoms gt shlvd n an xctd stat and a unabl to absob o sc adaton fom th gound stat. Aft th fst woks dctd to undstand ths nomnon [14] numous thotcal wok has bn don n od to ncopoat th quantum umps nto th pto of yscs [15]. On th thotcal sd th ncluson of quantum umps has bn usd as an altnatv appoach n smulatons to poduc th tm voluton of th dnsty matx. In ths smulatons th avag systm s dscbd as a collcton of many ndvdual wav functons followng pods of contnuous voluton and andom quantum umps followd by th nomalzaton of th wav functon [16]. In spt of all ths dvlopmnts th man quston about th quantum umps mans th sam snc th tm of Boh namly: what tggs th ump? Tadtonal quantum mchancs s unabl to answ ths quston howv alstc thos can povd an answ at th tm thy a solvng th poblm of masumnt. Entanglmnt of th wav functon maks th quantum potntal dpndnt on th poston of all th patcls composng th systm. Whn on patcl nts som spcfc wav-packt that has no spatal ovlappng wth th oth ons th only pat of th wav functon flt by th manng patcls n th systm s th on dctly colatd to th on that s occupd. A smpl xplanaton offs by tslf n th cas of oton msson: f th oton lavng th atom s colatd only wth th atomc gound stat aft msson th lcton ss - 1 -
only th gound stat wav-functon and stops oscllatng. Ths should happn n a pod compaabl wth th tm t taks to th oton to lav th atom vcnty; w can say that a quantum ump has occud. III. PHOTON-WAVE FUNCTIONS In ths Scton w stat wth th dvlopmnt of ou fomalsm. In pat A w povd th psnt status about th consdaton of th lctomagntc flds and potntals as th oton-wav-functons. In pat B w consd th spaaton of th lctomagntc flds nto f and achd to th patcls. Ths spaaton wll pov to b of fundamntal mpotanc fo th gnaton of ntanglmnt btwn th lctomagntc flds and th atomc systms. A. Maxwll s quatons as th sngl oton Schödng quaton Inducd by th dvlopmnts dscbd n th pvous Scton w popos to consd th lctomagntc fld as psntng th wav functon fo otons. In suppot of ths da w mnton that Moss [87 17] was abl to show that Maxwll s quatons tak th fom: 1 Ψ Ψ 4πΦ t ad H ad Ψ s a fou-lmnt column matx constuctd wth th lctc and wh ( ) ad magntc flds Φ ( ) s a souc-cunt tm th Hamltonan H s gvn by 1 x y z H α and ( α α α ) α s a st of 4x4 matcs. Wth ths dfntons th smlaty wth th Dac quaton fo th lcton s vdnt and n th absnc of soucs th only dffnc s th xstnc of a fnt lcton mass. W can mnton that convsly th caton of an lcton-poston pa by a gamma ay also admts a dscpton allowng fo a souc tm [18 19]. Whn w consd th ntacton btwn m and lght w fnd that th Schödng quaton nvolvs th lctomagntc potntals n addton to th lctc and magntc flds. W nd thfo to fnd an xpsson wh ths potntals a avalabl at th sam tm than th lctc and magntc flds. In som sns th potntals sto nfomaton about th hstoy of th flds and dsv to b consdd as notd by Bohm and Ahaanov [11] on th sam footng as th lctomagntc flds. In od to kp tack of th flds as wll as th potntals Kmm [7] dvlopd n 1939 a spnoal psntaton of th Maxwll quatons ncludng not only th lctc and magntc flds but also th lctomagntc potntals n th Lontz gaug. Th tm voluton of ths spno was govnd by th followng Schödng-lk quaton: - 11 -
( Ψad h H Ψad ( + J ( (1) t wh J ( was a nw souc functon. In Kmm s appoach th Schödng-lk quaton psntd th Maxwll s quatons: E B A φ B E E φ A t t t t wh th last quaton s th Lontz condton fo th potntals. W mak h that q. (1) holds as an altnatv fomulaton of th classcal Maxwll quatons and as th Hsnbg quaton of moton fulflld by th lctomagntc fld opatos n quantum lctodynamcs. Kmm also studd th latvstc nvaanc popts of th gvn matx H but w a not gong nto th dtals h. W only mnton that ths psntaton s known as th Kmm-Duffn-Hash-Chanda fomalsm [7]. Of cous on can wt Maxwll s quatons followng ths fomalsm n th Coulomb gaug at th pc of losng Lontz nvaanc. B. Spaaton nto achd and f flds It s wll known that lctomagntc flds dvg at th poston of th patcls. Thfo th lctomagntc flds a not nomalzabl and an nfnt amount of ngy s assgnd to thm. Ths s n dstncton wth th wav functons fo lctons whch a squa ntgabl ov th full spac and allow fo an asy manpulaton nsd th Hlbt spac. Th pupos of ths scton s to show how t s possbl to spaat th lctomagntc fld n two pats. On pat achd to th patcl wll b th sponsbl fo th dvgncs and non-nomalzablty of th lctomagntc flds. Th oth pat assocatd wth th adatd ngy wll b nomalzabl and ncopoatd nto th fst quantzaton fomalsm of th Hlbt spac. In laton to th quston of ngy contnt of th systm w can avod th dvgnc poblms of quantum lctodynamcs by assgnng th momntum and ngy contnts of th systm to th patcls and not to th flds. Th achd flds wll play a ol analogous to potntal ngs and thfo can dvg to nfnty n som gons wthout mplyng an nfnt ngy fo th yscal systm. W wll ncount stuatons wh th numb of patcls psnt n ou systm changs wth tm. It s cla that th caton of a patcl modfs th numb of dgs of fdom of th systm by addng to th poston and vlocts of th alady xstng patcls thos cospondng to th nwly catd on. W can howv duc th us of a vaabl numb of otons by assumng that otons that hav bn cohntly scd stll suvvd confnd nsd th flds achd to th chagd patcls. Ths pont of vw s n ln wth th ognal appoach followd by Dac [66] wh th - 1 -
vacuum stat was ust th gound stat fo th oscllatos psntng th adaton fld. In that appoach th act of mttng a oton mly changd th stat but not th numb of dgs of fdom of th systm. Anoth advantag of consdng achd flds s that thy wll allow us to gnat ntanglmnt btwn th dffnt stats of th adaton fld and th dffnt stats of th atomc systm. In od to dvlop ths da futh lt s consd th Maxwll s quaton fo th lctc fld: 1 J ( ) ( t c t E µ t It s cla that ths quaton dscbs two subsystms th patcl psntd by th cunt tm and th flds. Lt s wt th lctc fld as th sum of two pats whch w mght call th achd flds lnkd to th patcl and th f flds popagatng n f spac as E ( E ( + E f( wth E ( t ) and E f ( t ) th achd and f flds spctvly. Now w can wt th Maxwll quaton as follows: 1 J ( ( ) ( )) ( t + f t c t E E µ. t Nobody can pvnt us fom dfnng an ffctv cunt dnsty by th followng xpsson: Jf( J ( 1 µ µ E ( t t c t In tms of ths ffctv cunt dnsty w can wt th Maxwll quaton fo th f flds as 1 Jf( f( c t E µ. t On can ntpt ths quaton as dscbng agan two subsystms but now thy a th ffctv cunt souc fo th f flds (whch w can ntpt as th ognal lcton cunt togth wth th achd flds) and th f adaton flds. Ths s not a nw da; n fact dffnt modls hav poposd dffnt lns spaatng achd fom adatd flds. A fst xampl s ust gvn by th Coulomb gaug wh th longtudnal lctostatc flds play th ol of achd flds and th tansvs flds a consdd adatd o f [5]. Th souc cunt fo th f flds bcoms th wll-known tansvs cunt. Kams [111] Paul and Fz [11] ncludd th magntostatc flds dvd fom th statonay pat of th vlocty n th achd flds. Indd th da of mass nomalzaton of quantum lctodynamcs can b ntptd as a latvstc gnalzaton of th Kams tansfomaton [113 114 115] wh th ngy assocatd wth th magntostatc achd flds s ncopoatd nto - 13 -
th xpmntally obsvd lcton mass. Fnally Kll ncludd a tansvs achd fld n th dscpton of an oscllatng dpol and povdd a untay tansfomaton ladng to mov t [116]. Fo an oscllatng dpol w can s that all modls daw th dvdng ln at a dstanc of about on wavlngth fom th souc. In ths wok w wll dfn th f flds as th acclaton flds and th achd flds as th vlocty flds. Wth ths dfnton w vfy that th achd flds a domnant at a dstanc small than a wavlngth fom th atomc systm whl th f flds a domnant at a dstanc lag that a wav lngth fom th atomc cnt. W antcpat h that n th famwok of a alstc modl oton patcls should spond to both achd and adatd flds. As long as th oton s clos to th atom than on wavlngth th achd flds wll govn ts dynamcs so w can say that th oton blongs to th atomc systm t s an absobd oton. As soon as th oton lavs th poxmty of th atomc systm t wll b govnd by th adatd flds and bcom a f oton an mttd oton. Lat n Scton VII w wll analyz th obctons asd aganst th concpt of a fld achd to a patcl. Accodng to ou dfnton th achd flds fo chagd pont patcls a gvn by [117]: ( γ nˆ v E 3 and th f flds by E f ( nˆ c c B ( c) ( t ) [ nˆ E ( ] 1 nˆ t v t () {( nˆ v c) v& c} B f( t ) nˆ E f( ( 1 nˆ v c) t 3-14 - [ ] t (3) wh all th quantts and postons hav to b takn a t a tadd tm t' t c and fs to th adus vcto statng at th poston of th patcl and ndng at th fld poston. It s vy mpotant to not that th f flds dvg at most as 1 as w appoach th patcl: 1 1 E f B f. In ths way th nom and also th tadtonally assgnd ngy contnt man fnt aft ntgaton ov spac. W can say that ths patton nto achd and f flds s latvstcally nvaant. Th vlocty flds tansfom among thmslvs wthout mxng wth th acclaton flds. F and achd flds satsfy th quatons: E B ( ( f f
1 f E f ( B ( ( E & f + c B & f ( E f( and E t 4 B ( ) πρ ( 1 E & t B t E + ( ) ( ) ( c B & t E t ( ) ( ) Wh w dfn E f E f t ' t δ ' d' E ( ) ( ) ( ) E ( ( ' δ ( ' ) d' f wh δ ( ' ) δ ( ' ) δ ( ' ) tansvs and longtudnal dlta functons n a smla way to th dfnton of th Ths quatons can b wttn n matcal fom. Th popts and symmts of ths matcs a smla to thos consdd bfo fo th Kmm-Duffn-Hash-Chanda fomalsm [7] th only dffnc bng that th soucs a assocatd wth th achd and f flds spctvly. In th dpol appoxmaton th tms popotonal to th fst tm dvatv of th dpol and th scond tm dvatv of th dpol cospond to th vlocty and acclaton flds found n th non-latvstc appoxmaton. Th tm popotonal to th dpol cosponds to th dvaton of th poston of th chag fom th ogn of coodnats. H p () t psnts th magntud of th dpol at tm t assumd to pont n th z dcton and to b locatd at th ogn of th systm of coodnats. W fnd that th achd flds a: p( t - /c) p& ( t - /c) E ( cos( ) ˆ sn( ) θˆ d + θ + θ ) 3 c p& ( t - /c) B sn( )φˆ d θ c and th f flds a: p( t c) sn( θ ) θˆ Ed f & c p( t / c) B sn( )φˆ d f & θ c - 15 -
In od to fnd th souc tm fo th adaton flds n th dpol appoxmaton w notc that thy a always ppndcula to th adus vcto connctng th dpol wth th tst pont. Thfo what w nd s ust th pocton of th flds and th souc nto th -ppndcula dcton. Ths s th sam as th tansvs dlta functon but now n -spac ths s f() () δ δ. IV. THE SCHRŐDINGER-MAXWELL EQUATION In ths scton w popos a nw schm fo th assocaton of lctodynamcs and quantum mchancs by povdng a combnd Schödng-Maxwll quaton dfnd n a gnalzd confguaton spac. Ths spac ncluds not only th tadtonal patcls as lctons and potons but also th oton patcls as wll. Th gnalzd quaton povds a unfd fst quantzd thoy fo all thos patcls n spac. W stat wth th obsvaton that n tadtonal many-patcl quantum mchancs n fst quantzaton th Schödng quaton ads F(... ) 1 n Kˆ Vˆ h + ( ) F( 1... n ) (4) t wh F ( 1... n ) s th wav functon dfnd n confguaton spac. Th Kˆ Vˆ s th Hamltonan wh th fst tm opato + ( ) 1 m psnts th sum ov th kntc ngy of th patcls and th Vˆ th sum ov all ntacton ngs. Th wav functon can b Kˆ scond tm ( ) wttn n gnal as th sum ov poducts of th wav functons fo ndvdual patcls: F ( ) b ψ ( ψ ( Lψ ( 1 n k 1 k n stats L L (5) Compason of q. (1) and q. (4) nducs us to consd th dntfcaton of th oton hamltonan H fom q. (1) wth th oton kntc ngy and J ( wth th oton ntacton ngy. Ths allows us to wt a gnalzd wav quaton as: F ( L ) ( L ) 1 n h HF 1 n t wh F ( ) ˆ 1 L n s th wav functon dfnd n th gnalzd confguaton spac and w dfn Ĥ as th Schödng-Maxwll Hamltonan - 16 - (6)
Kˆ ( ) + Vˆ ( ) + Hˆ 1443 ( ) ˆ ( ) 144 443. (7) H ˆ + Schödng Maxwll a sum of th Schödng and th Maxwll opatos ths last on n th Kmm-Duffn- Hash-Chanda fomalsm wh to smplfy th notaton w hav consdd th ntacton of a sngl oton wth an atomc systm. Fo th wav functon w wt n analogy wth q. (5) poducts of th fom F ( 1 L n ) b Lk l m nψ ( 1 ψ ( Lψ k ( n stats ad ad ad Ψ ( Ψ ( LΨ ( l 1 wh now ( ψ ( ψ ( m ψ 1 L k n psnts th poduct of lcton-wav-functons ad ad ad and Ψl ( ( 1 Ψm L Ψn ( o th poduct of oton-wav-functons. In suppot of ths modl w can mnton that th Hsnbg quaton of moton fo th lcton-wav-functon opato ψˆ ( t ) n scond quantzaton s dntcal wth th Schödng quaton fo a sngl lcton ψ ( t ). Also th Hsnbg quaton of Ψ ˆ n quantum lctodynamcs s moton fo th lctomagntc fld opato ad ( th sam as th Maxwll quatons fo th classcal flds ( ) Ψ ad t n th Kmm fomalsm as gvn n q. (1). In th sam way th quaton of moton fo th poduct of th opatos ψ ˆ( Ψˆ ad ( t ) wll b gvn by ( ψˆ ( Ψˆ ( ) ( ) ( Ψˆ ( ) ( ad Ψˆ ˆ ψ ad h ad t h + ˆ ψ t h t t t ad ( Hˆ ˆ ( + ψˆ ( Hˆ Ψˆ ( ( Hˆ + Hˆ ) ˆ ( Ψˆ ( Ψˆ ψ Sch ψ (9) ad wh w hav usd th Hsnbg quaton of moton fo th opatos ψˆ ( t ) ˆ ( ) Ψ ad t and th fact that thy commut at qual tms. W can s that th xpsson povdd by q. (9) s th sam as q. (6) and q. (7). Th oton-wav-functons appang n q. (8) can b f o achd. W can fnd f tms of th fom Ψ l ( Ψm ( wh both functons sha th sam oton coodnat. Th ul w a gong to us s that f oton-wav-functons wll cospond to th xstnc of al otons n spac fulfllng ngy qumnts and thfo multplyng lcton stats cospondng to th low stat of th tanston ognatng th msson of adaton. Attachd stats wll b consdd as blongng to th al chagd patcl gvn s to th flds. Thy wll multply both stats cospondng to th tanston takng plac. Sch n o ad (8) and - 17 -
Th solutons to quaton (6) hav th countpat on th dffnt stats of adaton and m n quantum optcs. An mpotant ol n quantum optcs s playd by th so calld vacuum stats namly th stats of th adaton fld wth no otons psnt. W wll ntoduc th quvalnt ot vacuum stats also n ou dscpton povdd by q. f (8). Thy wll b gvn by tms of th fom Ψ vac ( null ) and Ψ vac ( null ) wh th fst on cosponds to mttd flds and th scond on to achd flds. Th dffnc wll b that ths functons a not assocatd wth any al oton at all f thfo thy hav an agumnt povdd by null. Th functons Ψ vac ( null ) may appa multplyng th upp stat of ngy consvng tanstons o both stats fom ngy non-consvng tanstons. Th functons Ψ vac ( null ) wll multply both stats of ngy non-consvng tanstons. To xmplfy th pvous consdatons lt s wt down th wav functon fo th cas of a sngl lcton and a sngl oton n th pocss of msson of adaton: f f F( [ aψ ( Ψvac ( null ) + bψ g ( Ψad ( ] Ψad ( t ψ t a th atomc xctd and gound stat spctvly. wh ψ ( ) and ( ) g W wll adopt fo th flds th multpol xpanson. W not that th adopton of th multpol fomalsm s quvalnt to th dscpton povdd by th Pow-Znau- Wooly tansfomaton [118 119]. W adopt ths dscpton bcaus t movs th statc longtudnal flds that a dstngushabl fom th oscllatng flds whl mantanng full causalty. Th statc poton of th achd flds s automatcally ncludd n th Coulomb potntal. Nglctng tms quadatc n th vcto potntal q. (6) can b wttn n mo dtal as: F f h ( H ˆ at + Hˆ I + Hˆ Hˆ )F t + f wh Ĥ at s th usual atomc Hamltonan. Th tms Ĥ and Ĥ psnt matcs fo f flds and achd flds spctvly. Th ntacton Hamltonan Ĥ I has two tms Hˆ ˆ I H ˆ f + ad I at th fst tm ˆ ˆ + ˆ gvs s to th souc cunt ad ad ad appang n th Maxwll s quatons fo th f and achd flds and th scond tm H ˆ I at cosponds to th standad ntacton tm n th patcl Hamltonan dscussd n many txt books. Ths two tms connct wav functons wth f flds to wav functons wth and wthout f flds ths last on n a smla way as th thoy by Landau and Pls [69]. Th xplct fom fo th souc cunt wll b dscussd lat n Scton VI. Fo th scond tm w popos th followng fom: Hˆ q p Aˆ + Aˆ δ I at ( ( ) ( ) ( ) - 18 -
ˆ + psnts th total (achd plus f) vcto potntal opatos and p h s th usual momntum opato fo th -patcl. Th sum xtnds ov all chagd patcls wth coodnats and s th coodnat n confguaton spac assgnd to th oton patcl th Dac s dlta nsus that th fld s takn at th poston of th patcl. wh A( ) Aˆ ( ) Th oton-wav-functons followng th Kmm-Duffn-Hash-Chanda fomalsm E ( B ( A( ) t φ a gvn by ( ) ( Ψ t and fo th oton vacuum stat w dfn 1 1 1 1 E ( null B( null A( ) null t φ ( ) ( null Ψvac null. W assum that no al oton s v assocatd wth th 1 1 1 1 coodnat null howv vacuum flds can b latd to th Casm focs and oth assocatd ffcts. Ths dfntons hold fo both fo th f and fo th achd flds. Ex ( H w hav dfnd E( Ey ( and smlaly fo B( E ( and A ( and z 1 1 1. Smlaly w dfn th adunct oton-wav-functons 1 s s s s s s * * * * Ψ ( ( E ( B ( A ( φ ( 1 1 1 1) and s s s Ψ t E t B t A t φ t 1 1 1 1. Wh vac ( ) ( ) ( ) ( ) ( ) ( ) null null null null null - 19 -
- - ( ) * * * * z y x E E E E s and ( ) 1 1 1 1 s. Ths dfntons allow us to dfn th adunct wav functons ( ) t F. Wth ths dfntons th opato f Ĥ taks th xplct fom ( ) ( ) ( ) ( ) 1 ˆ f H t wh ( ) s / / / / / / x y x z y z ( ) z y x / / / and ( ) ( ) z y x / / /. Th ( ) Â and ( ) A ˆ opatos fo f flds can b gvn xplctly by th matcs: ( ) 1 ˆ t A and ( ) 1 ˆ t A. It can b vfd that th opato ( ) Â appld to th oton-wav-functon Ψ gvs
A ˆ A Ψ( ) so that Ψ ( ) Aˆ Ψ( ) A( ) and ( ) A ˆ * Ψ ( ) A ( ) ( ) usual n quantum optcs. Th opato pˆ ˆ ( ) vac - 1 - Ψ as A sandwchd btwn two wav functons gvs Fvac ( Aˆ * p ˆ( ) F ( ( ) dd A( ( ( pˆ 1 δ ψ ψ1( ) d as qud by th ntacton tm n th patcl Hamltonan. Also ths opatos fulfll th followng popts: ˆ A ( ) Ψ ( ) Ψ ( ) Aˆ ( ). In th cas vac null vac null of havng many otons th total opato sn by th atomc systm wll b  wh  wll act only on th -oton. On any oth oton wll poduc zo. Th sam appls fo A ˆ. Its gnalzaton fo achd flds and total flds s staghtfowad. Smla dfntons can b gvn fo th opatos fo th lctc and magntc flds spctvly. nut Lt s dfn th nutal-dot opato fo otons by Ψ ( ( ) Ψ( 1 fom: 1 nut nut nut ( ). W s mmdatly that ( ) ( ) nut nut nut ( ) ( ) ( ) nut nut ( ) f and a nutal-dot opato actng on achd wav functons ( ) dfn th oton ngy-scala poduct by Ψ ( ( ) Ψ( E B vac n n matx and. W dfn a nutal-dot opato actng on f wav functons +. Lt s.g. th
lctomagntc ngy dnsty and th oton coss poduct opato ( ) * Ψ ( ( ) Ψ( E B - - by ths s th coss poduct btwn th lctc and magntc flds popotonal to th Poyntng vcto. In both cass th flds a th total flds f plus achd. Explctly thy a wttn as: nut nut nut nut Eˆ + Eˆ Eˆ + Eˆ + [ ][ ( ) ( ) ] ( ) f ( ) ( ) f f f nut nut nut nut + [ Bˆ f( ) + Bˆ ( ) ][ Bˆ f( ) + Bˆ ( ) ] f f and nut nut nut nut nut ( ) [ Eˆ f ( ) + Eˆ ( ) ]( ) Bˆ f ( ) + Bˆ ( ) f nut wh th opato ( nut ) s gvn by ( ) [ ] s wth 1 1 s ˆ 1 ˆ + + 1 kˆ. W mak that th sult of 1 1 Ψ t Ψ t s a vcto n th-dmnsonal spac. ( )( ) ( ) W can dfn th nom-scala poduct btwn two oton-wav-functons as Ψ Ψ Φ Ψ Ψ Ψ f f f ( ( ) Φ ( f ( ( ) Ψ ( d d f and th nom of th functon s gvn by. Bcaus w dfn th nom as gvn xclusvly by th f flds ths nom s fnt and allows us to wok nsd th Hlbt spac wth th oton-wav-functons. In th cas whn an ncdnt adaton fld s psnt th msson fom th atomc dpol may psnt two componnts: on whch has th sam fquncy and s cohnt wth th ncdnt bam and a scond whch has a dffnt fquncy and thfo s ncohnt wth th ncdnt bam. Th otons blongng to th cohnt componnt a ndstngushabl fom th otons fom th ncdnt bam. Thfo thy must hav th sam coodnat as th otons fom th ncdnt bam. On th oth hand th ncohnt otons a n pncpl dstngushabl fom th otons fom th ncdnt bam; thfo must hav a dffnt coodnat not psnt n th ncdnt bam.
Accodngly th souc quaton can b spaatd n two dpndng on th cohnc popts of th souc. Fo th cohnt msson w hav: scat Ψcoh ( scat coh h Hˆ Ψ ( ˆ ( coh + t wh s th coodnat of a oton psnt n th ncdnt bam fo th cunt dnsty componnt at th sam fquncy as th ncdnt fld. Ths cohnt scng s th sponsbl fo facton and flcton of optcal bams as wll as th dffacton and xtncton of x-ays n cystals. On th oth hand fo th ncohnt msson w hav ( scat Ψnc s scat nc h Hˆ Ψ ( ( s nc ˆ + s s t wh s a coodnat not psnt n th ncdnt bam fo th cunt dnsty s componnt at a dffnt fquncy than th ncdnt bam. Th ncohnt cunt wll b th souc fo nw otons catd n th scng pocss absnt fom th ognal ncdnt bam. In th psnc of many otons w hav: coh nc ˆ ˆ( ) ˆ ( ) + ˆ coh ( ) wh ˆ ( ) ( ) ad s s s ˆ s δ s ω ω ˆ s ( ) s ( ) ˆ( )( 1 ) δ δ and nc δ s s ω ω 1 s f w assum th systm s contand n a fnt volum and on can hav a fnt numb of dffnt fquncs. wh δω ω fom Konck whch s qual to on f th scd fquncy quals th ncdnt s s th dlta fquncy and qual to zo othws. In th cas of an ncdnt bam wth fnt wdth ω th δ ω ω can b placd wth th δ ω ω ω whch sgnfs thos fquncs alady psnt n th spctum of th ncdnt bam. δ s s th dlta fom Konck whch s qual to on f th scd oton s on of th ncdnt otons and qual to zo f th scd oton s nwly catd namly was not psnt n th ncdnt bam. On can assum that th oton mass s th paamt that can b usd to dfn th dstngushablty of th oton. If adaton changs fquncy bcaus of scng ths wll b flctd n ts assocatd oton mass m hν. Thfo on has to c chang th coodnats dscbng th scd oton: t s not th sam oton t was ncdnt on th systm bcaus ts mass s dffnt now. Th cunt dnsty opato s gvn n matcal fom by - 3 -
- 4 - ˆ A B E t t t so that ( ) Ψ ˆ A B E ad povds pcsly th cunt dnsts ndd fo vy fld n th nhomognous Maxwll s quatons. Fo chagd pont patcls w hav: ( ) ( ) f f E q t v δ ( ) ( ) E q t v δ ( ) t B ( ) t A In ths scton w hav compltd ou fst task namly th psntaton of a fst quantzaton fomalsm that wll allow us to postulat gudng condtons fo th matal patcls ncludng otons and to obtan a alstc dscpton fo th atom-adaton ntacton. Bfo gong nto th scond task w consd n th followng Scton th gnalzaton to th cas of many patcls. V. GENERAL CASE MANY PARTICLES A mult-patcl mult-oton nonlna gnalzaton fo th Hamltonan can b th followng: ( ) ( ) ( ) { } [ ] ( ) { } + + + + + + + ' 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ n n n f f ad q H H V m H A A δ actng on th functons ( ) ( ) ( ) ( ) Ψ Ψ Ψ ' 1 1 1 ' 1 1 n n k k k n n c F L L K K K K ψ ψ wh th -sum uns ov th dffnt lcton s coodnats and th -sum uns ov th dffnt oton coodnats. Ths quaton dscbs thos cass wh mo than
on lcton patcpat actvly and cohntly n th msson o absopton pocss and on thos cass of mult-oton msson o absopton common n nonlna optcs. Fo non-ntactng f patcls th wav functon must b poply symmtzd and antsymmtzd followng th postulats of ndstngushablty of dntcal patcls of quantum mchancs. In ths way th poply nomalzd wav functon fo bosons ads [1 11] F P ( ψ ( ) ψ ( ) Lψ ( ) 1 α β N! n1! n! L N υ wh α β L υ a th dffnt patcls composng th systm 1 N a th stats wh ths patcls can b found th sum s pfomd ov all possbl pmutatons of th patcl coodnats and n n 1 L a th numb of patcls n stat 1 ; fnally! mans th factoal as customay. W mak that ths slcton fo th nomalzaton also povds th ght xpssons fo absopton and spontanous and stmulatd msson n th cas of th ntacton of otons wth m [1]. W should mmb that n th cas of otons ndstngushablty s assumd only fo otons havng th sam fquncy ths s fo otons nsd th sam ngy shll [13]. Ths s consstnt wth th consdaton of th oton mass as an obsvabl abl to dstngush btwn otons of dffnt fquncs. Fnally w mnton that n ou modl w can povd an altnatv ntptaton fo th symmtzd wav functon gvn n q. (1). Instad of callng th pncpl of ndstngushablty of dntcal patcls w can qually wll say that ths xpsson s gvn as an ntal condton fo th wav functon gnatng ntangld stats wth stong non-local colatons. In ths cas th Bos-Enstn dstbuton would not b a consqunc of th ndstngushablty of th patcls but of th colatons xstng among thm [14]. (1) In th cas of ntactng patcls on cannot apply navly th symmtzaton postulats as sn fom th followng consdatons. Th tm dvatv fo any quantum mchancal opato s gvn by th commutato of ths opato wth th Hamltonan. Intgaton of that quaton povds th tm-dpndnc of that opato. In th spcfc cas of th caton and dstucton opatos fo th lctomagntc fld ths tm dpndnc wll b a functon of f spac adaton opatos and n th psnc of ntacton wth atomc systms of atomc opatos as wll. Th atomc opatos a n gnal not boson opatos. Thfo th adaton opatos aft ntacton wth m hav no mo pu bosonc commutaton popts. Thy psnt a mxtu of bosonc fmonc and spn popts. Bcaus of that th oton-wavfunctons wll not b stctly symmtcal. In gnal th typ of wav functon wll dpnd on th pocss of caton of th otons and th ndstngushablty of th oton paths [15]. - 5 -
W mnton h that on has th fdom dpndng on th gomty and constants of th poblm to choos as ndpndnt vaabls any lna combnaton of lcton and oton coodnats Ths wll povd multpl possblts fo ntanglmnt btwn lcton among thmslvs btwn otons among thmslvs and btwn lctons and otons. Th possblty fo th gnaton of ntanglmnt dffntats ou appoach fom smclasscal modls whch povd no way fo ntanglmnt wth otons. Fo solatd atoms scal symmty s th most appopat and th tchnqus dvlopd n nucla yscs [16] and atomc yscs [17] (fo xampl th hypscal modl) may b appld to fnd th solutons to th gnalzd wav quaton usng th appopat mult-patcl functons fo both lctons and otons. Fo las bams mpngng on nonlna cystals th us of cylndcal gomty mght b mo usful and th wav functons mght dpnd on th sum and dffnc of oton postons as s th cas fo th boton [18]. If w hav mo than on atom adatd smultanously by th sam lght bam all atoms may sha th absobd oton and may stat oscllatng cohntly n a collctv fashon. In ths cas th oton-wav-functon may hav soucs on all atoms gnatng nducd msson wth dctonal popts as n th cas of an aay of mttng antnnas. Ths s th cas of amplfyng las mda and cystals wth facton and flcton at th ntfacs. W can say that whn th dstanc btwn scng cnts s lag than a wavlngth thy a dstngushabl fo th ncdnt adaton and scd otons hav ndpndnt wav functons gnatng ncohnt scng. If th dstanc btwn scng cnts s shot than a wavlngth thy a ndstngushabl fo th ncdnt adaton gnatng cohnt scng. VI. ELECTRON AND PHOTON IN REAL SPACE H w stat wth th scond pat of ou wok namly w mak th assumpton that th lcton as a patcl has al xstnc n spac as n Bohman s mchancs and futhmo w xtnd ths assumpton to th otons. W assum addtonally that th total lctomagntc flds and not th f lctomagntc flds alon gnat th gudng condton fo th otons. In ths way w can concl th flux of ngy n th patcl modl wth th classcal Poyntng vcto. W wll postulat that th oton vlocty s gvn by th spd of popagaton of ngy; ths s th ato btwn th Poyntng vcto and th lctomagntc ngy dnsty as poposd by Wsly [19]. Ths appoach assgns to th oton-patcl th spd of lght n vacuum whn t s fa fom any chagd patcl but dcass fom that valu n th poxmts of chags du to th psnc of th scd flds. A. Souc cunts n th gnalzd wav quaton. - 6 -
In smclasscal modls [13 1] t s customay to tak th souc fo th lctomagntc flds as th cunt obtand fom th Schödng chag dnsty as ognally poposd by Schödng hmslf [13] and showd to b ncoct by Hsnbg [131]. Instad w popos h that th souc cunts a th al chagd patcls actually psnt n spac. W a nducd to tak ths poston aft consdng th cas of an ulta-latvstc lcton appoachng th ath fom out spac. If w tak th lcton-wav-functon squad (.g. th lcton pobablty dnsty) as th souc fo th lctomagntc flds t may cov a volum of phaps galactc dmnsons gnatng an almost unfom dnsty and lctomagntc flds of mmatal magntud. Any masumnt howv wll fnd th flds cntd at th poston of th lcton wth a stngth qual to that of any patcl confnd to a gvn locaton on ath. W mak howv that w kp th usual momntum p h n th xpsson fo th ntacton atomc ngy. Ths mans th lcton wav functon wll act to an ncomng lctomagntc wav vn n th cas whn th lcton s not oscllatng. B Total flds. Gudanc condtons. Fom th pont of vw of alstc thos fo th lctomagntc fld th hav bn hstocally two appoachs: th fst on du to Bohm [3 13 133 134] and th scond poposd by d Bogl [1 135 136]. Bohm s appoach consdd th xstnc of a sup-potntal actng on th lctomagntc flds n all spac and not on th oton patcls. D Bogl appoach consdd th applcaton of th gudanc fomula to spn-on-patcls of fnt mass dscbd by th Poca quaton but n th lmt fo nfntsmally small mass. Indpndntly of ths modls Wsly [19] poposd that th oton vlocty should b gvn by th ato btwn th Poyntng vcto and th ngy dnsty namly th goup vlocty o vlocty wth whch ngy popagats n a mdum. In a nonlatvstc appoxmaton w wll follow Wsly [19] who taks th gudng E B condton fo th oton [137] as v whch can b shown to povd th E + B coct valu fo th vlocty of popagaton of th ngy vn n nonlna cystals [138] and can b obtand by followng th Kmm-Duffn-Hash-Chanda fomalsm [139]. A latvstc gnalzaton can b found by followng th latvstc dfntons of momntum and oton dnsty [14 141]. W can not howv that ou modl wll povd a vy good appoxmaton to th latvstc soluton unlss th souc patcl s movng wth a spd compaabl wth th spd of lght. An mpotant pont s that th gudng condton s a functon of th poston of th oton and th poston of th lcton as wll gnatng a nonlocal colaton o ntacton among thm. - 7 -
W dfn th gudng condton fo th oton as a gnalzaton of Wsly condton: F*( ( ) F( v. F*( ( ) F( dfnd pvously. In a sns smla to Wh w us th two opatos ( ) and ( ) Bohman mchancs fo lctons on can assocat wth ths gudng condton a oton quantum potntal gnatd by th lctomagntc flds and sponsbl fo th moton of th otons. In som sns th achd flds smbl soucs of classcal potntals bcaus thy a always assocatd to th patcls gnatng thm. Howv th non-classcal natu s manfstd by th fact that thy can ntf wth th quantum flds gnatd by th f lctomagntc flds. W wll dfn th absopton of th oton whn ts tactoy achs th poston of th souc of th fld. On th oth hand th Quantum Potntal fo th lcton now ads: 1/ h ( F*( ( ) F( ) Q(. m F*( F( 1/ ( ( ) ) wh w mak xplct us agan of th opato ( ). Ths quaton s quvalnt to th Bohman gudng condton: ( ) ( v t. ρ( namly th local cunt dnsty dvdd by th patcl dnsty. In th psnc of lctomagntc flds th cunt dnsty s gvn by [14] h * * R [ ψ ψ ( ψ ) ψ ] A m mc wh R s th pobablty dnsty and A th vcto potntal. W can dfn an avag vlocty smoothng out th vaatons n vlocty ampltud gvn by th local valu of th wav functon and consd th avag ov th volum wh th patcl s movng: ( ) ( ) ( ) ( v ( ) ( ) ( ) t v t ρ t d d ρ t d d t d d t. ρ In th cas wh th wav functon can b factozd as th poduct of an lcton-wavfuncton and a oton-wav-functon th ntgaton ov th oton coodnats gvs ust a nomalzaton constant and w a lft wth th valu of th total cunt as gvn by th ntacton matx lmnt n th Schödng quaton. Th flds whch a poducd by th al cunts wll follow on th avag th dstbuton of th Schödng flds. In th dpol appoxmaton both xpssons should gv pactcally th sam sult. At ths pont w hav fnshd th dscpton of ou alstc modl. In th followng Scton w wll consd ts applcaton to a two lvl systm. - 8 -