The Quantum Oscillatory Modulated Potential Electric Field Wave Packets Produced by Electrons

Transcription

1 Jural f Mder Physics, 05, 6, Published Olie Nvember 05 i SciRes. The Quatum Oscillatry Mdulated Ptetial Electric Field Wave Packets Prduced by Electrs Waldemar Wley Filh Istitut de Flsica, Uiversidade Federal de Giás, Giâia, Brasil Received 3 Jauary 05; accepted 3 Nvember 05; published 6 Nvember 05 Cpyright 05 by authr ad Scietific Research Publishig Ic. This wrk is licesed uder the Creative Cmms Attributi Iteratial Licese (CC BY). Abstract I this wrk we are frmulatig a ew thery fr describig the wavig ature f a micrscpic electric particle. Based the predictis f the Quatum Oscillatry Mdulated Ptetial QOMP, fr describig the iteracti betwee tw micrscpic electric particles, electr-electr, fr istace, we are pstulatig that the wavig behavir f these particles may be a attribute f the charges f the particles ad t their masses as curretly accepted. Fr a micrscpic electric charge, we are presetig ew argumets shwig that the electric field i the viciity f a micrscpic charge is spatially wavig ad ca be determied as the gradiet per uit f charge f this ew quatum iteracti ptetial, with use f a apprpriated phase factr t accut fr the behavir f a ubud electr. Differetly f what is predicted by the classical Culmb electric field, whe a charged particle is mvig uder the acti f a ptetial f V vlts, the ew electric field existig arud the charge has the frm f a wave packet. Fr typical values f the ptetial V, the wavelegth f the wavig electric field is i very gd agreemet with thse experimetally bserved with diffracti f electrs i crystallie slids. Keywrds Ptetial, Electr, Wavelegth, Wave Packet, Bsic Atm. Itrducti I this paper we are revisitig a ld, but itriguig ad challegig tpic f the mder physics the pheme f particle-wave duality. Eve csiderig that the pheme has bee ivestigated by may physicists, fr mre tha a cetury, it seems t us that the subject still presets sme bscure pits ad deserves fur- Hw t cite this paper: Filh, W.W. (05) The Quatum Oscillatry Mdulated Ptetial Electric Field Wave Packets Prduced by Electrs. Jural f Mder Physics, 6,

2 ther ivestigati. I the develpmet f this wrk, it was able t bserve that the ew ptetial, used t describe the iteracti betwee tw micrscpic electric charges, tw electrs fr example, presets surprisig but strg evideces fr a pssible frmati f electr pairs i a evirmet f a dese cluster r a dese beam f electrs. Otherwise, bservig this pssibility frm ather pit f views, we are cjecturig that the electr-electr iteracti idicates a pssible existece f a kid f a bsic atm, a atm with characteristics similar t thse f the hydrge atm but with psitive eergy iteracti. Ather perspective t predict the existece f this ew type f atm is the frmati f a pair f electrs, such as the Cper pair used i the BCS thery, i the supercductivity. I rder t update the reader with the subject, withut gig it t much details, first we will make a brief verview f tw imprtat physical mdels, develped by emiet physicists f the last cetury. I the last years f the 0 decade, Eurpea physicists wrked ethusiastically t give a explaati fr this seemigly irreccilable prblem the particle-wave behavir. I the particular case f the electr, e f the first attempts t explai this pheme was made by Schrödiger, de Brglie ad Br []. I a duble-slit experimet, with the use f a beam f electrs, these physicists tried t explai the pheme, assciatig a prbabilistic wave t each electr f the beam, lkig fr a explaati t the bserved iterferece patters i crystallie slids. Here, as metied befre, we are t gig t etry i details f the methd. But, i geeral way, the results f the methd were i accrdace with the experimetal bservatis. What was t clear i their prpsed physical mdel was the essece r, i ther wrds, what was the real cause fr the frmati f the prbabilistic waves assciated t the electr. Adptig ather pit f view t slve the puzzle, years later a differet apprach was idealized ad frmulated by the America physicist Richard Feyma [] []. Based the results f experimets already available ad csiderig that at the ed f the 0 decade, the physicists have cme t the cclusi that ay attempt made t determie, simultaeusly, the ambiguus characteristic f the electr, particle r wave, the iterferece f the bserver i the prcess ivalidates the experiece. Later, i Secti 4, we will retur t discuss this pit. I a duble-slit experimet, Feyma, tryig t elucidate the prblem, tk a rather radical attitude. Accrdig t his pit f view, there is eed t ji the electr a wave f prbability. He argued that the electr, leavig the surce culd pass simultaeusly frm either f the tw slits f the apparatus, fllwig a ifiity umber f pssible paths util reachig the phsphrescet scree t prduce the iterferece patter. Accrdig t the view f Feyma, the electr siffs, simultaeusly, all paths cectig the begiig t the ed f the trip ad chse the mst cveiet e. Based i this premise Feyma preseted a ew apprach t the prblem. Accrdig t the perspective f Feyma, there is eed t create a wave f prbabilities fr the electr. Frm his pit f view, the electr, always see as a particle, may reach a certai pit the scree, ad is the result f the cmbied effect f all pssible ways t get there. This methd is kw as sum ver trajectries ad it is e f the majr ctributis f Feyma fr the develpmet f quatum mechaics []. Hwever, eve frm his pit f view, the pssibility f a electr t take simultaeusly differet paths, mre tha this, t take a ifiite umber f paths, cstitutes a legitimate bjecti t ur cmm sese. Mre tha that, this seems absurd. Other prmiet physicists, icludig Albert Eistei, ever agreed with the prbabilistic pit f view f quatum mechaics. By the ed f his life, Eistei was still believig that there was a differet way, a -prbabilistic maer, t explai the quata phemea. O the ther had, the scietific cmmuity recgizes that i the micrscpic wrld, the quatum mechaics presets a large umber f surprises. The results f the calculatis preseted by Feyma, based his methd, agree with the results f the methd previusly metied. Als, it agrees with the experimetal results. Thus, fr the majrity f the scietific cmmuity, this prblem seems t be slved ad thig else remais t be de. Hwever, this is t cmpletely true. As already stated previusly, sme bscure pits with respect t this subject, still persist ad deserve a ew apprach i a attempt t fid a mre cvicig explaati. Related t the subject uder discussi, at this pit, it is quite lgical t ask the fllwig questis: What physical prperty f the electr is respsible fr geeratig the, s called, wave f prbability assciated t this particle? Is it the mass f the electr? Is it the charge? Or is it the effect caused by a electric r a magetic field assciated t the particle? The electr has mass, has charge, has spi ad, csequetly, we like it r t, there is a electrmagetic field assciated t it. I this paper we are tryig t fid a aswer t these questis, adptig a quite differet pit f view. This subject will be ivestigated with details i the fifth part f this series f wrk. 094

3 Nwadays, e f the theretical mdels mre acceptable t explai the dual behavir f a micrscpic particle, such as the behavir f a electr i the duble-slit experimet, is t use the priciple f superpsiti f a ifiite umber f plae waves, t built up a wave packet arud the particle. Simple plae waves ca be btaied frm the sluti f the Schrödiger s wave equati fr a free particle. Hwever, we must bserve that the ly prperty f a free ucharged particle is its mass. Thus, the eergy peratr i the Schrödiger s wave equati, ly ivlves the kietic part f the Hamiltia peratr. S, fr a free particle mvig i space, the Schrödiger wave equati takes the frm Ψ ( r, t) = i Ψ( r, t). () m t It sluti is a simple plae wave, writte as ( ) (, ) e i kr ω Ψ r t = A t () where A is a arbitrary cstat. Hwever, a plae wave like this, ccupies the etire space. Thus, it becmes impssible t determie the psiti f the particle, whse behavir is described by this wave fucti, because the particle ca be aywhere i the space. S, t vercme the difficulty t lcalize particle, the physicists, artificially, created a wave packet arud the particle. This wave packet iteds t cfie the particle iside a small regi f space. The frmati f a wave packet ca be accmplished by makig the superpsiti f a large umber r, better yet, a ifiite umber f plae waves arud a specified mmetum p = k f the particle. Thus, usig this prcedure t built up a wave packet, the ly physical prperty f the particle that takes part i the prcess is the mass f the particle. I this case, the ly frce field that ca be assciated t the particle i mvemet is the gravitatial field. Hwever, fr a small mass, cmparable t the mass f the 30 0 kg electr, which is apprximately, the gravitatial field assciated t this particle is cmpletely egligible ad ca be discarded. Therefre, what is r what are ther prperties f the particle that ca be csidered fr the frmati f a wave packet? Let s try t aswer this questi i the ext paragraphs. I actual experimets f electrs diffracti, the kietic eergy f a beam f electrs is depedet bth, the mass ad the charge f the particle, ad csequetly, the electric field existig i its eighbrhd. Regardless f the csideratis r cjectures take attemptig t explai the dual behavir f a charged particle like electr, the itrisic electric field i the eighbrhd f the particle cat be disregarded. I priciple, accrdig t the Culmb ptetial, the frmati f a arrw ad dese beam f electrs shuld be diverget due t the repulsive electr-electr iteracti. I real experimets usig a thi ad dese beam f electrs, the electric frce betwee these particles must be repulsive ad, i priciple, the electr beam shuld be diverget. Hwever this is t bserved r, the divergece effect is very small ad ca be eglected Experimets with beams f electrs (charged particles) ad X-rays (waves) scattered by thi metallic fils shw diffracti patters with great similarities. The similarity f the diffracti patters shws that electrs ca behave like X-rays, displayig wave-like prperties. I 94, the Frech Physicist Luis de Brglie pstulated that assciated t the particle there is a wave packet, give t it wave-like prperties. Frm the special thery f relativity, Eistei have shw that the wavelegth assciated t a pht ca be writte as [3] h λ =, (3) p where h is the Plack cstat ad p is the mmetum f the pht. The wavig ature f the pht r its particle-like behavir appears whe a electr i a atm chages its quatum eergy state frm E i t E j, with Ej > Ei. Whe this quatum trasiti ccurs, the electr i the atm drps frm a quatum state f higher eergy t a quatum state f lwer eergy, emittig the excess f eergy i frm f electrmagetic radiati, a pht a tiy electrmagetic wave packet. Sice fr a bud electr, bth eergies E i ad E j are egatives, E i is less egative tha E j, a ecessary cditi fr a pht emissi. Let us attribute the relativistic liear mmetum f the emitted pht t the variati f the relativistic quatized liear mmetum p ij f the electr i the atm. This variati f the liear mmetum f the electr may be writte i the frm, mm pij = mvij = v, ij (4) v ij c 095

4 where we are csiderig m m as the miimum relativistic mass f the electr. I the Equati (4), the miimum relativistic mass takes place whe vij 0, what makes m m m. I a atm this electr trasiti ccurs betwee tw quata states with large quatum umbers i ad j. Frm the abve equati we ca bserve that the relativistic mass f the electr, whe ccupyig a particular eergy quatum state E f the atm, may be writte as mm m =, (5) v c which icreases whe the velcity v f the electr icreases. O the ther had, it is kw that, i a atm, the velcity v f the electr, fr a specified eergy quatum state E, may be expressed by [4]. v = 4π Ze where Z is the atmic umber. This equati idicates that the largest velcity f the electr ccurs whe it is ccupyig the grud state f the atm, =. The Equatis (5) ad (6) idicate that fr a bud electr, the relativistic mass f the particle decreases whe the quatum umber icreases. The, the smallest value f the mass f the electr ccurs whe. That is, whe the electr becmes free ad the relativistic mass m becmes equal t its miimum value, m= mm. The reader must bserves that i the Equati (4), we are t usig the ccept f rest mass fr the electr because, accrdig t the Heiseberg ucertaity priciple, x p =, fr micrscpic particle such as the mass f electr, strictly, this particle ever will be at rest i the labratry frame f referece LFR. Beig s, rigrusly, the rest mass fr a micrscpic particle like the electr, des t exist. Hwever, fr what is called a relativistic electr, that is the case whe its velcity vij c, we ca make the apprximati mm m0, what is called rest mass f the electr. Uder this cditi, the variati f the liear mmetum f the electr, whe the atm chages the quatum state frm E i t E j, ca be writte i the frm Ei Ej pij = mv ij =. (7) c But, i a mre geeral way, takig it accut trasitis ccurrig i heavy atms, where v ij may be csidered large, this equati may be writte i a slightly differet frm mv ij hf ij h = = c c c τ, ij h Eij where m is the relativistic mass f the electr ad, c τ = c beig the relativistic mass f the pht, τ ij is ij the trasit time fr the electr t mve betwee tw quata states f the atm. Durig this time iterval, the atm is lsig eergy, a pht is beig created ad gaiig the relativistic quatum eergy Eij = Ei E j = hfij. As it ca be see, the quatum relativistic Equati (8), has a simple but a prfud physical sigificace. It ifrms us, i a quite clear maer, the ature f the dual character f the particle. The equati shws that the dual behavir f a charged particle, the electr fr istace, is a pure electrmagetic pheme assciated t the particle. This ca be see i the fllwig way; the pheme f wave particle duality is due t a electric field assciated t the charge ad a magetic field assciated t the spi f the particle, i this case, the spi f the electr. I the Equati (8) we ca bserve that the left side f the equati is a prduct f the relativistic mass f the electr times the trasit velcity vij whe the electr i the atm jumps back frm a excited eergy state E i t a lwer eergy state E j. The argumets preseted abve, shw that the electr carries arud it a electrmagetic field that chages its iteracti cfigurati with the ucleus f the atm durig the time f flight f the particle, r equivaletly, durig the time the electr is tuelig thrugh the ptetial barriers as predicted by the ew ptetial, the Quatum Oscillatry Mdulated Ptetial QOMP [5] [6], which is give by the frmula (6) (8) 096

5 e 4π r ( ) = cs ( ξ r) U r where ξ is a space frequecy, it is a quatized parameter, give by πµ e ξ = 4π 0 0 where µ is the reduced mass f the system prt-electr i the hydrge atm, which is (9) (0). µ = m + m () p e Fr the hydrge atm, accrdig t the ptetial predicted by Equati (9), the chages f the iteracti cfigurati is shw i Figure, whe the electr i the atm chages its eergy frm E t E. The right side f the Equati (8) is the prduct f the relativistic mass f the pht ad its velcity c. The, by a questi f symmetry r cciliati, bth sides f the equati must preset, simultaeusly, bth aspects, relativistic mass ad wave. Frm a purely mathematical pit f view, the Equati (8) reflects the particle-wave dual behavir fr the electr. Accrdig t what we have see previusly, we have a clear idicati that assciated t a bud electr there is a quatum electrmagetic field ad, assciated t a free electr, we are pstulatig the existece f a electrmagetic field i the frm f a wave packet, mvig with the velcity particle. Because f this we ca, apprpriately, say that the electr is a electrmagetic-particle-wave maifestati f ature, sice the electr carries, simultaeusly, bth aspects, particle ad wave. As we will see i the ext paragraph, this maifestati will be idetified as the aget respsible fr the bserved particle-wave duality as it is bserved i experimets with diffracti f electrs i slids. The argumets preseted abve may be exteded t ther atmic particles, like the prt ad thers kw charged particles. Hwever, we must bserve that the eutr, a ucharged particle, als presets similar effect ad is frequetly used i experimets f eutrs diffracti. Beig s, hw this argumet culd be accepted fr explaiig the eutrs diffracti, beig these ucharged particles? We believe that the aswer t this questi is the fact that the eutr, as well as the hydrge atm, that als preset similar effect, are cmpsite particles. The eutr, as prpsed by Gell-Ma ad G. Zweig [7] is built up f three kids quarks, e labeled up with charge +/3 (u) ad tw thers labeled dw with charge /3 (d) ad /3 (d). The, ather way t represet the eutr is (udd). These particles must have assciated t them a electric diple mmet the s-called EDM. I this case, they will preset similar electrmagetic effects, like the electrs. Curretly there are at least fur experimets tryig t measure the EDM [8]. Theretical typical predictis fr the eutr electric diple mmet rage betwee 0 7 Cm ad 0 30 Cm [9]. Fr the hydrge atm, the QOMP metied befre, predicts that the electr, whe ccupyig the fudametal state f the atm, is separated frm the prt with 0 a average radial distace + 3 r = m [5] [6], what predicts the existece f a average electric 9 diple mmet p = cm fr the hydrge atm. The cfigurati f this atm is shw i Figure. The existece f this electric diple mmet is t revealed whe the Culmb ptetial is used t describe the iteracti prt-electr f the atm. The existece f the hydrge atm electric diple mmet (hedm), may cstitutes a great break-thrugh t explai the frmati r eve the existece f the H mlecule. This idea will be explred further elsewhere. Accrdig t what we have see previusly, it is quite clear that, exists assciated t the particle a electrmagetic field arud it. Because f this, the wavig behavir f the particle must be a attribute f its charge, creatig arud them wavig electric fields. Thus, fr the micrscpic particles, like the electr, the prt ad thers kw particles, a better term t describe them wuld be particle-waves because, i fact, they are electrmagetic wave particles, sice, bth aspects are iheret t these atmic physical bjects. At this pit, a iterestig questi is: What is the frm f the electric field assciated t a free electr i mvemet? This questi will be aswered i the ext sectis. I what fllws, fr sake f simplicity, we will csider the electr as a spiless particle. With this At preset, we are ivestigatig the pssibility that a iteracti electric diple-diple ca explai i a very simple way the frmati f the hydrge mlecule. 097

6 Figure. Chage f the prt-electr eergy cfigurati i the hydrge atm fr the electr ccupyig the grud state f the atm =, r = 0.53 Å, E = 3.60 ev, the black lie. Ad =, r =. Å, E = 3.40 ev, the red lie. Als is shw the eergy f the quatum trasiti E E. csiderati, the magetic part f the electrmagetic field, that due t the spi f the electr, will t be csidered fr the determiati f the frm f the wave packet we are lkig fr. Here, ur ccer will be slely directed t fid the frm ad the characteristics f the electric field assciated t the particle. As we have see previusly, a pht is geerated whe the electr i a atm makes a quatum trasiti. I this wrk we are arguig that trasitis ccur whe the electrs are tuelig thrugh fiite ptetial barriers as created by the ew iteracti ptetial, the QOMP, previusly described. Differetly f the classical Culmb ptetial, whe this ew ptetial is used t describe the iteracti prt electr i the hydrge atm, the electr is t rbitig the ucleus f the atm, but mvig back ad frth scillatig radially betwee tw classical turig pits R ad R, like a quatum harmic scillatr. See Figure. The, eve i the absece f ay ther field, there is a fluctuati f the electric field iside the quatum wells f the atm, with the electr, evetually, fallig t a lwer eergy state by tuelig effect, causig a pht emissi by the atm, as previusly described. Of curse, the iverse f this prcess is the pht absrpti by the atm. I 93, the same Luis de Brglie pstulated the existece f ather imprtat relati i which, the liear mmetum p f the particle is als related t the space frequecy k by π p = k, with k =. () λ Mrever, it is als kw that a relativistic charged particle with mass m, mvig with velcity v, has kietic eergy E k, give by p E ( ) k = p = mek. (3) m Frm Equatis () ad (3) we have the well kw frmula h λ =. (4) ( me ) k I particular, charged particles like prts r electrs, accelerated thrugh a ptetial f V vlts, have kietic eergy equal t Ek = ev. The wavelegth assciated t these particles, is predicted by h λ =. (5) mev ( ) 098

7 This equati tell us that the de Brglie wavelegth, besides its depedece up the mass f the particle, it is als depedet its charge ad the acceleratig ptetial V, which is a exteral aget actig the particle.. A Electric Field Wave Packet As explaied previusly, the Equati (3) predicts the wavelegth assciated t a pht, a electrmagetic wave packet. The Equati (5) predicts the wavelegth assciated t a electric particle, the electr fr istace, accelerated thrugh a ptetial f V 0 vlts. A beam f X-rays ad a beam f electrs presets similar effects whe used t prduce diffracti i crystallie slids. A X-ray pht ca be see as a electrmagetic wave packet, mvig with the velcity f light c. I this wrk, we are pstulatig that the free electr als must be see as a electrmagetic wave packet mvig with the velcity f the particle. Fr this argumets t becme mre csistet ad realistic it is ecessary that the electric field assciated t the particle i mvemet be spatially wavig, frmig a wave packet with size ad wavelegth apprpriated t be diffracted as it is experimetally bserved. A electric field with these characteristics may be frmulated takig the gradiet per uit f charge f the ew iteracti ptetial betwee tw electrs. Similar t the Equati (9), the ew ptetial may be writte as e 4π r ( ) = cs ( χ r) U r 0 where χ is the space frequecy. Als, similarly t the iteracti betwee the prt ad the electr i the hydrge atm, χ i the Equati (6) is a quatized parameter expressed by πµ e χ = 4π 0 with µ beig the reduced mass f the electr-electr iteractig system. Here it is represeted by µ = m (8) which, fr a relativistic cditi, the reduced mass µ is e-half f the electr rest mass m. At this pit we will pe a paretheses i ur presetati t make sme cmmets abut the electr-electr iteracti ptetial as expressed by Equati (6). This ptetial presets a uexpected ad a quite iterestig iterpretati. First, as it ca be see i Figure, the ptetial is t always repulsive as predicted by the classical Culmb ptetial, als displayed i the figure. The ew ptetial is ull at every pits where the quatizati relati χ r = π is satisfied, with beig a iteger umber, differet frm zer. It is easy t see that this quatizati relati is a direct csequece f the Heiseberg ucertaity priciple. The ew ptetial presets psitive decreasig barriers with the /r depedece. See this ptetial i ather way; it presets barriers heights that decrease mdulated by the classical Culmb ptetial as shw i the same figure. A quite iterestig aspect f this ptetial is that it predicts the existece f kid f electr-electr bsic particle. A kid f atm frmed f tw iteractig electrs. That is a atm frmed f tw fermiic particles f spi /, prducig a atm with spi S = (the tw electr spis paired) r spi S = 0 (the tw spis upaired). Because f this attribute we are callig it a electr-electr bsic atm. Figure als shws the first three cfiguratis fr the electr-electr iteracti ad the fur first eergy levels f this ew kid f atm. As metied previusly, ather way t see this kid f atm, is the pssible frmati f a kid f a electr pair, similar t the existece f Cper pairs i the state f super-cductivity fr sme metals ad metals allys at very lw temperatures, temperatures belw T c [0] [] 3. I Secti 4, we will see hw this ew kid f atm ca affect the diffracti patters i slids. Havig called the atteti f the reader fr these particularities f the electr-electr iteracti, let us retur t ur iitial presetati, sice this iterestig subject will be ivestigated elsewhere, with great details. Fcalizig ur atteti just up e electr f a beam, as previusly explaied, we ca admit the existece f a electric wave packet mvig with the same velcity f the electrs f the beam. Hwever, fr this argumets t becme mre realistic, it is ecessary that the electric field assciated t the particle i mvemet be 3 This iterestig subject will be detailed discussed i the fifth paper f this series. (6) (7) 099

8 Figure. The electr-electr bidig eergy as predicted by the QOMP. The red hriztal lie represets the bidig eergy fr the grud state f the ew atm, E = 6.80 ev. The eergies f the first three excited states f the ew atm are als shw i the figure. The spi f the atm may be S =, spis parallel, S = 0 spis atiparallel. I the grud state f the atm, the electr is squeezed betwee the classical turig pits P. spatially wavig, with a wavelegth value apprpriated t be diffracted as it is experimetally bserved i the crystallie slids. A electric field with these characteristics may be frmulated takig the defiiti f the electric field as U( r) E( r) = lim q e. (9) q The reas t use the limit ( lim q e ) i the defiiti f the electric field, is because the electric charge e is the smallest electric charge existig i ature. The, frm Equati (6) we btai the expressi E e ζ ( r) = cs ( ζr) + si ( ζr) r, 4π r r u (0) where ζ is the space-frequecy assciated t a free electr, which, due t the uusual peridicity f the ew electric field, it may be writte as π ζ = ( ) mev () h ad u r i the Equati (0), is a uit vectr take alg the radial directi. Curretly, it is accepted that the electric field existig at a distace r f a egative electric charge, a electr fr example, is predicted by the well kw classical Culmb frmula e E( r) =. () 4π r This classical electric field, ear the electr, may be btaied frm Equati (0), takig ζ = 0, what makes the liear mmetum pr = ζ f the electr t be ull. As stated befre, this makes the electr t be at rest i the LFR. This cditi ctradicts ur iitial suppsiti that the electr was mvig with a velcity v, havig kietic eergy Ek = ev. Besides, as explaied previusly, this cditi als ctradicts the Heiseberg ucertaity priciple r p. Csequetly, the classical Culmb electric field, as frmulated by r 00

9 Equati (), is t apprpriate t represet the electric field i the viciity f a micrscpic electric particle. The same ccurs with the classical Culmb ptetial that is t suitable fr describig the eergy iteracti betwee tw micrscpic electric charges [6]. This is a rather surprisig revelati, sice this iteracti ptetial has bee used i physics ad chemistry fr mre tha a cetury. O the ther had, as we will see belw, the wavig electric field, as frmulated by Equati (0), des t preset such, let s say, icveiece. The electric field wave packet as predicted by Equati (0) is quite differet f the wave packets btaied with a superpsiti f a ifiite umber f plae waves, as briefly discussed previusly ad, with details, i the physics literature [3] [4] []. The wave packets btaied with the use f a superpsiti f plae waves fr a free ucharged particle, fails t explai the rigi f the wavig aspect f the particle ad, if the particle is r depedece, which is a characteristic f the electric field charged like the electr, it des t shw the existig arud the charge. I the ext secti we will discuss the use f this ew electric field wave packet, usig a beam f electrs, expectig t give a aswer t this ad ther questis ccered t this subject. 3. The Frm f the New Electric Field i Oe Dimesi Fr sake f simplicity, let us csider the case f a electric field prduced by a relativistic electr, mvig thrugh a ptetial f V 0 vlts, i the x-directi. I this particular case the ew electric field takes the frm e ζ E( x) = cs ( ζ x) + si ( ζ x). 4π x x (3) Csiderig egative ad psitive values f x, this electric field has the frm f a e-dimesial wave packet with the charge, the electr, ccupyig the ceter f the packet. We are amig this ew electric field by Electric Field Wave Packet EFWP. Nrelativistically, this electric wave packet mves with the velcity f the particle, v ev 0 = As a typical example fr the predicti f the ew electric field, let us csider the case f a electr, that has bee accelerated thrugh a ptetial V = 00 vlts. Thus, accrdig t Equatis () ad (3), the electric field assciated t this particle is give by the particular expressi m ( x) cs E( x) = A + si (.88 x ) x x (5) 0 where A = N C. Figure 3 shws that this ew electric field has the frm f a wave packet as previusly described. The de Brglie wavelegth measured i the graphic is λ =.3 Å, which is i very gd agreemet with the value as predicted by Equati (5), λ =. Å. As shw i the same Figure 3, the effective size width f the electric wave packet is f the rder f 0 Å, what is abut the same rder f magitude f the iter-atmic space i slid crystals. What seems dd abut the shape f the wave packet shw i Figure 3, is the fact that, mathematically, the itesity f the electric field predicted by Equati (5) has a ifiite value at the rigi, the ceter f the packet. A particle with zer dimesi is ly a mathematical idealizati Ulike this, the wave packet frmed by the superpsiti f a ifiite umber f plae waves, has a fiite value at this psiti, meaig that the r t fid the particle at the ceter f the wave packet is higher tha it is elsewhere [5] []. Des this cstitutes a failure t the ewly created mdel? Our aswer t this questi is. I what fllws, we ited t aswer this questi makig sme csideratis tryig t shw why this respse is egative. O the ctrary, istead f beig a prblem t be slved, this apparet ctradicti, will help us t better uderstad the pheme f particle-wave duality. Firstly we eed t bserve that Equati (0), as it is frmulated, it describes the electric field f a pit charged particle with zer space dimesi. This is t the case fr the electr, this particle has a fiite size ad, csidered as a particle, its diameter is equal t the Cmpt wavelegth. Thus, if we csider the electr as slid sphere with a uifrm charge desity, we ca use the Gauss law t estimate the value f the electric field desity f prbability Ψ (, t) Ψ( rt, ) existig i its surface. With the use f Equati () we ca calculate the phase factr ζ r f the scillatry. (4) 0

10 Figure 3. The wavig electric field-wefp E( x ) fr a electr that has bee accelerated thrugh V = 00 vlts. The measured wavelegth is λ =.3 Å. The predicted value f de Brglie wavelegth is λ =. Å. 3 terms i Equati (0). Takig the radius f the electr as R =.93 0 m (half f Cmpt wavelegth) ad perfrmig the calculati we fid ζ R = radias. This small value makes cs ( ζ R) ad si ( ζ R) 0. Thus, accrdig t Equati (0), the electric field the electr surface, r eve very clsed t it, ceases t be scillatig, becmig equal t the classical electric field, with the r depedece. With the ifrmati btaied abve, we ca use the Gauss law t calculate the electric field iside ad the surface f the electr, as stated previusly, csidered as a slid sphere f radius R. Accrdigly, this electric field is predicted by the equati [7] er E( r < R) =. (6) 3 4π R This shws that the electric field i the ceter f the electr, r = 0 is zer. The value f the electric field at 6 the surface f the particle is determied takig r = R. Its value is E( r = R) = N C. This represets a very large but, t a ifiite value fr electric field at the psiti f the electr. I the graph shw i Figure 3, we ca see that the value f the electric field at the psiti f the secd 3 mst itese peak is abut 500 N/C. This electric field is apprximately 8 0 times smaller tha the electric field ear the surface f the electr. This ifrmati tells us that i rder t btai a patter f electr diffracti i a duble-slit experimet, it is ecessary that the umber f electrs i the beam be f the rder f trillis. A small umber f electrs i the beam will prduce ly a radm distributi f small spts the surface f the film, frmed by the cetral peaks f the electric field. Thus, we ca say that the disctiuities f the electric field as predicted by the Equati (5) ad shw i Figure 3, becmes iterrupted by the presece f the electr at the ceter f the wave packet. Therefre, we ca cclude that the tw physical mdels, i this particular aspect, are i agreemet. Otherwise, as it will be discussed i Secti 4, the great differece betwee the itesities f the cetral ad the lateral peaks, is the crucial aspect t be bserved fr the cmprehesi f the dual behavir f the electr. 4. Predictis f the EFWP ad Experimetal Results The crrect predicti f the ew electric field wave packet is i agreemet with the results f may experimets reprted i physics literature [3] [4]. The first successful experimet with a beam f elctrs icidet a crystal was prduced i 97 by Daviss ad Germer. I this experimet was used a parallel beam f meergetic lw eergy electrs icidet the surface f a Ni crystal with Bragg plaes spaced by d = 0.9 Å, 0 0

11 btaied frm previus X-ray scatterig experimets crystallie ickel [3]. I this experimet the applied acceleratig vltage f the electrs beam was 54 vlts, ad the de Brglie wavelegth btaied via Bragg s law was λ =.65 Å. Accrdig t the EFWP predictis, Equati (3), the de Brglie wavelegth measured i the graphic is λ =.66 Å, what is i very gd agreemet with the experimetal result. See Figure 4. The results displayed i Figure 3 ad Figure 4 shw strg evideces that the electric field wave packet assciated t a electr is the aget respsible fr the wavig aspect assciated t the electr. 5. The EFWP ad Yug s Duble-Slit Experimet Armed with the ideas ad the ifrmatis btaied i the previus sectis, we are w gig t discuss the applicability f the EFWP, i a duble-slit experimet, fr explaiig the behavir ad results predicted by this ew electric field wave packet, takig a very qualitative prcedure t face the prblem. Fr dig this, let us csider a surce F emittig a arrw beam f electrs strikig a paque plate S, where tw arrw slits S ad S were made. A scree P, a phtgraphic film fr example, is apprpriately placed i frt the plate S. This arragemet is called Yug s duble-slit experimet ad it is described ad illustrated with details i may quatum mechaics text bks [] [] [] ad, we are t reprducig it here. Withut eterig it details f the experimetal arragemet, we firstly csider a situati fr which the slit S is blcked ad S is pe. I a situati like this we will btai the scree P a distributi f friges f the electric field E ( x ) with a itesity I ( ) ( ) x = E x, represetig the diffracti patter f electrs passig thrugh S. Fr example, usig the Equati (3), a electr accelerated thrugh a ptetial f 00 V, passig thrugh the slit I x represeted by the expressi S, will preset the scree P, the itesity ( ) ( x) cs I ( x) = B + si (.88 x ) x x 0 where B = N C. The diffracti patter, as predicted by Equati (7) is shw i Figure 5, where we ca bserve that, due t the x depedece f the electric field, the cetral peak f the diffracti patter, at the psiti where the electr is mre likely t strike the scree, x = 0, is much mre itese tha the lateral peaks. The itesity f the electric field, as predicted by Equati (7), is mathematically diverget at the psiti f the particle, what is a characteristic f the ptetial used. But, frm a physical pit f view, (7) Figure 4. The electric field E( x ) fr a ubud electr that has bee accelerated thrugh V = 54 vlts. The de Brglie wavelegth measured i the graphic is λ =.66 Å. Fr Ni crystal, the Daviss ad Germer experimetal value is λ =.65 Å. 03

12 Figure 5. A diffracti patter prduced by e electr accelerated thrugh 00 vlts strikig the scree P f the duble-slit apparatus,csiderig just e slit pe ad a film with high sesibility. Due t the /r depedece f the electric field wave packet, the cetral peak f the diffracti patter is much mre itese tha the lateral peaks. I x, as x 0, ca be very large ideed but havig a fiite value as have bee shw i the ed part f Secti. The heights f the lateral peaks decrease as mdulated by the classical Culmb electric field. As we will see belw, this characteristic f the electric field, will allw us t iterpret the particle-wave duality behavir f the electr, usig a quite differet pit f view. A similar diffracti patter, I ( ) ( ) x = E x, f electric field E ( x ), will be btaied if S is blcked ad S is pe. Nw, if we csider that the umber f electrs per uit f time, passig thrugh the tw slits, t simultaeusly pe, is the same, the tw diffracti patters becme equals ad they will appear as displayed i Figure 5, but with a itesity that will deped the umber f electrs per uit f time are strikig the film surface. Hwever, whe the tw slits S ad S are pe, simultaeusly, ad csiderig a differece f phase φ ad φ betwee the fields E ( x ) ad E ( x ), what we will bserve the scree P is a patter f iterferece f friges E( x) = E( x) + E( x), which will t be the sum f the itesities I ( x ) ad I ( x ) as btaied separately. Istead, the itesity I( x ) f the tw fields is represeted by the expressi [5] [] because the electr has a very small but a fiite size, the itesity ( ) Ix ( x) E( x) E( x) E( x) E( x) E( x) E( x) ( φ φ) The term E( x) E( x) cs( φ φ) E ( x ) ad E ( x ). = + = + + cs. (8) i the Equati (8) represets the iterferece f the tw fields Nw, let us csider a situati i which the surce F emits electrs, practically e by e. With this cditi, three distict situatis may ccur: ) Let us assume that ly e f the slits, lets say, S is pe. Csiderig that the sesibility f the film is very... very high, accrdig t Equati (5), what we must bserve the scree P is a diffracti patter f e electr, with very... very small but t vaishig itesity f the lateral peaks. I this case, we wuld say that the electr presets the character f wave. This situati is shw i Figure 5. By ther side, icreasig the umber f electrs f the beam per uit f time, r equivaletly, icreasig the time f expsure f the film, the diffracti patter will becme mre, ad mre accetuated, withut chagig its wavig aspect fr the prducti a diffracti patter. This cditi is shw i Figure 6, where it was csidered that, 5 ad 50 electrs per uit f time were strikig the surface f the film f very... very high sesibility. Icreasig the umber f electrs i the beam, we ca see that the diffracti patter will be preserved. ) Let us take ather csiderati, if the sesibility f the film is t ecessarily very high, what we will 04

13 Figure 6. The diffracti patter f beam f, 5 ad 50 electrs per uit f time passig thrugh the duble-slit experimet, with just e slit pe. It is bserved that the itesity f the lateral peaks f the electric field is quite sesible t the umber f electrs i the beam. expect t bserve the scree P is ly the sigal prduced by the cetral peak f the diffracti patter fr sigle electr, which due t the character f the electric field it is f the rder f 0 3 mre itese tha the lateral peaks. I a situati like this, what we expect t see the scree P is a small spt, similar t the mark prduced by a sigle particle strikig the film surface. A simulati f this situati is displayed i Figure 7, where the itesities f the lateral peaks is csidered t small t be bserved. I a situati like this, we wuld say that the electr presets the character f a particle. 3) Fr last, let us csider a prbabilistic iterpretati fr bservig the iterferece pheme f electrs i the duble-slit experimet. As we ca see i Figure 3, the effective size f the electric wave packet arud the electr is abut 0 Å. Thus, if we csider the aperture f the slits S ad S is aythig larger tha this value, let s say 5 Å ad the slits S ad S pe, what we will expect t bserve the film is a certai umber f electrs f the tw beams cllidig with the edges f the slits. These electrs will be scattered radmly withi a certai agle θ befre reachig the plae f the film. The remaider electrs f the beams will pass thrugh the slits S ad S withut beig scattered by the brders. Nw, let us csider that the umber f electrs i the beams is very small, passig e by e thrugh the slits. The, accrdig t the situati umber, described abve, what we expect t see is a radm distributi f small spts the film surface, prducig a patter f at least three strips f spts. Tw f them, with larger ccetrati prduced by electrs t scattered by the edges f the slits. A third e appearig betwee these tw, strips, prduced by crss scatterig f electrs, ccurrig the upper ad the lwer edges f the tw slits. Furthermre, csiderig a small umber f electrs i the beam, what must be bserved i this case is a radm distributi f spts the film, but preservig the behavir f particles i the patter. See Ref. [4]. Otherwise, icreasig csiderably the umber f the electrs i the beam, the lateral peaks f idividual electr diffracti patter begi t icrease i certai regis f the film, due t cstructive iterferece f the electric field ad decreasig i ther regis due t destructive iterferece, i such a way that a cmplete iterferece patter begis t be frmed the film surface. I a situati like this, we wuld say that the electrs behave as wave. Ather iterestig cclusi we ca draw frm this iterpretati is the ctrversial discussi abut the chage f character f the particle, be it a particle r be it a wave, ca be affected by the iterveti f the bserver i the experimet. Accrdig t the iterpretati preseted abve, ur aswer is. T give the egative aswer t this questi, will csider tw differet situatis: a) Let s assume that the umber f electrs passig thrugh either slit is t very high, i such a way that 05

14 Figure 7. Simulati f the diffracti patter f a electr, strikig a film f very lw sesibility. Oly the cetral peak f the patter is appearig as a spt the film. The spt is the mark left by the electr behavig as a particle. we ca cut them usig a detectr. The fact f placig a detectr ear e f the slit, t see if the electr wet thrugh that particular slit, will t affect the electrs that are gig thrugh the ther slit. Thus, eve csiderig that the detectr placed ext t e f the slit ca blck it, prevetig the electrs frm reachig the phsphrescet scree, will t ivalidate the experimet ad will t chage the character f the particle. Because, as the umber f electrs i the beam have bee csidered small, the itesity f electric field the scree als will be small. Thus, the radmess f the scatterig f electrs at the edges f the slit t blcked, will cause ly a radm distributi f small spts the scree, prduced by the large cetral peak f the electric field itesity. Thus, as explaied previusly, the electrs will be bserved as particles. b) Nw, let us csider situati f (a) but with the umber f electrs i the beam beig very... very large. I this case, what we expect t bserve the scree is diffracti patter prduced by the superpsiti f the radm distributi f cetral ad lateral peaks, iterferig cstructively r destructively, t frm the diffracti patter. I a situati like this we wuld say that the electrs behave as waves. Thus, the presece f a detectr ear e f the slits is will t affect the results f the experiece as predicted by ther iterpretatis f the pheme. ) Nw let us csider that bth slits are pe ad withut the presece f the detectr. Let us als assume that the umber f electrs i the beam is large, r equivaletly, the umber f electrs ca be csidered small but the expsure time is large. The result will be the same. The cstructive r destructive iterferece f cetral ad side peaks f electric field itesity i the phsphrescet scree, will prduce a iterferece patter. I this case, ur cclusi is that the electrs will behave like a waves. Thus, accrdig t this ew mdel, what are the facts mre imprtat t be bserved i the experimet, t cclude whether the electr is particle r wave. First, the umber f electrs i the beam r the time f expsure f the film. Secd, the sesibility f the film is als imprtat. I Secti, we called the reader s atteti t the pssible existece f a bsic atm i a dese beam f electr, similar t the existece f electr Cper pairs fr sme metals i the state f supercductivity. This ew atm culd be see as a particle with twice the mass f the electr, mba = me ad havig twice its electric charge, qba = e. As see i Figure, the grud state f this ew atm is frmed f tw electrs separated frm each ther by a average distace d =.06 Å. At this distace, the electric frce betwee them is ull ad, if they are i a state f spi S = 0 (upaired spis), what exist betwee tw electrs is a attractive magetic frce, give t the system a certai degree f stability, frmig what we are callig bsic atms prduced by the iteracti f tw electrs. Thus, if we assume that i a duble-slit experimet, with the 06

15 use f a dese beam f particles, electrs ad bsic atms ad the beam is subjected t a give acceleratig ptetial f V 0, what kid f diffracti patter will be bserved i the phsphrescet scree f the device? I Equati (5) we ca bserve that due t its larger mass the bsic atm, twice times larger, shuld be twice times slwer tha the islated electr. At the same time, it shuld be twice times faster due t its larger charge, twice times larger. S, what the bsic atm lses by e side is cmpesate by the ther. Beig s, the time that bth particles take frm leavig the surce ad reachig the ed f the trip is the same because they have the same velcity. See Equati (4) t make sure f this statemet. Fially, let us csider what kid f chage may ccur i the diffracti patter f electrs admittig the presece f bsic atms i the electr beam. Figure shws us that i the grud ad the first excited states f this ew kid f atms the tw electrs i the atm are separated frm each ther, respectively, by a average distace f.06 Å ad 4.4 Å. Thus, fr a give umber f electrs arrivig the film surface, what we expect t see i the diffracti patter is certai umber f islated spts prduced by islated electrs cllidig with the film ad ther umber f pair f spts radmly distributed with regular distaces, cmpatible with the separati f the tw electrs i the existig pairs i the beam f electrs. T ur surprise ad delight, a experimet t illustrate ur frecast has bee carried ut by E. R. Huggis, Physics I. The diffracti patter i a duble slit experimet is reprduced i the bk Mder Physics P. A. Tipler ad R. A. Llewelly, W. H. Freema ad Cmpay, N. Y. 04 [4]. The experimet cducted by Huggis, clearly shws the evluti f the mechaism fr the passage frm a radm scatterig f particles (radm frmati f spts the phtgraphic film) t the frmati f beautiful iterferece patter, whe the umber f electrs i the beam is greatly icreased. By carefully bservig the evluti f the figures btaied i the experimet, we ca clearly see the existece f pairs f spts, radmly distributed the film. Des it cstitute a prf fr the existece f the bsic atms i a dese beam f electrs? This simple qualitative iterpretati preseted abve, was strgly suggested r iduced by the results experimetally bserved with electrs scatterig i may crystallie slids. I a situati like this, the figure bserved the plae f the film must be ccetric bright ad dark alteratig rigs f decreasig itesities frm the ceter t the brder f the patter. Due t the r depedece f the electric field wave packet, the cetral peak f the diffracti patter must be mre bright tha the thers. I fact, this is the patter experimetally bserved with electrs diffracti i may cryslallie slids. The explaati give abve, eve beig csidered very simple, it cstitutes a differet pit f view fr a prbabilistic iterpretati f the pheme f diffracti r iterferece prduced by electrs i slids. Accrdig t this iterpretati, the acceptace f the electr as a particle r as wave will deped hw the experimet is perfrmed. Ather aspect that is als clear is that the iterveti f the bserver, is t a determiig factr t say whether the electr is a particle r a wave. This is a irrelevat cditi t the prcess f bservati. What is clear with this iterpretati is that, eve cexistig at the same time, bth characters, wave r particle, cat be bserved simultaeusly. This is a fact kw sice lg time ag. What is ew i this presetati is the use f a electric field wave packet, which eabled us t preset a ew ad a quite differet iterpretati fr the pheme f the particle-wave duality ad, we believe, mre palatable mdel tha thers, frmulated previusly, t achieve this purpse. 6. Cclusi ad Remarks I this wrk, we have used the Quatum Oscillatry Mdulated Ptetial t ivestigate the pheme f wave particle duality. With this ew ptetial, we have shw that the electric field existig i the viciity f a micrscpic electric particle i mvemet, has the frm f a wave packet surrudig the particle. The ew electric field wave packet allwed us t preset a ew iterpretati fr the duble-slit experimet with a beam f electrs, i a much mre simple way. We have shw that the magitude f the de Brglie wavelegth assciated t this electric field wave packet is depedet the mass f the particle ad its charge. As it is well kw sice lg ag the wavelegth, decreases whe the mass f particle icreases. Hwever, accrdig t the argumets ad calculatis preseted i this wrk, the rigi ad the wavig behavir f the particle are beig ascribed t the electric field prduced by the charge ad t t the mass f the particle. Accrdig t this mdel ad based experimetal results, a ew iterpretati was preseted t explai the diffracti patter f electrs i a duble slit experimet. Besides its simplicity, the results predicted by the ew mdel f electric 07

16 field wave packet are i very gd agreemet with the experimetal data, btaied with diffracti f electrs i may crystallie slids. Furthermre, as a additial ifrmati, usig a pit f view which ca be csidered rather audacius, the wrk pits ut idicatis fr the existece f a kid f bsic atms i a arrw ad dese beam f electrs as well as i a sea f electrs ear the Fermi surface i supercductig metals at temperatures belw T c, frmig what is called Cper pairs f electrs. The bserved Cper pairs i Jsephs effects with supercurret tuelig als may be a idicat fr the existece f the ew kid f bsic atm frmed by the iteracti f tw electrs. Refereces [] Greee, B.R. (04) The Elegat Uiverse: Superstrig ad Hidde Dimesis. Title i Prtuguese: O Uivers Elegate. Traslated by Jsé Viegas Filh, Editra Schwarcz S.A. [] Feyma, R.P., Leight, R.B. ad Sads, M. (966) Lectures Physics. Addis Wesley, Readig. [3] McQuarrie, D.A. (983) Quatum Chemistry. Uiversity Sciece Bks. [4] Eisberg, R.M. (96) Fudametals f Mder Physics. Jh Wiley & Ss, Ic., New Yrk. [5] Wley Filh, W. (04) Mecâica Quâtica. d Editi, Editra da Uiversidade Federal de Giás, Giâia. [6] Wley Filh, W. (0) Jural f Mder Physics, 3, [7] Ohaia, H.C. (985) Physics. Nrt & Cmpay, Ic., N.Y. [8] Altarev, I.S., et al. (980) Nuclear Physics A, 34, [9] Pspelv, M. (005) Aals f Physics, 38, 9. [0] Kittel, C. (986) Itrducti t Slid State Physics. Jh Willey & Ss, New Yrk. [] Bardee, J., Cper, L.M. ad Sherieffer, J.R. (957) Physical Review, 5, [] Che-Taudji, C., Diu, B. ad Lale, F. (977) Quatum Mechaics. Vl., Jh Willey & Ss, New Yrk. [3] Jhss, T.H. (93) Physical Review, 37, [4] Tipler, P.A. ad Llewelly, R.A. (04) Fsica Mdera, sexta ediçã, Traduçã: Rald S. Biasi, LTC, Ri de Jaeir. 08