A New Wave Equation of the Electron

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1 Journal of Modrn Physis,,, -6 doi:.436/jmp..9 Publishd Onlin Spmbr (hp:// A Nw Wav Equaion of h Elron Absra Arbab I. Arbab Dparmn of Physis, Fauly of Sin, Univrsiy of Kharoum, Kharoum, Sudan arbab.ibrahim@gmail.om, aiarbab@uofk.du Rivd July 7, ; rvisd April, ; apd April 8, A nw form of Dira quaion of a sond ordr parial diffrnial quaion is found. Wih his wav quaion h quivring moion (Zirbwgung) is saisfaorily xplaind. A quarnioni analogu of Dira quaion is prsnd ompard wih h ordinary Dira quaion. Th wo quaions bom h sam if w rpla h paril rs mass, m, in h lar by im. Nw spa im ransformaions in whih hs wo quaions rprsn a masslss paril ar found. Th invarian of Klin-Gordon quaion undr hs ransformaions yilds h Dira quaion. Th lron is found o b rprsnd by a suprposiion of wo wavs wih a group vloiy quals o spd of ligh in vauum. Kywords: Dira Equaion, Zirbwgung, Univrsal Quanum Wav Equaion, Quarnion Quanum Mhanis. Inroduion In 98 Dira inrodud his rlaivisi wav quaion []. H had wand o find an quaion for h lron ha would b onsisn wih spial rlaiviy dsrib h known fin sruur frquny sprum of hydrogn. In hnial rms h quaion was Lornz-ovarian. I orrly dsribd h fin sruur of hydrogn. Whn Dira xamind h soluions o his quaion, h found ha, muh o his surpris, i also prdid h lron s spin is magni momn. Prhaps vn mor surprising, his quaion also suggsd a posiivly hargd paril wih h mass of h lron ould xis. Basd on his quaion, in 93 Dira prdid wha h alld h ani-lron (a posiron) onfirmd xprimnally in 93. In 933 h shard wih Shrodingr h Nobl Priz for physis for his work in quanum mhanis. In dvloping his quaion, Dira assumd ha h lron was a poin-lik paril wih an lri harg qual o h xprimnally masurd harg of h lron. Dira howvr did no ry o dvlop visualizabl physial modls as aids o piuring or undrsing his mahmaial rsuls. Thr ar svral rsuls of Dira s quaion whih wr problmaial, dspi h quaion s gnral suss. I was mniond ha Dira did no in his priod amp o mak a physial modl of h lron did no suppor suh amps. Howvr, many of h propris of h Dira lron, inluding hs problmaial propris, lnd hmslvs o a physial modling approah, whih may hlp rsolv som of hs problmaial propris. Th Zirbwgung is a horial rapid moion of lmnary parils, in pariular lrons, ha oby h Dira quaion. Th xisn of suh moion was firs proposd by Erwin Shrodingr in 93 as a rsul of his analysis of h wav pak soluions of h Dira quaion for rlaivisi lrons in fr spa, in whih an inrfrn bwn posiiv ngaiv nrgy sas produs wha appars o b a fluuaion (a h spd of ligh) of h posiion of an lron around h mdian, wih a irular frquny of m, or approximaly.6 Hz. This vry rapid moion of h lron mans w anno loaliz h lron xrmly wll givs ris o h Darwin rm. This moion nvr bn obsrvd for a fr lron, bu h bhavior of suh a paril has bn simulad wih a rappd ion, by puing i in an nvironmn suh ha h non-rlaivisi Shrodingr quaion for h ion has h sam mahmaial form as h Dira quaion (alhough h physial siuaion is diffrn) [,3]. Thus, if w masur h vloiy omponn in any dirion, w should ihr g plus or minus. This sms qui surprising, bu w should no ha a omponn of h vloiy opraor dos no ommu wih momnum, h Hamilonian, or vn h ohr omponns of h vloiy opraor. If h lron wr masslss, vloiy opraors would om- Copyrigh SiRs.

2 A. I. ARBAB 3 mu wih momnum. In mor spulaiv horis of parils, lrons ar aually hough o b masslss, ging an ffiv mass from inraions wih parils prsn in h vauum sa. Baru Brakn [4] analyzd Shrodingr s Zirbwgung rsuls proposd a spaial dsripion of h lron whr h Zirbwgung would produ h lron s spin as h orbial angular momnum of h lron s inrnal sysm, whil h lron s rs mass would b h lron s inrnal nrgy in is rs fram. Th sas of dfini momnum ar no ign-sas of vloiy for a massiv lron. Th vloiy ign-sas mix posiiv ngaiv nrgy sas qually. Thus, whil momnum is a onsan of h moion for a fr lron bhavs as i did in non-quanum Mhanis, h vloiy bhavs vry srangly in h Dira hory, vn for a fr lron. Whil Dira quaion is a firs-ordr diffrnial quaion in spa im, Klin-Gordon quaions is a sond-ordr in spa im. Th wo quaions ar invarian undr Lornz ransformaions. W would lik in his papr o wri Dira quaion as a sond - ordr diffrnial quaion in spa im mimiking h sard wav quaion. In doing so, w hav found nw spa im ransformaions undr whih Dira quaion rprsns a paril wih zro mass. Morovr, h ordinary Dira quaion is invarian undr hs ransformaions. Inrsingly, h invarian of Klin-Gordon quaion undr hs ransformaion yilds Dira quaion. Th nw Dira quaion has many inrsing onsquns. As in h d Brogli hory, h lron is dsribd by a wavpak whos group vloiy is qual o h spd of ligh in vauum. Th naur of his wav may xplain h Zirbwgung xhibid by h lron as found by Shrodinr.. Quarnioni Quanum Mhanis: Dira-Lik Equaion Considr a paril dsribd by h quarnion wavfunion i, ψ. This is quivaln o spinor rprsnaion of ordinary quanum mhanis whih w hav rnly dvlopd [5]. Th voluion of his quarnion wavfunion is dfind by h hr quaions [5] m ψ, () ψ m ψ, () ψ. (3) Equaions () - (3) yild h wo wav quaions ψ m ψ m ψ ψ, (4) m m. (5) Using h ransformaion m Equaions () () bom ψ ψ,. (7), (6) Employing Equaion (6), Equaions (4) (5) ar ransformd ino h wav quaions, ψ,. (8) Th physial maning of h vor ψ is sill no lar. Bu I argu ha i an b assoiad wih fundamnal parils (g., quarks) ha making up all hadrons, sin i has hr omponns. In his way may rprsn a salar paril (boson). Noi ha Equaions (4) (5) an b obaind from h nrgy quaion E p, whr E E im, E is h oal nrgy of h paril, using h familiar quanum mhanial opraor rplamns, viz., pˆ i ˆ E i. E p is an nrgy quaion for a masslss paril. Thus, i is inrsing ha a massiv paril an b ransformd ino a masslss paril using Equaion (6). Sin nrgy is a ral quaniy, his quaion is physially apabl if i dsribs a paril wih imaginary mass. In his as h nrgy quaions spli ino wo pars; on wih E E m h ohr wih nrgy E E m. Suh nrgis an dsrib h sa of a paril aniparil. A hypohial paril wih an imaginary mass moving a a spd highr han spd of ligh in vauum is known as ahyon [6]. Hn, our abov quaion an b usd o ra h moion of ahyons. This implis ha our quaions, Equaions (4) (5), an b applid o ahyons. Som siniss propos ha nurino an b a ahyoni frmion [7]. This is in favor of h xprimnal finding ha h squard mass of nuriono is ngaiv [8]. W know ha Chrnkov radiaion is mid from a paril moving in a mdium wih spd grar han spd of ligh in vauum. Whn h spd xds h spd of Copyrigh SiRs.

3 4 A. I. ARBAB ligh in vauum, h xra nrgy aquird by h paril is ransformd in radiaion. This an happnd momnarily for a paril kping is oal nrgy onsrvd. Thus, h xss nrgy (spd) is suh ha i ompnsa h dissipaions. 3. Ordinary Quanum Mhanis: Dira s Equaion Dira s quaion an b wrin in h form [] im. (9) whr, σ, σ σ ar h Pauli maris. From Equaion (9) on has im. Squaring his quaion yilds, mi m. () Employing Equaion (9) on again yilds, mi m α. () This an b wrin as whr, m () i. (3) Equaion () an b obaind from Equaion (9) by squaring i. Ths forms of Dira quaion, Equaions (9) (), hav no bn obaind bfor. Thy xhibi larly h wav naur of h lron. I an b ompard wih h Klin-Gordon quaion of spin- parils m. (4) Equaions () () ar anohr form of Dira s quaion xhibiing h wav naur of spin- parils xpliily. Using h ransformaion m i Equaion () an b wrin as., (5) (6) This is a wav quaion for a masslss paril. Hn, an also rprsns a paril wih zro mass whn akn in im. Equaions () (6) show ha h paril is in sa of oninuous raion annihi- laion. This happns afr a im of a dis- m an of m whih is ravrsd wih spd, as vi- dn from Equaions (3) (5). I is inrsing o s ha h invarian of Klin- Gordon quaion undr h ransformaions dfind by Equaions (3) (5) yilds Dira quaion. Morovr, Dira quaion, viz., Equaion (9), is also invarian undr h ransformaions in Equaions (3) (5). Sin is a four omponns spinor, w an wri i in rms of wo omponns doubls, viz.,. Subsiuing hs domposd spinors in Equaion (), on obains h wo quaions mi m, (7) mi m, (8) Equaions (7) (8) imply wo nrgy soluions, on wih E E m h ohr wih nrgy E E m. Equaion () an b obaind from Equaions (4) or (5) by rplaing m by im. Thrfor, Equaions (4) (5) an rprsn imaginary massiv bodis (ahyons). D Brogli hypohsis ha all miroparils xhibi a wav naur. This is possibl if w dsrib a paril by a wavpak. In his way h group vloiy of h wavpak will b h vloiy of h moving paril. Howvr, a singl wav an dsrib a paril. In Dira s hory h lron is found o mov wih spd of ligh owing o Einsin s rlaiviy i mus hav a zro mass. Hn, his moion is problmai in Dira s hory. Th ngaiv nrgy quaion, i.., Equaion (8), an b wrin as m i, m m whih an b sn as a gnralizd Shrodingr s quaion. I an hus rprsn h moion of a posiron. I is inrsing o noi ha if w roa spa im, viz., i r ir, in Equaion (4) w will obain h Dira posiiv nrgy quaion, i.., Equaion (7). Similarly, for anilokwis roaion, viz., i r ir, Equaion (4) yilds h Dira ngaiv nrgy Copyrigh SiRs.

4 A. I. ARBAB 5 quaion, i.., Equaion (8). Hn, our Equaion (4) is quivaln o Dira quaion in imaginary spa-im. Th ransformaion quaions, Equaions (3) (5), an b ompard wih h ovarian ransformaion ha is dfind by i D A, (9) whr A is h gaug ponial. Hn, on onluds ha m m A α, A. () This analogy dias ha A o b a marix. I is rlad o Dira maris by h rlaion m m A γ, A. () Th plan wav soluion of Equaion (5) aks h form ikr r, A, A ons. () Subsiuing his soluion in Equaion () yilds h disprsion rlaion m k, (3) in Equaions (7) (8) yild h posiiv ngaiv frqunis m m k, k, (4) rspivly. This implis ha his plan wav is a ollion of wo wavs of wo diffrn frqunis bu sam wavlngh. I is hus a wavpak. W may assum ha a paril is a ombinaion of wo sas: on moving wih momnum k in on dirions h ohr wih momnum k is h opposi dirion wih vloiy of ligh,. This may imply an inrnal irular moion rsuling from h wo sas. Hn, in h lron rs fram h oal momnum is zro. Dos his irular moion giv ris o h spin of h lron? Dos his imply ha h lron is no fundamnal onsiss of som subsruur? Dira s hory of h rlaivisi lron did no inlud a modl of h lron islf, assumd h lron was a poin-lik paril. Noi ha sring hory posulad ha h lrons quarks wihin an aom ar no -dimnsional objs, bu rahr 3-dimnsional osillaing lins (srings). Th hory advoas ha hs srings an vibra, hus giving h obsrvd parils hir harg, mass spin. Aording o Equaion () a paril is in a posiiv nrgy sa whn is wavlngh is grar han h h Compon wavlngh,, ( > ), in a m ngaiv nrgy sa (aniparil) whn is wavlngh is smallr han h Compon wavlngh ( < ). Howvr, a paril xising in h ngaiv nrgy sa (aniparil) having will b in a sa of rs (rappd). Thus, h paril is in ssn a suprposiion bwn is mar animar sas, hs wo onradiory sids of is prsonaliy should inrfr, sing h paril quivring (Zirbwgung). Howvr, his phnomnon has nvr bn obsrvd xprimnally baus h moion is oo small oo fas o d in ral quanum sysms. Th group vloiy of a (non)-rlaivisi paril is qual o is vloiy. Equaions (9) () implis ha h group vloiy of h wavpak is givn by v g,. (5) k This implis ha no h lron as a whol ha osillas bu is onsiuns ha having zro mass. Hn ral parils ahyons ar parils mov wih spd of ligh. Howvr, h phas vloiy of ah wav omprising h wav pak is lss han spd of ligh in vauum, as vidn from Equaions () (). Th group vloiy v g dos no nssarily rquir a paril o hav m, as vidn from Equaion (). I is a ransin propry of h paril whn sn in a diffrn im fram ( ). Hn, h Dira s vloiy dilmma is now rsolvd. 4. Elromagni Wav in Conduing Mdium For a onduing mdium wih onduiviy, Maxwll quaions rad for lri magni filds, E, B h wav quaion B B B, E E E. (6) (7) Comparing his wih Dira quaion, Equaion (4), on obains h rlaion m m*, (8) 4 dfining h ffiv mass ( m* ) of h magni fild whn propagaing in a mdium. This is apabl if w mploy h wav-paril hypohsis of d Brogli. I is vidn ha his mass dpnds on h onduiviy of h Copyrigh SiRs.

5 6 A. I. ARBAB mdium. Hn, h magni fild insid a onduing mdium bhavs lik a paril wih mass of m *. Thrfor, w s ha whn an lromagni fild propagas in a onduing mdium, i loss par of is nrgy so ha h phoon aquirs a mass ha is dirly proporional o h onduiviy of h mdium. Morovr, if on now wris h las rm in Equaion (7) as m E, (9) so ha Equaion (7) will b similar o Equaion (7). If w ak h divrgn of boh sids of Equaion (9) us Gauss law, w will obain m *. (3) Considr h rflion of posiivly hargd spin- paril from a high Coulomb barrir for whih h Klin-Gordon quaion is [9] iv m V, (3) whr h ponial V is zro o h lf of h barrir a onsan o h righ. Hn, omparison wih Equaion (7) rvals ha whn m (masslss boson) in Equaion (3) h rsuling quaion will b similar o Equaion (7) for spin- on finds h mass of h spin- paril. In his as, paril will b m V. Hn, a masslss boson bhavs as a massiv Dira paril of an ffiv mass of m V. Thus, a masslss mson nounring a ponial of.5 MV will aquir a mass quals o h lron mass, hrfor, will bhav as an lron. 5. Conluding Rmarks W hav wrin Dira quaion in rms of a sond-ordr wav quaion. Spifi spa im ransformaions allow h mass rm o disappar. Whn hs ransformaions ar applid o Klin-Gordon quaion, Dira quaion is obaind. Th soluion of h Dira quaion yilds wo osillaing sas wih posiiv ngaiv nrgis. Ths sas mov wih a group vloiy quals o h spd of ligh in vauum. Th xisn of suh sas rsolvd h quivry moion of h lron (Zirbwgung) found by Shrodingr. This is baus no h lron as a whol ha osillas bu is onsiuns ha having zro mass. I is qui inrsing o noi ha h quarnion analogu of Dira quaion yilds h ordinary Dira quaion if w rpla m wih im. 6. Rfrns [] J. D. Bjorkn S. D. Drll, Rlaivisi Quanum Mhanis, MGraw-Hill, Boson, 964. [] L. Lamaa, J. Lón, T. Shäz E. Solano, Dira Equaion Quanum Rlaivisi Effs in a Singl Trappd Ion, Physial Rviw Lrs, Vol. 98, No. 5, 7, Aril ID: 535. doi:.3/physrvl [3] D. Walr H. Gis, Probing h Quanum Vauum: Prurbaiv Effiv Aion Approah, Springr Vrlang, Brlin,. [4] A. O. Baru A. J. Brakn, Zirbwgung h Inrnal Gomry of h Elron, Physial Rviw D, Vol. 3, No., 98, pp doi:.3/physrvd [5] A. I. Arbab, Th Quarnioni Quanum Mhanis, arxiv: 3.75v,. [6] G. Finbrg, Possibiliy of Fasr-Than-Ligh Parils, Physial Rviw, Vol. 59, No. 5, 967, pp doi:.3/physrv [7] J. Ciborowski, Hypohsis of Tahyoni Nurinos, Aa Physisa Polonia B, Vol. 9, No. -, 998, pp. 3-. [8] R. G. H. Robrson, al., Limi on ν Mass Obsrvaion of h β Day of Molular Triium, Physial Rviw Lrs, Vol. 67, 99, pp doi:.3/physrvl [9] F. Gross, Rlaivisi Quanum Mhanis Fild Thory, John Wily & Sons, In., Hobokn, 993, p. 97. Copyrigh SiRs.