A Dynamic Approach to De Broglie's Theory

Transcription

1 Apeiron, Vol., No. 3, Jly 5 74 A Dynami Approah o De Broglie's Theory Nizar Hamdan Deparmen of Physis, Uniersiy of Aleppo P.O. Box 83, Aleppo, SYRIA nhamdan59@homail.om Einsein's relaiiy (SRT) [], raised an imporan qesion: Do is laws apply o maer/energy or o he frame of referene onaining i. In oher words, is SRT a propery haraerizing maer/energy or is i an imposed dynamis as expressed by he Lorenz Transformaion (LT). This arile examines he sble differene beween he wo. I proes ha maer/energy is he fndamenal realiy, while he referene frame, alhogh i exiss, exers no fore. Therefore, we an sar from Newon's seond law (NSL) o rebild SRT []. This will allow s o derie he de Broglie relaions from NSL while irmening he well-known onradiions beween SRT and de Broglie's qanm wae heory. Keywords: Newon seond law, de Broglie maer-wae; speial relaiiy heory.. Inrodion Einsein obiosly sbsribed o he hypohesis ha physial aspes of he spae-ime oninm are he priniple fores in LT and obey he relaiiy priniple hrogh spae onraion and ime dilaion. 5 C. Roy Keys In. hp://redshif.if.om

2 Apeiron, Vol., No. 3, Jly 5 75 SRT is iewed hisorially as a brillian ahieemen. Howeer, here are sill a few physiiss who are skepial abo is fndamenal logi. The heory liberaed s from he eher myh, only o hrow s ino he darkness of spae-ime. Maer is no longer free as is properies are now deermined by he oninm. SRT has been presened as a niqe solion, ye ens of alernaie heories are p forward o replae i. These are largely ignored sine i is belieed ha hey hae nohing new o offer. The heories lak he glamor and fame of SRT for wo main reasons: - There is only a sble differene beween he onep of he primay of maer and ha of is frame of referene. - SRT is a nifying heory of spae and ime, maer and energy, and has beome he basis of oher nifying heories. To be sessfl, alernaie heories shold demonsrae he shoromings of SRT and show i does no in fa nify. SRT has remoed he barrier beween maer and energy, b i reaed a new one beween he domains of non-relaiisi and relaiisi phenomena. The physial laws of lassial physis an no ransend his barrier. Relaiisi physis is able o represen lassial physis only hrogh approximaion. The logial approah wold be o sar wih he laws of lassial physis and make hem appliable o all parile eloiies, i.e. o expand he appropriaeness of hese laws o deal wih he relaiisi domain. This an no be ahieed nless we reer o he inariane of physial laws among inerial frames regardless of he oordinae ransformaions. In his way, SRT old be formlaed from a mehanial, raher han eleromagnei base. In doing his, we ms exend NSL by desribing he moing parile as a wae and irmen he onradiions beween de Broglie relaions and SRT. 5 C. Roy Keys In. hp://redshif.if.om

3 Apeiron, Vol., No. 3, Jly Energy (Mass), Momenm, Veloiy and Fore Transformaion Relaions Le s onsider wo moing inerial sysem S and S wih a relaie eloiy ox beween hem. The iniial laws in lassial mehanis are NSL and is expression for fore: d = p d F, = F (a,b) d d Where Eq.(b) an be deried on Newonian mehanial gronds[], b in lassial mehanis he kinei energy is m = T = wih m= m. In his seond paper regarding SRT, Einsein [3] proposed he famos eqaion, = m. Howeer, many ex books ofen deoe grea effor o disssing he proess of an elasi ollision beween wo pariles in deriing = m and he relaiisi mass m= γ m. Le s ry o esablish a law for oal energy and relaiisi mass wiho sing LT or SRT preeps. As demonsraed in [] and aording o relaiiy priniple NSL implies ha mass is ariable. The oal energy is dependen on his. One ndersands relaiisi mehanis as a modifiaion (orreion) of lassial mehanis. The same modifiaion an be obained if we reer o NSL and se hange in mass raher han spae-ime as in SRT. We will demonsrae ha Eqs.() and he relaiiy priniple are more naral for desribing he physis of relaiisi mehanis. We an hen go frher and derie relaiisi mass as well as all of SRT's relaions by his approah. The Caresian omponens of Eq. () in frame S are: 5 C. Roy Keys In. hp://redshif.if.om

4 Apeiron, Vol., No. 3, Jly 5 77 dp dp x y dpz = Fx, = Fy, = Fz (a,b,) d d d And d = F + x x F + y y F (d) z z d Applying he relaiiy priniple o Eq. (), we hae dp dp x y dp z = F x, = F y, = F z (3a,b,) d d d And d = F x x + F y y + F z z (3d) d Now mliplying Eq. (d) by and hen sbraing he resl from Eq. (a), diiding he resl by, we obain d px / ( Fyy + F z z = ) Fx d Mliplying and diiding he link hand side wih he salar faorγ, hen omparing wih Eq. (3a), we hae p x = γ px, d = γ d (4a,b) And 5 C. Roy Keys In. hp://redshif.if.om

5 Apeiron, Vol., No. 3, Jly 5 78 F y y F z z F x = Fx (5a) x Diiding Eq. (b) by γ hen omparing wih Eq. (5b), we hae, Fy p y = py, F = y (4b, 5b) x γ The same resl an be deried from he z and z fores and momenm Fz p z = pz, F = z (4, 5 ) x γ Saring from Eq. (a) mliplying by and hen sbraing he resl from Eq.(d), diiding he resl by, we obain ( ) ( ) ( F y y + F z z) d px x = F x d / ( Fyy + F z z) Adding and sbraing, we hae 5 C. Roy Keys In. hp://redshif.if.om

6 Apeiron, Vol., No. 3, Jly 5 79 ( ) ( x ) Fx + ( Fyy + F z z) d p x = d ( F y y + F z z) ( F y y + F z z) + Or d px x = + x x x d ( ) ( ) Fx F ( F y y + F z z) Mliplying and diiding he lef hand side wih he salar faor γ, we obain 5 C. Roy Keys In. hp://redshif.if.om

7 Apeiron, Vol., No. 3, Jly 5 8 ( ) ( ) ( F F F dγ p + γ d y y x x y y z z = Fx + x x x x x ( ) Fz z + Comparing wih Eq.(3d), we hae And Fy F y =, Fz F z =, ( p ) = γ (4d) x y x z x =, y =, z = (6a,b,) x x The salar faor an be fixed by applying he relaiiy priniple on Eqs.(6a) and (5b). So Eqs.( 6a ) and ( 5b ) old be wrien in frame S as 5 C. Roy Keys In. hp://redshif.if.om

8 Apeiron, Vol., No. 3, Jly 5 8 x + F y x =, Fy = x x + γ + Hene we an sbsie from Eq. (6a) ino (7a) gies x + = Similarly, sbsiing Eq. (5b) ino (7b) gies x γ + = Eqs. ( 8 ) and (9 ) lead o he deerminaion of γ i.e., (7a,b) γ =, hene γ = () We may wrie Eqs. (6) as x x = () x I se Eq.(6b) ino(4b) o ge m = γ m () Mliplying he Eq. () wih m, and omparing i wih (), we dede (8) (9) 5 C. Roy Keys In. hp://redshif.if.om

9 Apeiron, Vol., No. 3, Jly 5 8 m m m =, m = From Eq.(b), he oal energy is gien by d = Fd = d( m) = dm+ md And from Eq.(3a), we hae md dm = i.. e md = dm (3a,b) (4) (5) Sbsiing Eq.(5) in Eq.(4), we ge d = dm (6) By inegraion, from o, we ge = m (7) In he parilar ase, if = and he kinei energy k, i.e. So he qaniies =, hen shold eqal k = m = m m (8) m and m are he oal energy and in frames S and S respeiely. I is simple o proe, ha Eqs. (6 ) and () lead o 4 = = m Or P P (9) 5 C. Roy Keys In. hp://redshif.if.om

10 Apeiron, Vol., No. 3, Jly = P + m, = P + m (a,b) 4 The dynamis of a moing parile are bil ino SRT o aommodae LT. Therefore, we inroded an alernaie mehod whih is no based on LT o derie all he menioned relaions in his seion. Depending on his formlaion we onine o derie he de Broglie relaions for parile-wae daliy b wiho reqiring SRT or any relaiisi assmpion sh as LT. 3- De Broglie Theory and SRT Afer he reaion of he eleromagnei heory of ligh, i beome possible o formlae he laws of he orpslar properies of radiaion and he wae properies of he orpslar as h = hf = w, p = = k () λ De Broglie [4], poslaed he alidiy of relaion () for a parile wih res mass m hrogh his hypohesis of he periodi phenomenon, i.e., hf = m () When Eq. () is wrien wih respe o he frame S, hen Eq. () akes he form m hf = (3) Aording o Eq.(), 5 C. Roy Keys In. hp://redshif.if.om

11 Apeiron, Vol., No. 3, Jly 5 84 f = f (4a) Howeer, as is well known in SRT, ha if he lok has a freqeny f in he res frame of he parile, is freqeny a eloiy in frame S, (aording o he so-alled ime dilaion), is f = f (4b) Eidenly, Eq.(4b) is js he opposie of Eq.(4a). Indeed, aoning for ime dilaion leads o a slowing down in he freqeny of he moing lok, Eq.(4b), while aoning for he energy inrease of a moing parile yields an inreased freqeny, Eq.(4a). Ths, i is lear ha some addiional assmpion is needed o oerome sh a fndamenal onradiion. Namely he phase of he periodi phenomenon, i.e., π f = w shold be obained sing phase onseraion wih respe o LT, i.e. π m x m w = π = h (5) h = w kx From Eq. (5) we ge Eq.(4a). On he oher hand, sine Eq.(5) ms hold for eery x and eery, i also beomes he well-known formla for onneing a parile's momenm wih is waelengh. 5 C. Roy Keys In. hp://redshif.if.om

12 As well as k Apeiron, Vol., No. 3, Jly 5 85 π m h = = p, i.e. p h λ =, = (6a) m = (6b) As sal, he phase eloiy of a wae is w p = = (6) p k Whereas he parile eloiy eqals o he grop eloiy of wae dw = g = (6d) dk An obios onradiion exiss beween de Broglie's heory and SRT (hrogh whih i was formlaed). For insane, aording o SRT, Eq. (6b) has he onradiion ha he eloiy p is differen from he mehanial eloiy for he same parile, herefore he sperlminal eloiy p is said o be deoid of any physial meaning [5]. Alhogh of his, he Lorenz ransformaion for wae eor and freqeny, i.e. w w k k = γ k k, w = γ γ = ( w k) = γw k w are expressed by he phase eloiy as: p k γ k =, w = γ w p 5 C. Roy Keys In. hp://redshif.if.om (7a,b)

13 Apeiron, Vol., No. 3, Jly 5 86 For a ligh wae =, Eq (7b) is a longidinal Doppler shif formla, while for a maer wae, Eq (7b) is a non-longidinal Doppler shif formla. So he nion of SRT and de Broglie s wae formalism has always been prearios bease of he differen eloiies for he same parile. De o he differene beween phase eloiy and grop eloiy of de Broglie waes, de Broglie hrogh Eq. (6a), deeloped he onep of a wae assoiaed wih maerial pariles and remoed he poin parile phenomena o a maer wae phenomen. This led o a sienifi onroersy ha was sbje o mh disssion and aemps a resolion. The firs of hese aemps an be aribed o J. Wesely [6], who spposed a real wae fnion insead of he omplex wae fnion in radiional qanm heory. He old proe ha he phase eloiy eqals he parile eloiy. Anoher aemp in his onex was M. Wolff [7]. The wae srre of moing eleron is analyzed on he basis of spherial waes. He frees SRT from he sal onradiion and onldes SRT and de Broglie's heory are ompaible. 4- Deriaion of de Broglie Relaions for he Moing Daliy on a Dynamial Basis Reenly R. Ferber[8], showed ha Eq. (6b) is a resl of sing LT, and no a resl of de Broglie s hypohesis. Therefore, o deal wih hese onradiions, we ms re-derie all relaions in seion 3 wiho LT. To remoe he kinemaial onradiion in he de Broglie formalism, we firs derie Eq.(6a) on a dynamial basis saring wih Eq. (a), i.e. = p + m The las relaion yields: 4 5 C. Roy Keys In. hp://redshif.if.om

14 Apeiron, Vol., No. 3, Jly 5 87 d = pdp, i.e.; d = dp Using he Plank Einsein relaion, ha = h = w, yields dw = dp Now sing he definiion of grop eloiy, i.e.; Eq.(6d), we hae dp dk = By inegraion, = i.e. k =, we ge p k = i.e. h λ = (8) p We an now remoe he onradiion in Eq.(7) if we ake ino onsideraion ha Eq.(7) old be wrien as = m + m (9) The new form of oal energy, Eq. (9), is ery imporan, bease i shows s a new hidden ariable, whih is he relaie kinei energy = m (3) Eq. (3) akes he name of he relaie kinei energy bease i relaes o oal energy as = m = Now in addiion o he righ relaion = hf = m, we an speify he following relaion 5 C. Roy Keys In. hp://redshif.if.om

15 Eq. (3) helps s proe hap and (8) in Eq. (6) Apeiron, Vol., No. 3, Jly 5 88 = hf = m (3) =, if we sbsie boh Eqs. (3) w m / = p k = m/ = (3) Now seing Eqs.(8 ) and (6) in Eq.(4a), we hae: p k = γ k Using Eq.(3) in he las relaion, we ge x k = γ k Or k x = γ kx x Aording o Eqs.(3) and (6) he eloiy of a moing daliy in frames S and S is w= xk, w = xk, so we hae : w γ w = (33) Eq.(33) is now a longidinal Doppler shif formla for a moing maer-wae daliy, and redes o a longidinal Doppler shif formla for a phoon-wae daliy, = [9]. 5 C. Roy Keys In. hp://redshif.if.om

16 Conlsion Apeiron, Vol., No. 3, Jly 5 89 The inompaibiliy beween SRT and parile dynamis arise bease he LT and is kinemaial effes hae primay oer he physial aspes in deriing he relaiisi dynamial qaniies and in he inerpreaion of relaiisi phenomena [,a,b]. De Broglie formlaed his heory of qanm waes, hrogh LT, and he hypohesis of periodi phenomenon. Therefore de Broglie's formalism was he bes model o show ha kinemaial effes in SRT are no ompaible wih he dynamis of he moing parile. To deal wih hese onradiions, we reformlaed de Broglie's heory on a dynamial basis hrogh NSL and he relaiiy priniple. Depending on his approah, we an also find he inrinsi energy of a parile, whih allows s o onsr he framework of de Broglie's wae heory and SRT wiho he sal onradiions. Referenes [] A. Einsein, On he Elerodynamis of Moing Bodies Ann. Phys. 7, 89 (95). [] N. Hamdan, Newon s Seond Law is a Relaiisi law wiho Einsein s Relaiiy, Galilean Elerodynamis Jly/Ags (5). [3] A. Einsein, Does he Ineria of a Body Depend pon is Energy-Conen, Ann. Phys. 8, p (95). [4] L. De Broglie, The rren Inerpreaion of Wae Mehanis, A riial Sdy(Elseier, Amserdam,964). [5] P. Bergman, Inrodion o he Theory of Relaiiy: Doer, New York, p.45, (976). [6] J.Wesley, Classial Qanm Theory, Apeiron, Vol., Nr., 7 3 (995). [7] M. Wolff, Beyond he Poin Parile A wae srre for he eleron, GED, 6, 83 9 (995). 5 C. Roy Keys In. hp://redshif.if.om

17 Apeiron, Vol., No. 3, Jly 5 9 [8] R. Ferber, A Missing Link: Wha is behind de Broglie s periodi phenomenon?, Fondaion of physis Leers, Vol. 9, No. 6, (996). [9] N. Hamdan, Deriaion of he Relaiisi Doppler Effe from he Lorenz Fore, Apeiron, Vol, No., Jan.(5). [a] N. Hamdan, Abandoning he Ideas of Lengh Conraion and ime dilaion, GED, 4, (3). [b] N. Hamdan, On he Inariane of Maxwell's Field Eqaions nder Lorenz Transformaions, o appear in Galilean Elerodynamis magazine. 5 C. Roy Keys In. hp://redshif.if.om