Big Crash of Basic Concepts of Physics of the 20 th Century?

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1 Cllege Par, MD 013 PROCEEDINGS f he NPA 1 Big Crash f Basic Cnceps f Physics f he 0 h Cenury? Peer Šuja Hradesinsa 60, Prague, Czech Republic peer.suja@ .cz This paper analyzes he quaniies f energy and mmenum in he definiinal relainship f relaivisic mechanics, in he de Brglie mmenum hyphesis and in he Klein-Grdn, Dirac and Schrdinger equain. The resuls f analysis shws ha Planc cnsan and relaivisic relainships n he lengh cnracin and increase in mass are a reflecin f he same physical principles in naure, ha λ designaed in he de Brglie hyphesis λ =hmv as he wave f maer wih res sae value λ = mus be cnneced wih a real dimensin f a paricle wih res sae value λ = l = mc and ha n his basis we can cme he fundamenal equains f quanum mechanics ha are he Klein-Grdn, Dirac and Schrdinger equain wihu he necessiy f he wave funcins. 1. Inrducin The relainships f quanum physics fr energy E h and mmenum p h f a phn and he relainship f relaivisic mechanics fr al energy he al energy relainship E m c m c m v c E E we can deduce ha he ineic i.e. added energy he res energy E mc is E mc m v c m c are he basic relainships f cnemprary physics. Frm he quadraic frm f E mvc p We can als derive his relain frm he al relaivisic energy as E m c m c m c 1 1 v c 1 m c c c v 1 m c v c v 4 4 m vc c c v m vc 1 v c mvc p (1) In quanum mechanics (QM) as in relaivisic mechanics (RM) fr ineic energy we wrie direcly cnsequenly we bain E mvc mc m c and hus we can wrie E mvc m c 4 wrie he square r f he sum f squares E m v c m c E E. E mc m c E E and s in cnras al energy in RM where we This way we bain he relain fr he rai f ineic energy mmenum as E p mvc mv c and fr he rai f al energy mmenum as E p mc mv c v r E pc E E mcc mvc c v. Cnsequenly, as he speed f an bjec v appraches he speed f ligh c v c is mmenum, muliplied by c, appraches al energy pc E. RM esablishes a differen definiin f ineic energy E mvc mc m c frm classical mechanics (CM) E 1 mv. In RM, he relain f classical ineic energy is subsequenly seen as an apprximain f he relaivisic ineic energy relain and he classical relain can be fund by expanding he relaivisic relain in Taylr series E mc m c m c v c m v m v c m v. By his expansin, we change he definiin saus f ineic energy frm a linear funcinaliy in RM, a he speed appraches he speed f ligh, in a quadraic funcinaliy in CM, a a speed much slwer han he speed f ligh in a vacuum.. Energy and Mmenum f a Phn. Energy and Mmenum in Relaivisic Mechanics The relain fr relaivisic ineic energy E mc m c mvc, hwever, has a general frce and is acive a any speed, hus als a speeds much slwer han he speed f ligh. (The firs psulae f RM - physical laws are he same frm a any inerial sysems f crdinaes in unifrm ranslary min). The relains f a phn's energy E h and relaivisic energy E mc m v c m c are ied by he ineracin f phns wih paricles, such as he phelecric effec, he scaering effec 4

2 Šuja: Big Crash r elecrn psirn pair prducin (EPP). Fr EPP, we wrie he relain beween phn's energy and ineic energy f an elecrn and a psirn as h h m v m v and fr he elecrn's prin, we can wrie 1 1 e p he minimal frequency f phn energy necessary fr EPP and equals he inernal energy f an elecrn h h mev. The frequency ν is 1 1 h hc mc. This frequency ν crrespnds he Cmpn wavelengh f an elecrn h m c and hus we can reasnably suppse ha λ crrespnds he maximum radius l h m c f creaed elecrns. The energy balance relain fr EPP is based n he phelecric effec explanain, where he difference beween inciden energy f phns hν and binding energy f elecrns hν equals he ineic energy f elecrns E 1 E h h m v emied frm ams. The classical mechanics relainship fr elecrn's ineic energy 1 m v in he phelecric effec relain 1 E h h m v is, hwever, an apprximae and accrding RM, we wrie he relain h h hc hc mc m c mvc p This relain is in accrdance wih he relaivisic undersanding f energy E pc, since by dividing hc hc mvc by c, we ge he relain fr mmenum a he phelecric effec h c h c h h h mc m c mv p p p. This mmenum relain has he same frm in CM and RM and, in cnras he energy relain, here is n w-faced wriing r apprximain f is righ hand side. Fllwing his mmenum relain we can arrive a he idea ha, in he same way as we undersand ineic, al and inernal energy in RM, we can cnsider mmenum p, al mmenum p and inernal res mmenum p. We can imagine he inernal res mmenum p as rain, r as he angular mmenum spin insalled in QM. Thus we can al abu Einsein's definiin f energy (EDE), where mmenum muliplied by c equals energy E p In EDE, he al mmenum f a phn and paricle p h h c mc, muliplied by c is he al energy E p c hc hc c mc The al energy f he phn begins frm zer energy, s zer mass hen frm zer frequency 0 and frm infiniy phn wavelengh. The paricle al energy begins frm res energy, hen res mass wavelengh and frequency f a phn's energy required fr EPP. In EDE mmenum, E m c hc h, where λ and ν are he p mv mc v c h h h h h v c muliplied by c, prduces a change in energy E pc mvc h h hc hc v c mcc v The relainship E p c in which is he ineic energy EDE is valid hwever, jus fr he rai f crrespnding quaniies, ha is fr al quaniies fr he change in he quaniies E p h h The rai fr he nn-crrespnding quaniies is 1 E p h h mc mc c, 1 E p mvc mv h h c h h c and fr he res sae quaniies E p mc mv h h h hc h c c c v c v and als E E mcc mvc p p c v. The subsanial relain valid in RM, as well as in QM, pc mvc hen energy pc E as he speed v c. Then we can inerpre he relain E pc mcc mvc p c pc c v s ha fr v ccnse- quenly pc mc v c Ev c hen E p c v, resuling als frm p c m c m c m v c hen 4 4 E p c v, is inerpreed in a way ha, mmenum muliplied by c, appraches al pc E since ineic energy appraches al energy mvc mcc and since mmenum appraches al mmenum mv mc s p p. The same way as in he relain E mc mc mvc fr v c, res energy becmes negligible and ineic energy appraches al energy mvc mcc, s in a relain p mc mc mv, res mmenum becmes negligible and mmenum appraches al mmenum mv m In RM and in QM we al abu al energy f paricles E mc and abu al energy f a phn E h hc mc. Als, i is necessary sress ha in he relainship fr phn's mmenum p h h c mc, we al abu phn's al mmenum and fr a change in phn's mmenum we use he relain h c h c h h e.g. a Cmpn's effec. The phn mmenum and energy is equally expressed using ν r λ. Cmpn preferred fr mmenum he relain h c. Tal energy and mmenum f a phn, as well as he mass equivalen and frequency f a phn are running frm he zer values and a wavelengh frm an infi- nie value. Tal relaivisic energy fr a paricle is running frm res energy m c h hc by adding ineic energy. Thus, if we wan frmulae he energy relain f a paricle similarly he phn's relain E h hc mc, hen he ineic energy can

3 Cllege Par, MD 013 PROCEEDINGS f he NPA 3 mc m c mvc h h hc hc hc mcc v c p v pc and dividing his be wrien in he relain relain by c, we can bain he relain fr mmenum mc m c mv h c h c h h h mc v c p. Thus we can accep ha, if we wish ransfer he phn's relain E h and p h n an elecrn and, if we cnsider h h mv as mmenum f a paricle, jus as we ae ineic energy in relain als cnsider as he al mmenum f a paricle h rai f he al energy mmenum wrien as mc and al energy and rai f he ineic energy m E p mc mv h h h hc h c c. c v c v menum as E p mc mc mv mvc mv hc mc m c h h hc hc and mc hen we can bain he a nn-cnrversial h h c Subsequenly, we mus 1 accep inequaliy f he rais E p mc mv E p h h and he accurae rais are E p mc mv h h h c v and E p mc mc h h 3. Is he De Brglie Hyphesis h mv Accurae? De Brglie inrduced he presumpins ha he phn's relainships can be ransferred n a paricle as p h mv and E h mc where he res energy f a paricle is assciaed wih he frequency wih a paricle's wavelengh h mv, 1) whse value is infinie fr zer mmenum mv 0 Brglie cmes w differen rais f energy mmenum E p mc mv c v h mc and paricle mmenum is assciaed 1 E p h h hc h c. The paradx is clearly seen if we inser h mv in he al energy relain E mc hc hcmv h mvc, s we ge E mc mvc, which is valid nly when c v. Frm hese presumpins, de and, cncurrenly, energy wha is als valid fr a free paricle in QM. De Brglie wred u his paradx by he phase velciy E mc h hc, hen, s if he al energy equals he ineic w c v and he grup velciy f a paricle. Cnsequenly, a phase velciy is always higher han he speed f ligh c and he phase velciy is infiniy fr a speed v 0 and, a a speed v c, he phase velciy appraches frm infiniy c. The enire aing ver he de Brglie's frmalism f he wave prpery f maer by QM leads he wave funcin and he wave prbabiliy f paricles prpagain. Up day, he meaning and rle f phase velciy and wave funcin is unexplained. Bu frm he freging cnsiderains, we see ha he discrepancy 1 f he rais E p h h c and E p mc mv c v insead f real validiy f he rais E p mc mc h h hc h c and c c v s c v resuls frm he simulaneus apparen validiy E p mc mv h h hc h c c c v c v. The paradx als disappears if we pu in p h mc s h mc in E mc h hc hcmc h mc and hus arrive a Thus we see ha we can wrie he rai f energy and mmenum fr al energy energy E p h h mv mc m c mv mvc mv c by aing energy frm 0 E h h mv E mc m E p h mv mc mv c v, r fr change in E h 0 mc r frm s v s S in he case f he de Brglie's ransfer f phn mmenum relain p h mc n he mmenum f he paricle, his ransfer mus be perfrmed frm he limi values f wavelenghs by h h mv p s as added value he inernal mmenum f paricles h h mc m c mcv c mv p. 4. Lengh Cnracin and Increase in Effecive Mass Operae Inseparably Based upn he cnsiderains in his paper, we can be cnvinced ha fr ransferring he phn's relain f mmenum p h and energy E h n an elecrn we mus crrecly ransfer he quaniies f al, added and res energy and mmenum, as well as ransfer he dynamics f increase in he spaial energy cncenrain expressed in a raising phn's frequency, jinly wih shren-

4 4 Šuja: Big Crash ing he phn's dimensin expressed in shrening f he phn wavelengh. Thus in he same way cnsider abu shrening in he elecrn dimensin by increasing in elecrn's energy. We can regard he principle f spaial shrining f mass-energy by increasing mass-energy as a universal principle f naure. S as a he phn, s as in nuclear physics, s as in asrphysics, he greaer accumulain f mass represens greaer energy and leads is smaller spaial lcalizain. This principle in fac als predicae he relaivisic relainships n he lengh cnracin and increase in effecive mass, wih increasing energy resuling frm increasing speed. If we cnsider he relaivisic relains f increases in mass m m v c 1 and he lengh cnracin 1, hen we can wrie 1 v c m m l l l l v c r rewrie i r m l c m l c c v c v. Thus fr any difference in c v m c m c v l c l c v he prducs m l c m c l h remain cnsan, s he prduc f mass and is spaial layu remains cnsan m l ml h c 7 m g his leads Cmpn wavelengh f prn g.m. Fr prn mass 15 l m. Aferwards, agains he calibrain basis h m l c r h l m c and als an added r change in he value as h l h l m c m h l m c, we may express al value as Cnsequenly, as an elecrn increases in energy we mus cnsider he decrease in is radius frm he res value l a res energy m c h hc l required fr EPP, s frm he Cmpn wavelengh l h m c in cmpliance wih he lengh cnracin and increase in effecive mass. Then fr speeds appraching c, energy appraches infiniy and radius appraches zer, hus he speed f he paricle cann equal c, since is radius wuld be zer. Tday we accep as a naural ha fr he grea cncenrains f maer, s fr he grea cncenrains f energy, he dimensins apprach zer e.g. fr he blac hles, bu we fail cnsider he grea changes f he speed,energies and he penials f paricles in micr-wrld in he same naural way. T he presen-day, physics has n ye arrived a a specific value f an elecrn's dimensin and in a number f publicains i expresses a large radius fr slw elecrns and a shr ne fr fas elecrns, ) s we may believe in he relain beween he changing f an elecrn's dimensin wih is energy. Thus we may cnsider ha if we use res mass in relain h mc, we ge he Cmpn wavelengh f an elecrn ha is he res 1 15 diameer f a free elecrn l h m c.4310 m. Fr a free prn we ge l h m c m. S, if he QM res energy f he paricle is assciaed wih he frequency mc hen we can reasnably suppse ha his res energy is als assciaed wih he dimensin h m c h hc l. This frequency m c h hus crrespnds he Cmpn wavelengh f an elecrn h m c herefre he ulimae diameer l h m c f a creaed elecrn in EPP. (If we hin, ha EPP is unclear prcess and requires cndiins f a nucleus field and mmenum, hen we can equally cnsider he elecrn psirn annihilain prcess which des n require hese cndiins). By his means, he ransfer f he phn's mmenum relain p h n a paricle, represens a change in paricle dimensin frm he Cmpn wavelengh value h mc, fllwing he change in mmenum f a paricle wih increasing is speed v, as h l h l h l l l l mc v c mv p. Accrding he demnsraed cnvicin f change in elecrn dimensin wih a change in elecrn energy, we rus ha he experimens demnsraing he wave prpery f elecrns can be explained by a real change in he dimensin f elecrns. In his meaning we can cnsider he Bragg's x-rays inerference law n dsin, where he inerference f ligh as an inerference f phns is firmly lined wih he phn's wavelengh λ and herefre wih phn dimensin. Subsequenly, we can accep ha experimens fr elecrns presened suppr he de Brglie wave hyphesis fr example he Davissn-Germer's experimen (where he relain n dsin is accuned fr an explanain) can be inerpreed as he acual change f elecrn dimensin wih a change in is energy. 3) Thus we can reasnably suppse ha in he same way as fr change in mmenum f a phn he same is rue fr change in mmenum f elecrn frm he res sae h 1 h h l h l mv where l is he acual dimensin f an elecrn ha becmes shrer as is energy increases. Cnsequenly, he classical ineic energy f an elecrn is expressed in he relainship wrien as E 1 mv p m h m h m l h m l h m l l l l. Mrever, if we cnsider he relaivisic relains f he lengh cnracin m m v c l l v c 1 v c l l 1 s 1 we can wrie elecrn ineic energy as s v c l l l and increases in mass h m l l l l h m l l l l h m l v c () and fr h m l c we ge

5 Cllege Par, MD 013 PROCEEDINGS f he NPA 5 h m l v c m l c m l v c m c m 1 v c v c m v m 1 v c m v m p m mv. 1 If we direcly apply relaivisic mmenum in he classical ineic energy frm E 1 1 mv p m m v m v c hen frm classical ineic energy we arrive a classical relaivisic ineic energy (CRKE). As in RM in he same manner m v m m c m m c m is valid and fr al energy we may wrie (3), m v m c m c is valid, hen 1 m v m m c m m v m 1 v c m c m m c m v c v 1 s m m 1 1 v c hen m m 1 1 v c. 5. D he Relains E h and E mc Express he Quaniy f Energy? m c m 1 v c m c m (4) Frm he freging cnsiderains, we psi ha he classical ineic energy (CKE) relainship E mv p m m v m m v m v c m c m v c m c m m c v c m c mus be perceived as a limi f he relain me m v m c m c p a speeds much slwer han he speed f ligh. In RM we derive he equain f al relaivisic energy frm he relain m m v c 1 and hen bringing i he square m c m v m c, we muliply i by c². Afer applying he square r, we ge E mc m v c m Then we can reasnably 4 expec ha he sep f muliplying by c² is unfunded in physics and is ineninal, in rder ensure he dimensin f energy afer he resuling square r and merely because f ha, we deermine energy as mmenum muliplied by c. Wihu muliplying m c m v m c by c² afer applying he square r we bain he relains fr mmenums m v m c m c v c m c m c v c 1 v c 1 m c v c v 1 mc and mv m c m c m c 1 1 v c 1 m c v c v mv. Cnsequenly we can wrie and res p mmenum f paricles. mc mv mc and mv mc mc p p p where mc and mc mus be idenified wih a al Aferwards, accrding he ransfer f he relains f phn's mmenum n a paricle presened in his paper, he relain fr CRKE m c m c m v m c v c harmnizes wih he relain h l h l h l l l l h l v c m c v c, as well as wih he relain fr frequency expressin we ge h c h c h c h c v c m c v c, where l is he res dimensin and ν he res spin frequency f a paricle a res mass m jined in he Cmpn wavelengh relain h l m c h S fr CRKE we can wrie me h l h l h c m c m c m v p and fr mmenum we can wrie p h l h l h c mc m c mv. Afer muliplying he las relain wih c we ge he phelecric effec explanain f Einsein. wrie he relain hc hc h hc l hc l mc m c mvc E Then fr he classical relaivisic ineic energy, we can me h l h l h h l l l l h l v c h c v c m c v c m v (5) p

6 6 Šuja: Big Crash The ineic energy, i.e. he added energy wrien as he res energy m c and his energy are lined res values h h l v c h c h c v c, runs fr an elecrn frm 1 l 1 l and runs frm l,. The classical ineic energy h l h c m c, where symbls and mean ha E 1 mv m v m m c m 1 v c m c m m c 1 v c m c is he limi f he relain m c m c m v m c v c and The difference f added energy, expressed equally by l r, equals zer h h c h l v c h c v c 0. If we wrie 0 0 is added energy cmpared he values l, and res energy m h h l h c h c m c and we mean ha values f, m run frm 0, m 0 and value l runs frm infiniy l, hen we signify he al energy. The difference f al energy expressed equally by l r h c h h c h l m c m c 0 equals zer, where all erms fr energy in he las relain increase frm zer and up he wn values f a paricle m c h l h c, his increase is he res energy f he paricle, as fr insance he energy f he phn needed fr EPP. We can wrie he al classical relaivisic energy (TCRE) E f a paricle as h h h l h m c h l v c m c m c v c m c m c (6) and fr frequencies expressin h c h c h c h c m c h c v c m c where l,, m are he Cmpn values. Cnsequenly we cann wrie he equain fr added values m c v c m c m c (7) h c h h c v c h l v c m c v c m c v c 0 m c as well as fr al values we cann wrie equain h c h h c h l m c m c 0 m Fr he difference f al and added energy h c h r h c h we can wrie h c h h c m c h h 0 c h l v c r h c h h c h m c h c v c h m c m c m c v c m c m v m c (8) h c v c h l m c v c m c m v m c m (9) Frm he las relain h c v c h l m c v c m c m v m c m c we see, ha if we subsiue by c v c s h c h l m v c, hus if we subsiue cnnecin l c by cnnecin l v c we can wrie 6. Where Is Energy and Mmenum in he Klein-Grdn and Dirac Equain? Fr l c we may wrie h c v c h l m c v c m c m c r he relain h c h l v c m c m c v c m c bu we cann wrie equain as c h c h l h c h m Bu if we ae (as de Brglie did) c v, we may hen wrie equain c h c v c h l h c h m c, r equain c h c h l v c h c h m

7 Cllege Par, MD 013 PROCEEDINGS f he NPA 7 4 Bu as in QM, he wave funcin Ψ prvides rai c v s c 1, hen using he wave funcin we can wrie equain c m c which is a wriing f he Klein-Grdn (K-G) equain. Cnsequenly we can hen wrie he Klein-Grdn equain, wihu he wave funcin, in frm h c v c h v c m c and accrding ur reques sar frm, and if we sar frm 0, l r l, (de Brglie anicipain), hen m c m c r frm 0, l l hen, l l hen m c 0. m c m c Thus we can rus ha if ν and l (r alernaively and x ), run frm muually crrespnding values, hen he d'alemberian is always zer 0. The K-G equain fr a free paricle wih he wave funcin m c can be wrien if ω and (r alernaively and x ), d n run frm he muually crrespnding value and hen he value f energy expressed by and ( r alernaively and x ), are muually shifed wih he cnsan c, whereby we subrac, r he crrecin c v whereby we add, value Similarly we can believe ha fr he mmenum f a paricle we can wrie m Using he wave funcin, we perfrm crrecin v m c values expressed by and (r alernaively and x ). mc mv mc hen p p p s h c h mc r h c h 0 mc where x and runs frm diverse values. Wih he wave funcin, we can wrie h c h 0mc 0 which represens he Dirac equain. 4) In he marix frm f he Dirac equain c imc 0 he wave funcin prvides shif ver mc ne r bh f he erms, c, c r reverse and marix ffers a relevan algebraic sign + r - and marix ffers a relevan algebraic sign fr mc r + r ffers mc D Kineic Energy f Elecrns in Phelecric Effec Equal E mvc r E mv? 1 Millian, 5) wh fr many years disagreed wih Einsein's undersanding f he phelecric effec, fund in his experimens he prprinal increase in ineic energy f elecrns released frm a meal surface wih he linear frequency increase f phns sriing ha surface. The Millian's experimens cnfirmed prprinaliy beween he sp emissin penials V wih phns frequencies a he phelecric effec in relain h ev mv p m and i is believed, ha als wih phn energy. Bu as mmenum f 1 a phn is h c hen Millian's experimens cnfirmed he same prprinaliy f energy E and mmenum p f a phn wih f an elecrn. Then mmenum p f an elecrn has be prprinal he square r f energy h c f a phn. Bu in Millian's experimens Millian wre in his char h ev mv fac ha h is merely prprinal ev 1 mv p h as well as f mmenum 1 bu in ex unequivcally highlighed. S bserving linearly frequency f a phn is energy can be prprinal. h c p h c mv p m and we 1 Then we believe ha if we bserve linearly frequency f a phn is energy is 1 can wrie h c ev mv. Thus Millian's experimens shuld be inerpreed in a way ha, wih linear increase in he frequency f phns, heir energy increases quadraically and his energy equals he quadraic increase in ineic energy f elecrns me h h l h l h l v c h c h c h c h c v c m c m c m c v c m v p. (10) The same way as in Millian's experimens, we bserve a linear increase f phn frequency, while he energy f he phn increases quadraically, we in classical physics als bserve a linear increase f he speed v f an elecrn while is energy increases quadraically as 1 mv. Frm he freging reasning in his paper, we cme he belief ha he al mmenum f a phn represens he relain p h h c mc, where runs frm infiniy and, m runs frm zer. Change in al mmenum f phn represens he relain p h 1 h h 1 c h c m1c mc. The mmenum f a paricle is

8 8 Šuja: Big Crash p h l h l h l v c h c h c v c mc mc mcv c mv, where l,, m run frm h l h c m The al energy (TCRE) f a phn represens he relain h h c m c, where runs frm infiniy and, m runs frm zer and fr he al energy f a paricle we can wrie h l v c h l h c v c h c m c v c m c, where l,, m runs frm h l h c m As he relain me h c represens he phn's energy, s fr he phelecric effec we mus wrie he relain h c h c m c m c m v, which fr mmenum equals he relain wrien as h c h c mc m c mv. If we muliplied he las relain by c (in EDE E pc ), hen i represens relain f Einsein's wriing fr energy a he phelecric effec h h mc m c mv We can bain his las wriing fr energy a he phelecric effec, als by muliplying he Dirac equain h c h m c 0 (he equain fr mmenum mc mv m c 0 ) wih c wha hen resuls in h hc m c. c h 0 hc l hc l m c mc mvc m c Where Is Energy and Mmenum in he Schrdinger Equain? Amic physics n he basis f bservain f quadraic changes a hydrgen amic line emissin specra, frmulaed in he Rydberg frmula 1 R 1 H n1 1 n, came cnclude ha he differences hc R 1 y 1 1 represen he ransiin beween differen energy levels f an am and, ha fr energy levels cmpared he maximum energy level, he relain n H y H H E E n R n hc n h n is valid. 6) This quadraic changes in energy levels f ams was explained in QM (neglecing changes in energy f a prn, which is 1836 imes greaer in mass han an elecrn) by quadraic changes in elecrn's energy f ams. This explanain was frmulaed in he sainary Schrdinger equain (SchrE) h me V ineic energy relain, 7) where he classical 1 mv p m h m and he de Brglie hyphesis p h mv is used. Obviusly he same cndi- ins are als valid fr hydrgen amic absrpin specra, s ha a quadraic change in he wavelengh 1 r frequency c f inciden phns, gives rise he quadraic change in elecrn energy f he am 1 h m p m mv. Bu we can see absrpin specra as a firs sage f he phelecric effec. S, quadraic changes in energy f an inciden phn are equal he quadraic changes in elecrn energy befre is emissin u f an am. Als, fr hese reasns we cnsider i as unreasnable change he relain f classic ineic energy in he phelecric effec frm he equain h h mv 1 in h h mc m c mvc, wih he view f cnservain a lineariy in he equain. In QM we declare SchrE as nn-relaivisic because f nn-lineariy f he ime dependen SchrE i m s nn-lineariy f he relain h h m ha can be wrien as 1 h h mc m c mvc pc p m m v m mv. Cnsequenly, we perfrm refrmain in he righ hand side f he SchrE in he linearized erm h c h (change f al mmenum h c h!) which resuls in he Dirac equain. On he cnrary, we can believe ha h h r des n represen a change in energy, bu he change in mmenum muliplied by c. Thus we can reasnably believe ha a change in energy f an elecrn a he phelecric effec, jus as a change in energy a SchrE represens he relain h c h c h c h m c m c m v. This relain is seen, a speeds much slwer han he speed f ligh c as he classical limi f ineic energy h m c h m c h m c h m l h m l h m m c m m c m m v m p m mv. 1 Cnsequenly, we believe ha if we wan indicae energy hen we have change he lef hand side f he phelecric effec equain h h mv 1 jus in he same manner as he lef hand side f he ime ShrE h represening energy in EDE (where E pc ) in i m h h and frm h c h c h c fr phelecric effec and h m c h m c h m c fr ime ShrE. Bu afer ha we are aling abu he K-G equain fr energy h c h m c, hen abu he RM equain 1 m c m v m c, r abu he CRKE m c m m c m h m m v m p m mv as he limi f he relain much slwer han he speed f ligh. me m v p, a speeds

9 Cllege Par, MD 013 PROCEEDINGS f he NPA 9 If we wan persis in he energy definiin by EDE ( E pc ) hen we mus change he righ hand side f he phelecric effec equain and als SchrE 1 mv h m in mv h h c and cnsequenly we ge he Dirac equain c imc 0 s equain fr mmenums ha we can wrie as h 0 c h mc s mc mv mc. Las relain afer muliplying by c represens he equain fr energy in he sysem f EDE, where mc m c mvc E. 9. Cnclusins mc mvc m c r Frm he freging reasning in his paper we believe ha -he Planc cnsan and relaivisic relainships n he lengh cnracin and increase in effecive mass is a reflecin f he same physical principle f naure expressed in he relain h l mc and s fr rigid paricles in he relain h mlc m l c 1 v c 1 v c m l c - in he de Brglie hyphesis h mv, insead f idenifying as he wave f maer wih he res sae value, mus be cnneced wih he real dimensin f paricle λ=l wih he res sae value h m c h The same way he phn wavelengh λ is cnneced in he relain hλ=mc wih dimensin f a phn and wih is al mmenum - In he case f he de Brglie's ransfer f phn mmenum relain p h mc n he mmenum f he paricle, his ransfer mus be perfrmed frm he limi values f wavelenghs by h h mv p s as added value he inernal mmenum f paricles h h mc m c mcv c mv p. -n his basis, if we carefully cnsider he relain amng al, added, res energies and mmenums, we can derive he fundamenal equain f QM ha is he Klein-Grdn, Dirac and Schrdinger equain wihu he necessiy f he wave funcin -he Klein Grdn equain represens he equain fr energy -he Dirac equain represens he equain fr mmenum -classical ineic energy E 1 mv m v m m c m m c m m c 1 v c m c is he limi f he relain speeds much slwer han he speed f ligh -energies in RM as mc, mvc, me m v p a mc and energy f a phn h d n represen quaniy f energy, bu quaniy f mmenum muliplied by c, s mc c, mv c, mc c, c h c and merely he dimensin f such quaniies equals in dimensin he quaniy f energy. References [ 1 ] L. De Brglie, Ann. Phys.(Paris) 10e [3] (195) (Transl. by A. Kraclauer (004)) [ ] D. Burilv, hep-ph00017 [ 3 ] C. Davissn and L. H. Germer, Phys. Rev. 30 (197) 705 [ 4 ] P. A. M. Dirac, Prc. R. Sc. Lndn A 118 (198) 351 [ 5 ] R. A. Millian, Phys. Rev. 7 (1916) 355 [ 6 ] N. Bhr, Phils. Mag. 6 (1913) 1 [ 7 ] E. Schrdinger, Phys. Rev. 8 (196) 1049 [ 8 ] A. S. Davydv, Quanum mechanics (Pergamn Press, Oxfrd, 1965) [ 9 ] R. Feynman, R. Leighn and M. Sands: The Feynman Lecures n Physics (Addisn Wesley, Reading, 1965) [ 10 ] P. Suja, P. Carny, Z. Pruza and J. Hermansa, Radiain precin dsimery 19 (1987) 179 [ 11 ] P. Suja, Nuclear racs and radiain measuremens 1 (1986 ) 565