Tolga Yarman. Okan University, Akfirat, Istanbul and Savronik, Sanayii Merkezi, Eskiehir, Turkey.

Transcription

1 Intenational Jounal of the Physial Sienes ol. 5(7, pp , 8 Deebe, Aailable online at ISSN Aadei Jounals Full Length Reseah Pape Wae-like inteation, ouing at supeluinal speeds, o the sae, de oglie elationship, as iposed by the law of enegy onseation: Eletially bound patiles (Pat Tolga Yaan Okan Uniesity, Akfiat, Istanbul and Saonik, Sanayii Mekezi, Eskiehi, Tukey. E-ail: tyaan@gail.o. Aepted Noebe, ased on the law of onseation of enegy, we hae shown that, the steady state eletoni otion aound a gien nuleus in a non-iula obit depits a est ass aiation, though the oeall elatiisti enegy eains onstant. This is, in no way, onfliting with the usual quantu ehanial appoah. On the ontay, it poides us with the possibility of bidging the speial theoy of elatiity and quantu ehanis, to finally ahiee a natual sybiosis between these two disiplines, and futheoe, eluidating the quantu ehanial weidness, siply based on the ee law of elatiisti onseation of enegy. Ou theoy was deeloped oiginally, is-à-is gaitational bodies in otion with egad to eah othe; hene, it is ofoting to hae both the atoi sale and the elestial sale desibed on just the sae oneptual basis. One way to oneie the est ass aiation of onen is to onside a jet effet. Aodingly, a patile on a gien obit though its jouney ust ejet a net ass fo its bak to aeleate, o ust pile up a net ass fo its font to deeleate, while its oeall elatiisti enegy stays onstant thoughout. In othe wods, a inial est ass is tansfoed into exta kineti enegy though an aeleation poess, and kineti enegy ondenses into exta est ass though a deeleation poess. The tangential speed of the jet in question, stikingly points to the de oglie waelength. This, on the whole, akes the jet speed supeluinal, yet exluding any tanspot of enegy. This onjetue oinides with the eent easueents (Kholetskii et al., 7; Salat et al., 8, and poides a lue fo the wae-patile duality, also insues thei onuent oexistene. An ipotant onsequene of the appoah pesented heein is that, eithe gaitationally inteating aosopi bodies, o eletially inteating iosopi objets, sense eah othe, with a speed uh geate than that of light, and this, in exatly the sae way. In whih ase, though, the inteation oing into play, exludes any enegy exhange; thus, we would like to all it, wae-like inteation. The pesent appoah ends up the existing shizophenia, between diffeent oneptions painting sepaately the io wold and the ao wold, and unites these two wolds with an unequal, single and siple oneption, based on just the law of elatiisti enegy onseation. Key wods: Speial theoy of elatiity, eleti inteation, tahyons, de oglie elationship, supeluinal o wae-like inteation, geneal theoy of elatiity, gaitation. INTRODUCTION Louis de oglie has antiipated that (de oglie, 95, fo an objet of ass at est, thee should be a ν peiodi phenoenon, depiting a fequeny, suh that: h ν (de oglie s definition of the peiodi phenoenon s fequeny inside the objet in hand, at est. (

2 68 Int. J. Phys. Si. Hee, h is the Plank s onstant, and the speed of light in epty spae. It is eident that, de oglie had enisaged in extee ase, whee the entie ass would be tansfoed into eletoagneti enegy. On the othe hand, it is eakable that he onsideed Equation ( at a tie een when, the annihilation poess of an eleton with a positon eained fa away fo been disoeed. Posing T as the waelength, and as the peiod, to be assoiated with the eletoagneti wae oing into play, by definition we hae T ν Equations ( and (, as usual, lead to h (Waelength of the eletoagneti adiation assoiated with the ass, as oiginally assigned by de oglie, to desibe the peiodi phenoenon inside the objet in hand. ν The fequeny and the ass wee tansfoed diffeently and the objet is bought to a unifo tanslational otion (Lohak, 5; elatiistially, the fequeny deeases while the ass ineases, wheeas aoding to Equation (, ass and fequeny should athe be alteed in the sae dietion. This obseation, as de oglie entions it, intigued hi, fo a long tie (de oglie, 95. He ended up with the intodution of a new waelength desibing the anifestation of the wae haate of the objet; thus fo an objet oing with the eloity, de oglie faed, in siilaity with the RHS of Equation (3, as h (de oglie elationship witten fo the objet in hand, bought to a tanslational otion; is the elatiisti ass of the oing objet, that is, (4 (5 ( (3 whee, is the de oglie waelength ia Equations (3, (4 and (5. The elationship an be witten in a staightfowad way as:, (6-a (de oglie waelength witten along with Equation (, in tes of, the waelength of the peiodi phenoenon displayed by the objet, at est between the two waelengths and, in question. Hee, peaution is taken to wite the de oglie waelength fo a non-zeo eloity, sine odinaily one would think that de oglie elationship ould only be defined, along with a otion. Howee, as will be disussed late, de oglie elationship an be defined fo a zeo eloity, as well. In this ase, beoes infinitely long. As we will soon gie detail, this then onstitutes the basis of an iediate ation at a distane without any ass o enegy exhange. Fo this eason, in what follows, we will dop the estition. It is inteesting to eall that ou onjetue is well opatible with the quantu ehanial unetainty piniple, sine fo, the oentu is zeo too, whih iplies that, as stange as it ay look at a fist stike, the unetainty about the loation, is infinite. Fo ou standpoint, this esult alone an be onsideed as a lue fo an iediate ation at a distane, whih we popose to oin as wae-like inteation. Note that, deeases as ineases, and fo, beoes null (Equation (6-a. This would ean that, the wae-like inteation eases when the elatie speed of the objet, with espet to its suounding, eahes the eiling speed that is, the speed. At any ate, the wae-like inteation does not of light inole any ass o enegy exhange; any inteation inoling ass o enegy exhange, as usual, annot ou with a speed aboe the speed of light. Note futhe, fo / that beoes. The ontated waelength along the dietion of the tanslational otion (as assessed by the outside / obsee, is defined as usual as fo ν Equation (6-a. The oiginal fequeny, is ν ν onuently edued into the usual elatiisti tie dilation. / (pointing to

3 Yaan 68 At the sae tie, based on Equation (, as the objet is bought to the unifo tanslational otion, in ode to ath the elatiisti ass tansfoation, that is / / ; the definition of anothe fequeny is ν ν / / eoked, that is,. Theefoe, de oglie had poen that, if the peiodi ν ν / / phenoenon of fequeny popagates with the eloity /, while the peiodi ν ν / phenoenon of fequeny popagates with the eloity, then, the two waes, ae onstantly in haony with eah othe (de oglie, 95 (Appendix A. It is in fat that the geneation of these two diffeent peiodi phenoena though the unifo tanslational otion that onfused de oglie fo a long tie (aoding to his own stateent, and led hi to the foulation of Equation (4. Thus, based on Equation ( (that is, wae eloity waelength fequeny, ia witing / instead of, that is, ν ν (6-b (Set up fo the de oglie elationship, in tes of the inflated fequeny ν, popagating with the eloity /, and pointing to the wae haate of the oing objet. Louis de oglie, ould indeed ey well end up with Equation (4 (o the sae, Equation (6-a. This whole idea sees to hae been fogotten; de oglie elationship takes plae in all elated textbooks, but how de oglie had aied at this idea, is patially nowhee aound. Hene, it is the eloity /, whih is assoiated with the de oglie waelength, if oupled with the fequeny ν. Note that, this eloity would beoe / /, still assoiated with the de oglie waelength, but then oupled with the oiginal est ν fequeny (Equation (6-b. In any ase, egadless of the fat that enegy annot be aied by a wae of suh eloity, de oglie waelength should still be onsideed as a fundaental physial quantity. Eleton diffation obseed aleady in 97, thus long ago had been a onete poof of this stateent (Daisson and Gee, 97. The peplexing esults of neuton diffation expeients, ahieed late on, lealy display both the wae haate and the patile haate of diffating neutons (ilinge et al., 988, onjointly (and not just one of these haates, at a tie. In othe wods, neutons lead to diffation, although the bea is known to onsist of sae neutons taeling in a ay, one afte the othe. And this is how pesuably they should hae gone though the diffation slits; yet they still display a wae haate. Neutons ae finally deteted on the seen, as patiles; hene hee again, they display thei patile haate. This outoe eidently sees to be pofound, and should athe be onsideed along with the popagation eloity / of the wae-like infoation faed by de oglie waelength. He, hiself, does not see to hae onsideed the ase whee, fo whih the de oglie waelength beoes infinite. In this ase, the eloity / too, beoes infinite. This is nothing else, but (as will be elaboated on, thoughout, the popagation eloity of an infoation ν assoiated with the wae of fequeny, that is, the oiginal fequeny of the peiodi phenoenon of the intenal dynais of the objet at est, as intodued by de oglie (Equation (. In othe wods, the intenal beats of the objets, aleady at est, see to be felt instantaneously, eeywhee in spae, without any tansfe of enegy. This is what we ean by popagation of infoation, thus without attibuting any futhe eaning to the wod, infoation. How an suh popagation take plae? We do not know. Pehaps it is a popety of spae that oes into play. Neetheless we ae, in effet, inlined to think that the intenal beats of the objet at est, ae soehow ight away sensed eeywhee in spae, and onesely the objet itself, ey pobably aptues, patially at one, all (atoisti, nulea, gaitational, o othe intenal beats, existing at all possible sales, in its suounding, that is, the entie spae. So in one way o the othe, it ust be a question of a uniesal netwok of inteation in between all existing substanes, exept tanspot of enegy. In addition, the elated inteation speed is a supeluinal speed. Thus, though fee of any enegy exhange, the inteation in question ous instantaneously between objets, at est. Reent easueents see to bak up this aesting dedution (Kholetskii et al., 7; Salat et al., 8. On the othe hand, thee ay be a lue fo the ysteious duality. The wae haate, to us, is lealy due to the intenal dynais, the objet in hand peiodially delineates. This, ost likely, auses a gien distubane in spae, whih is tansitted all oe, and this at one, if the objet is at est. Thus, an objet at est, o in otion, in a way, ontinuously aks the signatue of its being in elation to the uniese (again, without howee, any enegy tansfe. Suh infoation would obiously go though any diffation slits, eeted

4 68 Int. J. Phys. Si. on the way, and eentually, ould well yield intefeene. The de oglie waelength beoes though finite, only if the objet oes, aking the oesponding diffation easueent possible. Thus pehaps, it is not, the diffating single neuton itself, whih goes though both slits, but the infoation about the peiodi beatings of its intenal whole atiity that it eanates. It ay ey well be this piee infoation, whih soehow digs out befoehand, the spae hannels that will host little late, oing neutons, on the way. At any ate, the popagation eloity / assoiated with de oglie waelength (Equation (6-b, is uh geate than the objet displaeent speed. That is, in the fist plae, the de oglie wae, aying the wae-like infoation was intodued, and the patile does not eidently oe, at the sae speed (no atte what, the wae of ν ν / fequeny oesponding to the dilated peiod of tie displayed by the intenal dynais of the objet, oes with the sae speed as that of the objet. It is futhe inteesting to note that the wae-like haate would be destoyed though an enegy and oentu exhange poess. Indeed, it sees lea that a oentu shok eeied fo the exteio petubs the oiginal beating syste of the objet in hand, thus ost likely wiping out the anteio wae-like infoation togethe with the spae hannels that would hae been oiginally faed. This ay indeed onstitute a lue to the lassial wae-patile duality. Now, diiding T the two sides of Equation (6a, by (Equation (, o the sae, eaanging Equation (6b, yields T T U We define the LHS as : U T (7a (7b T and ae in fat, just like, Note that, both defined in elation to the outside fixed obsee (It is tue T that and ae tansfoed, with the otion, though the aboe definition stays pefetly alid. U, ia Equation (, beoes U (7 (eloity defined based on de oglie elationship and the peiod of the peiodi phenoenon of the objet at est. Equation (6b, o oe speifially Equation (7a, o (7 tells us that, though a otion suh as the oh otational otion of the eleton aound the poton in the hydogen ato, eah tie (as assessed by the outside obsee the inheent peiodi phenoenon of the eleton assued by de oglie, and desibed (at est by (Equations (3 beats; a wae eho beats, all the way though the stationay obit of peiete (Reall that fo the eleton, based on Equation (3, tuns out to be The de oglie waelength to be assoiated with the eleton at the gound state of the hydogen ato is about Thus, / 35. T, on the othe hand, is.8 - s. Louis de oglie ould, in effet, show in his dotoate thesis that, in ode to display a stationay otion on a gien iula obit, the waelength to be assoiated with the eleton otion (depiting a onstant eloity along Equation (4, should beoe equal to the obit peiete. This an be pitued soewhat as a snake with two heads, one ateial and the othe one iateial; eah tie the snake, togethe with the ateial head, akes a oe fowad, its iateial head athes its queue. Reall on the othe hand that, one a wae is onfined, the quantization of it follows fo lassial physis (Appendix. Hene, de oglie, landed at oh s angula oentu quantization assuption (oh, 93. It is a pity that this ahieeent is not ited in any textbooks. The thing is that, the eloity U indued by the LHS of Equation (7 tuns out to be geate than the speed of light; at the slowest, it is the speed of light; it an be infinite fo a zeo. Fo this eason, de oglie onsideed it ey ational, as a eloity not aying any enegy, and we do well by stiking to this intepetation. We will futhe show that, the eloity U, owing to the law of onseation of enegy, oes along with a est ass ejeted o eeied by the objet, unde a gien inteation - whee the stati binding enegy ay ay, thoughout the otion of onen oupled with a fatual speed, and that, the atio U / is equal to /. Thus, we will see that not only an one deie Equation (6a, that is, the de oglie waelength, based eely on the law of onseation of enegy, but also that the eloity depited by the LHS of Equation (7 tuns out to be quite physial, if stikingly things ae not onsideed patile-wise. ut at the sae tie, t, wae-wise, does not inole any enegy exhange with the exteio, as will be iplied by the eleti and gaitational inteations that we will deal with, in this wok. Thus, in fat, de oglie waelength does not indeed ay any enegy, but it etainly aies a gien infoation about the

5 Yaan 683 inteation in question. Aazingly, ou appoah an be applied to gaitational and eleti inteations, undelining the fat that both inteations wok exatly the sae way. In fat, ou appoah an een be applied to a non-linea otion, suh as the otion of an objet situated at the edge of a otating dis. It is astonishing that the instant inteation, lealy eoked by diffeent aspets of quantu ehanis (as will be disussed in this wok, was not only thought to be against the speial theoy of elatiity (STR (and we will find out if it is not, but also its oelation with the siple faewok of de oglie elationship, whih onstitutes the basis of the wae theoy of atte (o the sae, quantu ehanis, following onsideations (as will be elaboated on, heein, eaining within the ee fae of the STR (Appendix A, has been badly oelooked. Fo oneniene heein, we will handle just the eleti inteation. In the subsequent pat, we will handle the gaitational inteation. Chiefly, fo this pat of this wok, we will deal with eletially bound patiles, fo a oplete pesentation; it would hae been useful, to add to the dissetation, a disussion about how one should iew the onnetion, between lassially onsideed eleti hages, and the bound hages, the way it will be handled in this study. As we will find out, the otion equation of bound eleton that will be witten soon, within the faewok of the pesent appoah, indeed, dieges no atte ey little, but, oneptually speaking, still seiously, fo the standad otion equation, lassially oined fo a bound eleton. In any ase, one will aise the question that, the appoah we will pesent heein, soewhat negates the Maxwell equations. Then of ouse, one should be expeted to wite expliitly new field equations, whih ae opatible with the postulate, we will foulate below, in fat nothing else, but the elatiisti law of enegy onseation, though ebodying the ass and enegy equialene of the STR. O, een oe fundaentally, one would hae been expeted to wite a new expession fo the Lagangian density of the eletoagneti field, oing into play, and haged patiles, and using the aiation piniple, to find new field equations, and a new foe law, et. This ay een hae been the topi of a sepaate atile. Howee, thoughout the elapsed tie sine 6, when the ateial we will pesent below, was essentially all eady, Kholetskii et al, who faed the pue bound field theoy and naed it, in shot, fotunately handled this poble, PFT (Kholetskii et al.,. They thus ae out with new field equations and a new Loentz foe law, though in a ey diffeent eans than that pesented heein; neetheless, thei esults bak that of this pesent study, allowing us, now, in the fist plae, not to hae to undetake heein the pobles, just entioned. It is ipotant to ephasize that the PFT is not a ontoesial appoah at all. In fat it has to do with the ipleentation of the law of oentu onseation fo bound, thus non-adiating hages, based on quantu ehanis. The PFT, biefly, stays within the faewok of the standad appoah, but geas it:. With espet to a full onsisteny, is-a-is the law of oentu onseation,. and fo quantu ehanially bound, non-adiating eleti hages. Its ange of appliability, though, as entioned, is quantu ehanially, bound patiles. PFT, neetheless, wipes out the long lasting quest of how to bidge the lassial. Maxwell equations, and quantu ehanis, and foulate aodingly, a useful faewok, essentially fo non-adiating bound patiles, thus filling the gap between the lassial eletodynais and the standad quantu ehanial appoah. In any ase, ou standpoint is that, any inteation depits a est ass hange. Say in a fee fall, in a gaitational ediu, the objet at hand, aeleates, due to the tansfoation of a inial pat of its est ass into kineti enegy. Suh an undestanding bings up the question of, how an this take plae? The oupling of aeleation and est ass hange indues the thought that, in the exaple at hand, est ass is ejeted fo the bak of the objet, to ath the exta kineti enegy aquied by the objet, in fat just like in a oket. This pitue, finally, as we will see, based on the elatiisti enegy onseation, togethe with the law of oentu onseation, leads to the de oglie elationship, poiding us with an inaluable bidge and sybiosis, between the STR and quantu ehanis. elow, we onside two eletially inteating objets suh as the poton and the eleton. We will all the fist hage, say, the poton, assued to be at est, the soue hage, and we will all, the othe hage, say, the eleton, eithe at est o in otion, the test hage. PREIOUS WORK: A NOEL APPROACH TO THE EQUATION OF ELECTRIC MOTION, AND DISCUSSION AOUT THE NOTION OF FIELD In a peious wok (Yaan, 4, we pesented a opletely new appoah to the deiation of the elestial equation of otion, whih led to all uial end esults of the geneal theoy of elatiity (Yaan, 6. Moeoe we applied, the sae appoah to the atoi sale, whih led to the deiation of a new elatiisti quantu ehanial desiption well equialent to that established by Dia, if geaed alike (Yaan, Rozano, 6. It beoes inteesting to note that the faewok foulated so, happens to be equialent to that of the PFT foulated by Kholetskii et al. (. In shot, we had stated with the following postulate, essentially, in pefet ath with the elatiisti law of onseation of enegy, thus ebodying, in the boade sense, the

6 684 Int. J. Phys. Si. onept of ass, though we will hae to speify, aodingly, the notion of field, whih we fundaentally ejet, as will be elaboated on, below. Postulate: Fo any isolated syste of patiles, the est ass of an objet bound eithe gaitationally o eletially, aounts to less than its est ass easued in epty spae, the diffeene being, as uh as a ass, equal to the binding enegy is-à-is the field of onen - taking the speed of light unity. A ass defiieny, onesely, ia quantu ehanis, yields the stething of the size of the objet in hand, as well as the weakening of its intenal enegy, on the basis of quantu ehanial theoes poen elsewhee (Yaan, 999 still in full onfoity with the STR. An easy way to gasp this, is to onside Equation (. If the est ass is deeased due to binding, so will be the fequeny. Thus, one obtains, at one, the gaitational ed shift. Ou appoah, ia Equation (, ight away iplies that, the size is aodingly stethed. Suh an ouene an be expeientally heked, if say a uon is onsideed to be bound to a nuleus instead of the eleton. The deay ate of the bound uon is indeed etaded as opaed to the deay ate of a fee uon (Ledean and Weinih, 956; aett et al., 959; Hezog and Adle, 98; Gilinsky and Mathews, 96; Yoanoith, 96; Huff, 96; Yaan,, Yaan, 5. Peditions, we ade about this phenoenon, eain uh bette than any othe aailable peditions. Reently, Kholetskii et al. ( aied at the sae esult ia thei pue bound field theoy (PFT. INDING ENERGY Take fo instane a piee of stone on Eath. We an assue that Eath is infinitely oe assie than the stone. Then the stati binding enegy of the stone to Eath is the enegy we hae to funish to the stone in ode to bing it to infinity. The alulation of the gaitational stati binding enegy is peulia though, sine the est ass of the stone is ineased as uh as the enegy funished to it, on the way. A detailed study of this poble is funished in Yaan (9, and is suaized in Appendix C. As the stone falls fo a patially infinite distane onto Eath, the kineti enegy it would aquie at the oent it stikes Eath is equal to its stati binding enegy, as it oes to est at the loation it stikes Eath. The binding enegy of the eleton to the poton in the hydogen ato in its gound state (supposing fist, fo sipliity, that the poton is infinitely oe assie as opaed to the eleton is the enegy one has to funish to the eleton in ode to bing it fo its gound state to infinity. The poton will eain at est, thoughout. Note that hee we do not speak about just the stati binding enegy of the eleton to the poton, but gien that the eleton is in otion aound the nuleus, it is question of the oeall binding enegy. Yet still, one as to funish that thee is uh enegy in the eleton, in ode to bing it fo its stationay obit to infinity. Hene, gien that the poton is infinitely oe assie than the eleton, unde the gien iustanes, it is the eleton whih will pile up the aount of enegy, in onsideation (A detailed analysis on this will be pesented below. In othe wods, the elatiisti est enegy (the elatiisti equialent of the est ass of the fee eleton (thus at infinity weighs oe than the elatiisti enegy of the bound eleton, and this as uh as the eleton is binding enegy (in the hydogen ato. On the othe hand, when bound, and still unde the gien iustanes, this uh enegy ought to be etieed fo the elatiisti est enegy of the eleton. As tiial as this ay sound to ost eades, this is ipotant to onseatie eations, dieted to the pesent appoah; theefoe, we should insist a bit futhe on it. Thus, let us go bak in oe details. Had we not assued that the poton is infinitely oe assie than the eleton, then the oeall binding enegy is the enegy in the non-elatiisti ase, and in 4 π e µ CGS unit syste / h (e being the hage of the eleton o the poton, µ the edued ass of the poton and the eleton, and h the Plank Constant. One has to funish in ode to dissoiate the hydogen ato into the eleton and the poton (that is while alost all of this enegy is to be delieed to the eleton, a inial pat of it is to be delieed to the poton. This enegy, in othe wods, is the ionization enegy of the hydogen ato, about 3.6 e. Afte dissoiation, the pai of eleton and poton will weigh as uh as opaed to what the hydogen ato weighs. In eese tes, the ass and enegy equialene dien by the STR, togethe with the law of enegy onseation, equies that the total est ass of the poton and the eleton onsideed sepaately in a spae fee of field, shall weigh 3.6 e less, when bound in a hydogen ato. Thus again, the hydogen ato weighs 3.6 e less than the su of the est asses of the poton and the eleton onsideed in a spae fee of field. Then, how an this ass defiieny be aounted fo by the oiginal ass of the poton and that of the eleton? As explained, as a fist appoxiation, it is the eleton elatiisti enegy at est, onsideed in fee spae, that will undego, patially all of the ass defiieny in question. When a assie hage + and an eleton ae bound altogethe at est, the eleton s est ass easued at infinity is deeased as uh as the stati binding enegy oing into play Hee, we will wok out the stati binding enegy of a

7 Yaan 685 nuleus hage, + (oposed of Z potons and an eleton of hage intensity, e, altogethe at est, and situated at a gien distane fo eah othe. Supposing, again fo sipliity, that the nuleus in onsideation is infinitely oe assie than the eleton, the binding enegy of the nuleus and the eleton, situated at est, at a gien distane fo eah othe, is the enegy one has to funish to the eleton in ode to bing it fo its bound loation to infinity. Had we not assued that the nuleus is infinitely oe assie than the eleton, then the stati binding enegy in question is the enegy one has to funish in ode to dissoiate the pai of nuleus and eleton bound at est (situated at the gien distane fo eah othe, into the fee nuleus (of hage + and the fee eleton ( that is, while one oe, alost all of this enegy is to be delieed to the eleton, a inial pat of it only is to be delieed to the nuleus. We all this enegy stati binding enegy; it aounts (in CGS unit syste to / R, whee the nuleus and the eleton ae oiginally at a distane R fo eah othe (eleti hages, unlike the gaitational hages, ae not affeted by the enegy piled up, as they ae aied away fo eah othe. Just like in the ase of the hydogen ato, the nuleus of hage + and the eleton situated at a distane R fo eah othe (altogethe at est, owing to the ass and enegy equialene dawn by the STR, along with the law of onseation of enegy shall weigh E R egs less than the su of the est asses, of the fee nuleus and the fee eleton (see the footnote, at the botto of the peious page and note that the eleti hages ae not affeted thoughout, only the est asses ae. Let us stess that, the enegy in question, ought to be etieed patially fo the eleton, alone. The eason is siple. Let us assue that the eleton falls fo a suffiiently lage distane onto the nuleus in onsideation, and is oiginally at est. If this nuleus is infinitely oe assie than the eleton, then the law of linea oentu onseation equies that the nuleus stays patially in plae, while the eleton keeps on falling. An outside an inteene soehow (this point will be elaboated on below, and stop the eleton, at a gien distane to the nuleus. Then, the only enegy he would tap would be the kineti enegy, while the eleton would hae piled up on the way. Thus, the syste oiginally oposed of the nuleus of hage + and the eleton, when bound at the gien distane R, fo eah othe (and oiginally at est altogethe, will lose the kineti enegy the eleton would hae aquied on the way. E R That is the enegy egs, and sine the nuleus would itually not oe thoughout, this enegy ought to be extated fo the eleton est ass alone. Now, let us disuss what we ean by an eleton and a nuleus held at est, at a gien distane fo eah othe and how one an ahiee suh a pai. Hee, we gie an exaple on how this an take plae. Just onside a dipole, suh as a wate oleule, in whih, the oxygen ato (O attats, espetiely the two binding eletons of the hydogen (H atos, delineating an angle HOH of about 5º. This leads to positiely haged hydogen atos and negatiely haged oxygen ato. Thus, wate oleule an indeed be desibed by a dipole, ade of -e situated neaby the oxygen ato, and +e situated on the edian of the tiangle HOH, in between the hydogen atos (e is again, the eleton Dipole hage intensity. In this ase, is the distane between the two epesentatie hages +e and -e. This then oesponds to a situation whee the hages Dipole +e and -e ae held still at a distane fo eah othe. Thus ou answe to the aboe question is affiatie, that is one an indeed well oneie a dipole oposed of + and e, at a gien distane fo eah othe, and at est, sine this is oneptually not any diffeent than the dipole (oposed of +e and -e delineated by a wate oleule. The binding enegy of the wate oleule (assued at est, o the sae with that of the dipole ade of +e and -e, is the enegy one has to funish to it, in ode to ay these two hages, fa away fo eah othe. In othe wods, this is the enegy one has to funish to the wate oleule, in ode to dissoiate it into its oxygen ato and its two hydogen atos. This enegy, whih we all E H O, is about 9.5 e. Thus negleting the ibational enegy along the bonds of wate oleule, uh of enegy should be extated fo the su of the est asses, of espetiely the hydogen atos, and the oxygen ato, weighed sepaately fo eah othe, to get the est ass of wate oleule. Noting that the oxygen ato is uh oe assie than the hydogen ato, oughly speaking, 9.5 e (oe speifially, the ass equialent of this uh enegy ust be extated, fo the hydogen atos. Hene, the bound hydogen atos, in wate oleule, shall eah weigh less than the ass H of the fee hydogen ato. How uh less? Eah, about half of the E H O dissoiation enegy of the oleule. (Note that, fo quantities defined at infinity, noally we use the subsipt ; but beause this sybol an be onfused with O epesenting the oxygen ato in the oxygen oleule, hee, we pefe to use the sybol, instead. Thus, the ass of the hydogen ato bound to O, in a H O oleule, shall nealy weigh, E H - H O / ( This is yet an appoxiation, sine the ass atio of the hydogen ato to the oxygen ato is about /6. We an

8 686 Int. J. Phys. Si. do uh bette than that. Fo exaple, the ass atio of the hydogen ato to the telluiu (Te ato is about /8. Thus, when two H atos ae bound to a Te ato, in a H Te oleule, beaing the dissoiation enegy E H Te, one an with onfidene affi (though still oelooking the ibation enegy of the oleule along the bonds that, eah of the H atos will, to an E H Te aeptable peision, weigh / e less, as opaed to the H ato weighed at infinity; while the Te ato (owing to the law of linea oentu onseation, as explained aboe, patially eains untouhed. Thus, the ass of the hydogen ato bound to Te, in a H Te oleule, shall patially weigh, E H H - Te /(. Futheoe, suppose that the oleule H Te undetakes a outine otational otion. Sine Te is uh too heay as opaed to H, the otational otion shall take plae aound Te ato. Let Rot be the tangential eloity of the H atos, otating aound Te. The oeall elatiisti enegy of suh an H ato thus beoes - Te /( ]/ Rot [ H E H Thus, following ou disussion we onlude that when a hage + and an eleton ae bound altogethe at est (still supposing that is ey uh oe assie that the eleton, at a distane R fo eah othe, the eleton ass easued at infinity, is deeased as E uh as the stati binding enegy / R egs, oing into play, to beoe - /( R ; the ass of the heay nuleus (owing to the law of linea oentu onseation is not itually touhed. E The enegy / R is nothing else, but the agnitude of the lassial potential enegy. Howee, we aoid this denoination, fo easons that will beoe lea soon. That is within the peuliaities we hae intodued, as we will see, hiefly the total enegy annot be set equal to the su of kineti enegy and potential enegy, wee they lassially defined. Theeby the next ipotant question to be answeed is the following: What is the oeall elatiisti enegy of the eleton quasistatially bought nea the nuleus of hage, if it is futhe set to a otational otion of eloity aound? Is it ( - / R /, o / - / R ased on the lassial potential enegy onept, all textbooks we know of, would answe the seond question. Howee, ou appoah leads to the fist as we will disuss in detail below. Let us think of the total elatiisti enegy of a H ato in the oleule of H Te, set to a otational otion aound Te. As disussed aboe, it is ( H and not H - E H Te // / Rot Rot E Te - H /. If so then the oeall elatiisti enegy of the eleton quasi-statially bought nea the nuleus of hage, if it is futhe set to a otational otion of eloity aound this nuleus, ust be ( - / R / and not / - / R., In any ase, henefoth, we will solely opeate on the onept of elatiisti enegy (and nothing else, whih we an lealy define, and wok out, with egads to a gien patile, eithe at est (based on the ass and enegy equialene dawn by the STR, o in otion. Let us eall that, in ode to alulate the binding enegy oing into play, fo eletially bound patiles, we ake use of the Coulob foe, yet with the estition that, it an only be onsideed fo stati hages. As we will suaize quikly, Coulob foe well woks fo a pai of stati hages. eyond this, a pioi, we hae stitly no idea, whethe it will still hold o not, wee one of the hages is in otion with egads to the othe, and as we will soon deie that it does not. In effet, Coulob foe eigning between two stati hages is a equieent iposed by the STR. It is that we wee able to deie the /(distane dependeny of the Coulob foe between two stati hages, just based on the STR (Yaan, 8. The undelying eason is eely that, the quantity (foe x (ass x( distane 3 (beaing the diension of the squae of the Plank onstant is Loentz inaiant. Thus, suppose we take a dipole into a unifo tanslational otion. Conside fo sipliity, the ase whee the unifo tanslational otion takes plae along the dietion of the line joining the two poles. Let the ass in question be the ass of the dipole. Then, the quantity (ass x (distane is Loentz inaiant; fo this ase, aodingly [ that is in iew of the Loentz inaiane of the quantity (foe x (ass x (distane 3 ], the quantity (foe x (distane,

9 Yaan 687 is Loentz inaiant. The eleti hages on the othe hand, ae following obseations, Loentz inaiant. (Othewise, the Galilean piniple of elatiity would be boken, and eleti hages, just like the speed of light, ust eain Loentz inaiant. Theeby the foe eigning between the two poles, expessed as the [podut of stati eleti hages oing into play]/[distane]n, an only allow the exponent n. Theefoe, the STR, exlusiely iplies the stutue of the lassi Coulob foe eigning between two stati hages. Hene, the faewok we set up heein, fundaentally lies on, the STR. Note that below, just like we did aboe, we onside eely the losed syste ade of two hages (of opposite signs. This eans we will ontinue to takle, all the way though, with these two hages, soehow engaged with eah othe eelastingly. Thus, we exlude the possibility of haing to deal with one hage only up to a gien point of a possible poess, suddenly allowing the popping out of the seond hage, ight next to the fist one (whih an, fo instane, be ahieed ia haging at a gien oent the plates of a apaito, while the fist hage is lying in the inside of it. Poesses taking plae in aeleatos also fall in this ategoy, whih is that of an eleti hage expeiening in its fae of efeene the eation of an eleti field on its way. All this lies outside of the sope of the pesent dissetation. The equation of otion We define as the ass of the eleton, at infinity. When this is bound at est, to a nuleus of hage +, assued fo sipliity infinitely oe assie as opaed to, this latte ass, following the disussion we hae just pesented, will be diinished as uh as the stati binding enegy oing into play, though the binding poess, to beoe distane ( ( to the nuleus. So that, (ass of the bound eleton, at est κ( whee is κ ( (, at a As is well known, and as will be speified ight below, / is the stati binding enegy of the eleton, at the (8 (9 loation. In othe wods, this uh of enegy is equied to bing the eleton fo its loation, at the stage in onsideation, thus fo est, on the obit, to infinity (assuing that the effot in question leaes untouhed the poton, haing noted that it is patially infinitely oe assie than the eleton. Equation (9 theeby expesses nothing else, but a deease in the est ass of the eleton. Indeed the est ass of the eleton at the loation beoes ( ( Z e In the lassial field theoy, this latte equation, when ultiplied by, that is, in the fo of ( ( /, tells us that at this loation, we still hae at hand, the oiginal est ass [and, not (], while we duped fo the field, an enegy of /. Wheeas again, in ou appoah, the hange does not take plae in the suounding at all, but inside the eleton. We ae well awae that ou appoah is to alte the Loentz foe law. Fo one thing, as we will hae to fae below, we will hae to alte the lassial eleti foe te / witten fo the soue hage at est, and the test hage e, egadless whethe this latte hage is at est o in otion. Thus obiously, we ust be expeted to ay out the neessay disussion about the hange, we ae about to bing to the lassial eletodynais. Futheoe, we well ealize that suh a disussion ust be oupled with the intodution of an appopiate agneti foe te, next to the eleti foe te, we ae to alte. Fotunately as entioned, Kholetskii et al. filled this gap in thei wok on the pue bound field theoy (Kholetskii et al.,, whih allows us to skip, oe hee the elated effots. We would like to stess indeed that, though though ey diffeent paths, Kholetskii et al. land at exatly the sae esults, as those outlined by the pesent appoah, as fa as the alteation of the lassial quantu eletodynais is onened. Note that the distane between the eleton and the nuleus, when easued by an obsee bound to the eleton, and when easued by the distant obsee, does not point to the sae quantity, but in what follows we will oelook this detail. In othe wods, the est ass deeases (ia Equations ( and (3, as we will elaboate on, a little, below, altes the eti. Fotunately this ay not hae to be detailed fo the deiation we will now, offe. Neetheless, it should be eebeed that, in ode to suessfully ope with the expeiental esults, we should onside woking in the pope fae of efeene of the eleton, o if the eleton oes on a ile, then in the fae of an obsee situated at est in a gien loation on the obit of onen. We will all the latte fae, the loal fae of efeene. Note futhe /( that, Equation (9 sees to allow κ (, also κ( <.

10 688 Int. J. Phys. Si. It is that, as the eleton is quasistatially bought lose and lose to the nuleus, its est ass deeases oe and oe, until it oes to anish at, whih, fo Z, tuns out to be the lassial eleton adius, that is, e /(.8 x -3. efoe poeeding futhe, let us onside the possible easons why the eleton, o any othe siila hage, annot fall down any futhe beyond a speifi distane. Depending on the situation in question, one an find diffeent explanations. Fo instane, the dipole epesenting wate oleule annot go naowe than the distane it delineates, beause, the eletoni stutue of the atos does not allow it; at shote distanes the atos in onsideation, would epel eah othe oe and oe. On the othe hand, if one onsides an eleton falling onto a nuleus, it should be eebeed that the eleti hage of the nuleus taking plae in the expession of the foe exeted by the nuleus onto the eleton, deeases gadually, as the eleton goes beyond the nuleus wall, assuing that it an do so, without getting absobed, et. In this latte ase, obiously, Equation (9 should be efoulated. As we will show below, aoding to ou appoah, the eleton annot fall down into, say a poton, beyond a ange aking the RHS of Eq.(9 anish, beause thee would be no ass left to fuel the endeao. Futheoe, one should eall that Equation (8, along with Equation (9, is neessay iposed by the law of enegy onseation, in the boade sense of the onept of enegy, ebodying the ass and enegy equialene bought by the STR. If one bings quasistatially the eleton up to an obstale, situated at a gien distane R fo the heay nuleus in onsideation, one ust wok against the attation foe. To etun the eleton bak to infinity, one ust funish to the eleton, the aount of enegy equal to the wok one had to spend in ode to get it to R. Thus, when the eleton is bought bak to infinity, it ust be delieed to the eleton, the R enegy ( / d / R. This then lealy eans that the eleton, when sepaated fo the nuleus, elatiistially speaking, weighs oe, and this as uh as / R. In othe wods (beause, as disussed, the nuleus is supposed to be uh too heay as opaed to the eleton, and aodingly, owing to the law of linea oentu onseation, it will stay patially in plae, though, bak o foth, when statially bound, the eleton will expeiene a ass defiit, and this is, as uh as / R. Othewise, the law of enegy onseation would be iolated. Thene Equation (8, along with Equation (9, is as a ust iposed by the law of enegy onseation, in the boade sense of the onept of enegy, ebodying the ass and enegy equialene dawn by the STR. Now suppose that the eleton is engaged in a gien otion aound the nuleus; the otion in question an be oneied as, ade of two steps (Yaan, 4:. ing the eleton quasistatially, fo infinity to a gien loation, on its obit, but keep it still at est.. Delie to the eleton at the gien loation, its otion on the gien obit. The fist step yields a deease in the ass of as delineated by Equation (8 (The fat that the eleton is bought to the loation in onsideation, quasistatially, poides us with the faility of not haing to deal with the adiation poble that would aise othewise. The seond step in onsideation yields the Loentz dilation of ( the est ass, at the loation, so that the ( oeall elatiisti enegy, o the sae, the total elatiisti enegy of the eleton on the gien obit, beoes ( ( ( (Oeall elatiisti enegy of the bound eleton on the gien obit Whee, is the agnitude of the loal tangential eloity of the eleton at. The eloity is not to be onfused with an eentual eloity, the ato would delineate, when bought to a unifo tanslational otion. If so then, the oeall ( elatiisti ass would eidently beoe ( / / ( The total enegy of the eleton in obit [that is, ] ust eain onstant, so that fo the otion of the objet in a gien obit, one finally has (. (a (total enegy witten by the autho, fo the eleton in otion aound the nuleus.

11 Yaan 689 This elationship is in fat the integal fo of ou geneal equation of otion, gien below. One an notie that Equation (a is diffeent fo what one would wite lassially, that is, ( Constant: Inoet (b (total enegy one would wite lassially, fo the eleton in otion aound the nuleus. The diffeene in the elation gien in Equation (a is base on ou appoah and that gien in Equation (b stes fo the fat that Equation (b delineates a iolation of onseation of enegy. What is inoet in Equation (b? Altenatiely, what, aoding to ou appoah, exatly does iolate the law of enegy onseation, in this equation? It is that, if we follow ou shee ade of two onseutie steps we faed ight aboe, in ode to bing the eleton into its obital otion aound the poton; one we aied at, quasistatially fo infinity, what is bought into the obital otion, is not the est ass of the eleton at infinity, but the est ass ( ( Z e (Equations (8 and (9; this is the est ass of the eleton at infinity, deeased as uh as the stati binding enegy oing into play, at the loation on the obit. Again, in ou appoah, the hange, about the bound patile, does not take plae outside of the patile, in the so-alled field, that suounds it, but on the ontay in the intenal dynais of the patile, at hand. Thene, what is to be aounted fo, along with the Loentz onstant oing to be assoiated with the patile now bought into otion, at the gien loation (is not the est ass of the eleton at infinity is to be the deeased est ass ( ( Z e /( (in one single piee. And this is exatly what we wote along with Equation (-a. Haing failed to do this, Eq.(-b eains, unfotunately inoet. It it then fails to wholly satisfy the law of elatiisti enegy onseation, as explained, although fo alost all of us, it ight still bea a etain onsisteny. Suh an ipession, ost likely stes fo the fat that, we ae soewhat onditioned to beliee that Coulob Foe / eigning between the hages and e, is alid, whethe e is at est o in otion. In effet, one an show that, Equation (-b is alid, if Coulob foe eigning between the hages and e, holds in the ase whee is at est, but e ay be in otion. With the way we hae set up ou own appoah, we ae onfident that Coulob s foe / holds fo both /( and e, stitly at est, and this, is a equieent of the STR, as we hae disussed aboe. ased on suh an ouene, we do not a pioi know that, it still holds, if is at est, but e is oing, and one Equation (-a is set, we will see that Coulob foe does not eain the sae at all, if e is in otion with espet to. The onfidene ost of us hae, with espet to Equation (-b, stes futhe fo the belief that, the enegy deease aising within the faewok of Equations (8 and (9, that is, that delineated by the ( ( Z e /( equation, though binding, is due to a loss of an aount of enegy as uh as / in the suounding field. At that, it is still the est ass of the eleton, at infinity, whih is bought in otion at, in the gien field. Wheeas as speified in ou appoah aboe that the hange does not take plae in the suounding, but inside the eleton itself, we will futhe elaboate on this below. If indeed so, the estitution of the istake in question eidently, is to alte so ey any elated deiations. This ay be unfotunate, but that is, the way it is. To be peise, Equation (-a is witten, with espet to an obsee at est, situated on a gien loation, say whee we delieed its otion to the eleton on the obit. Conesely, Equation (b assues that the total elatiisti enegy is oposed as follows: the est elatiisti enegy (that is, the est ass ultiplied by + the elatiisiti kineti enegy + the potential enegy. In addition, what is inoet with this? To opose the total elatiisti enegy, that way, is what we all leaned aleady in high shool, and that is what ost of us keep on teahing. In ou appoah, on the othe hand, we do not ake use of, o efe to the onept of potential enegy. To us, it is a question of the stati binding enegy, and this uh of enegy is daped fo the est enegy (o, taking unity, the sae, est ass of the eleton, of the eleton, when this is bought quasistatially at a distane to the nuleus. In othe wods, the est enegy of the statially bound eleton, is deeased as uh as the stati binding enegy (f. the fist step we hae onsideed, in witing Equation (, and it is the eaining est enegy of the eleton, whih is dilated by the Loentz fato, while we delie to it its otion on the obit (f. the seond step we hae onsideed in witing Equation (. The esult we aied at, is not any diffeent than that we established in egads to the total elatiisti enegy of a H ato in the oleule of HTe, set to a otational otion aound Te. Thus, this enegy is ( H H - E H Te // / Rot Rot E Te - H /!, and not Anothe eason why Equation (b diffes fo Equation (a is that the foe assues that Coulob foe holds between a stati soue hage (the nuleus and a oing test hage (the eleton. Howee, ou

12 69 Int. J. Phys. Si. appoah onsides that Coulob s Law eigns in between only two stati hages, whih in fat, as we hae shown, tuns to be a equieent iposed by the STR. Hene, Coulob s Law does not hold in between the poton (assued at est, thoughout and the oing eleton (the way it holds, between the poton and the eleton at est. In shot, one annot ipose Coulob foe between a stati hage and a oing hage. It is to be noted that, ou onlusion as to Coulob s Law eigns in between, only two stati hages does not, in any way, tell us how the law of foe would look, if one of the hages oes. This is the uial point. In othe wods, as will be speified below, Equation (b would be alid, if Coulob s law wee alid between the poton and the oing eleton, the way it is witten fo the poton and the eleton, both at est. Howee, it is not, and Equation (b is only an appoxiation. This dislosue too, is to alte ey any elated deiations (Kholetskii et al.,, but that is the way it looks. One way o the othe, it is that, the ass of the bound eleton, is not the sae as the ass of the fee eleton, and as tiial as it ay look at this stage, this is what essentially had been oelooked thoughout the past entuy. Thus, although Equation (a looks staightfowad, to ou eolletion, it happens to be new. The way we wite it indues the need of elaboating on the onept of field, whih we aoid; this quest will be detail below. We show elsewhee that Equation (a futheoe onstitutes the basis of a elatiisti quantu ehanial desiption, well equialent to that of Dia, if geaed alike, yet established in an inopaably easie way (Yaan and Rozano, 6. We hae to stess that the appoah in question is in full haony with all the existing quantu eletodynaial data; in fat it sheds light on the sall but still easuable disepanies between easueent and the standad appoah (Kholetskii et al.,. The diffeentiation of Equation (-a leads to: d d (a (diffeential fo of Equation (a, equialent to the equation of otion One an tansfo Equation (a into a eto equation; the RHS is aodingly tansfoed into the aeleation (eto of the eleton on the obit. Thus, ealling that the LHS of Equation ( that is, one an wite: (, is onstant, d (t ( (b dt (etoial equation witten based on Equation (a, o the sae with equation of otion witten by the autho ia the law of onseation enegy, extended to oe the elatiisti ass and enegy equialene. Hee, is the adial eto of agnitude, dieted outwad, and is the eloity eto of the eleton, at t tie ; note that d and lie in opposite dietions; it is ipotant to eall that this equation is witten in the loal fae of efeene. Note that the aboe equation is in full onfoity with what is funished by the PFT deeloped by Kholetsii et al, if the oodinates oe into play, and ae tansfoed into those easued in the fae of the distant obsee. Fo a sall Z, a sall, the obit would be as ustoay elliptial; othewise it is open; in othe wods, the peihelion of it shall be poess thoughout the otion. Equation (-b has anyway the sae elationship as that poposed by oh, exept that the Coulob foe intensity is now deeased by the fato /, siila to what is epiially, but appoxiately poposed by Webe, by the end of nineteen entuy (Webe, 848; Webe, ; Phipps, 99, 99. Along this line, one an onsult elatiely eent atiles (Wesley, 99; Wesley, 99. Note that, a ealisti intepetation of Equation (b should onsist of onsideing the fato /, at the denoinato of the RHS of this equation. Then, it is as if the lassial foe now auses a geate equialent oentu hange ate. What we do is in no way in onflit with quantu. Quite on the ontay, though ou appoah soon we will land at the de oglie elationship, whih is the basis of quantu ehanis. At this stage, it sees useful to daw the following table displaying the diffeenes between ou appoah and the standad appoah. Disussion onening the total dynai enegy we poposed fo the eleton Sine Equations (b and (a signifiantly diffe fo those deied fo the onentional appoah, it is woth noting and disussing soe ipotant issues elated with the diffeenes. One appaent diffeene is the kineti enegy aquied by an eleton feely falling onto a poton at est. Consideing the eleton of ass easued in epty spae, let be the eleton eloity at, and the