Limitation of Applicability of Einstein s. Energy-Momentum Relationship

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1 Limitatio of Applicability of Eistei s Eergy-Mometum Relatioship Koshu Suto Koshu_suto19@mbr.ifty.com Abstract Whe a particle moves through macroscopic space, for a isolated system, as its velocity icreases, the kietic eergy ad hece total eergy of the particle will icrease. However, accordig to classical quatum theory, whe the mometum ad kietic eergy of a electro iside a hydroge atom icreases, total eergy decreases. From this truth, it is evidet that the equatio E =c p +E for Eistei s eergy-mometum relatioship does ot hold true iside a hydroge atom. I this paper, we will draw the followig relatioship: (E +E ) +c p = E, (=1,,, E < ). Key words: Special Theory of Relativity, Eistei s eergy-mometum relatioship, Klei-Gordo equatio, Dirac equatio. PACS codes: 3.3.+p, 3.65.Sq 1. Itroductio Geerally, the physics theory that best describes the behavior of electros iside atoms is thought to be quatum mechaics. This paper does ot disagree with this theory, but further poses the questio of whether Eistei s eergy-mometum relatioship holds true eve i the space iside atoms, ad further attempts to aswer to this questio. (See Appedix A) This questio should have bee first posed ad aswered whe Eistei aouced the abovemetioed relatioship that is, whe quatum mechaics was still 1

2 icomplete (aroud the 19s). Based o this backgroud, we shall cosider the argumet set forth i this paper, which does ot rely upo complete quatum mechaics, as valid. Oe of the importat relatioships i the Special Theory of Relativity (STR) is as follows. E = c p + E. E is the total eergy of a object or a particle, ad E is the rest mass eergy m c. The followig equatio is presumed to be true whe derivig Eq. (1.1) [1]. de vdp. = () I classical mechaics, the icrease of kietic eergy correspods to the work doe by exteral forces, ad we have: dk = Fdx (3a) dp = dx (3b) dt = vdp. (3c) Also, i this situatio, the particle s total eergy ad kietic eergy K icrease, but the icreases are equal. That is, de = dk. (4) Eq. (1.) ca be subsequetly derived from Eq. (1.3c) ad Eq. (1.4).. Derived relatioship betwee eergy ad mometum whe cosiderig potetial eergy However, i the case of electros iside a atom, we must cosider Eq. (B.4) (See Appedix B) I the cetral force field iside a hydroge atom, the amout of reduced potetial eergy ca be thought to be equivalet to the sum of icrease i the kietic eergy of a electro ad the eergy of photos emitted by the electro. (See Appedix C) First we assume that the reduced amout of potetial eergy of a hydroge atom is -ΔV(r). (whe ΔV(r)< ) Accordig to the eergy-mometum law of coservatio, the icreased amout of kietic eergy of a electro ΔK is equal to the emitted photoic eergy sum of the two is -ΔV(r). Thus, the followig relatioship is true. Δ V() r = Δ K + h ω. h ω, ad the

3 ΔV( r) = ΔK. Here, half of the reductio i potetial eergy is used i the form of work to icrease the kietic eergy of the electro. Because the other half is emitted outside the atom as photoic eergy, total eergy decreases. Meawhile, based o Eq. (B.6), ΔV(r)/ is equal to the reductio i total eergy. Namely, ΔV() r =ΔE. (whe ΔE< ) Also, as evidet from Eq. (B.5), the followig relatioship ca be derived from Eq. (.) ad Eq. (.3). de = dk. (4) Whe work is performed agaist the electro iside a hydroge atom ad the kietic eergy of the electro icreases, total eergy decreases. The followig relatioship ca be subsequetly derived from Eq. (1.3c) ad Eq. (.4). de = vdp. (5) From this fact, we must revise Eistei s relatioship for the space iside a hydroge atom (1.1). () (3) 3. Relatioship betwee eergy ad mometum of a electro iside a hydroge atom Referrig to a STR textbook, we derive the eergy- mometum relatioship of a electro iside a hydroge atom []. I classical mechaics, p m =. v Ad, i STR, E m =. () c If, further, we suppose that Eq. (3.) describes a uiversal equivalece of eergy ad iertial mass, we ca combie Eqs. (3.1) ad (3.) ito a sigle statemet: c p E =. (3) v Next, by multiplyig the left ad right sides of Eqs. (.5) ad (3.3), we obtai: 3

4 EdE = c pdp. (4) We itegrate this: E = c p + cost. (5) 4. Discussio A. Determiatio of the costat of itegratio. As show by Eq. (1.1), a electro at rest has rest mass eergy E. Similarly, whe a electro at rest a ifiite distace from a hydroge atom is absorbed ito a atom, the origiatio eergy ca be assumed to be E. The costat of itegratio Eq. (3.5) should ormally determied through experimetatio. However, from the aalogy of Eq. (1.1) of this discussio, the costat of itegratio Eq. (3.5) ca be assumed to be E. Thus, E + c p = E. B. Total eergy of a electro as defied from a absolute viewpoit. Referrig to classical quatum theory ad Eq. (B.5c), the relatioship betwee the total eergy ad kietic eergy of a electro iside a hydroge atom is: E me 1 e 4πε = h (a) E1 = (b) K, ( 1,,. 1 = = ) (c) Here, is a pricipal quatum umber. I this case, the total eergy of the electro has a egative value. Thus, the total eergy of Eq. (4.a) is ot a value obtaied from a absolute measuremet. I classical mechaics, we emphasize the differece i eergy, ot the absolute eergy. However, i order to derive the eergy-mometum relatioship established iside a atom, we must defie the absolute quatity of total eergy of the electro. Fortuately, E of Eq. (C.4) defies a absolute quatity, which icludes the electro s rest mass eergy. Therefore, a defiitio of Eq. (C.4) is a importat guidelie for total eergy as defied i this paper. Accordig to existig theory, the total eergy of a electro is cosidered to be zero 4

5 whe the electro is separated from the atomic ucleus by a distace of ifiity ad remais at rest i that locatio. The total eergy of Eq. (4.a) is the value obtaied from this perspective. However, eve if we place a electro at rest a ifiite distace from its ucleus, the absolute eergy of the electro is fudametally ot zero. Accordig to Eistei, a electro i this state should have rest mass eergy E. The electro tries to eter the regio of the hydroge atom. Durig this time, whe the electro trasits to a lower eergy state ad kietic eergy icreases, a amout of eergy equalig the icreased kietic eergy is released outside the atom. From this fact, Eq. (C.4), ad Eq. (4.c), i this paper, total eergy i absolute terms, E ab, for a electro iside a hydroge atom is defied as below. E = E + E (whe, =1,,, E < ). (3), ab, Here, E ab, is the total eergy as defied i absolute terms whe the pricipal quatum umber is. This defiitio ca be used to rewrite Eq.(4.1) as: ( ) E E c p E + + = (whe, =1,,, E < ). (4), Eq. (4.3) is a o-relativistic equatio, although substitutig this equatio for oe that is relativistic (4.1) raises doubts cocerig the mixture of relativistic ad o-relativistic equatios. However, Eq. (1.1) is ormally cosidered a relativistic equatio, ad ca eve actually be derived without some kid of relativistic request beig required. This is the most geeral equatio that ca be applied to particles movig at o-relativistic speeds. However, whe describig those movig at o-relativistic speeds, sice the approximatio E(v) E +(1/)(E /c )v is substituted, thigs add up eve i the absece of Eq. (1.1). Also, i the case of Eq. (4.1), the same logic is materialized. Thus, from Eqs. (4.1) ad (4.3), we obtai Eq. (4.4). Eq. (4.4) is Eq. (4.1), which icludes the pricipal quatum umber. This equatio represets the relatioship betwee the eergy ad mometum of a electro i a system i which the eergy level is degeeratig. 5. New Relatioal Expressio Quatizatio This paper has fulfilled its objective i the precedig chapter, but for future expressio, we shall attempt to quatize the ewly obtaied relatioship (4.1). 5

6 The fact that whe we perform quatizatio for Eq. (1.1), E i h, p ih. t we obtai the followig Klei-Gordo equatio is evidet. h = ψ h c m ψ t x1 x x3 cψ. () This equatio described the wave fuctio i relativistic terms, but this iterpretatio was icosistet with the iterpretatio accordig to the more commoly used Schrödiger equatio. Dirac surmised that a correct equatio to resolve this shortcomig must take the followig form [3]. ih i c mc t x x x ψ= h α1 + α + α3 + β ψ. (3) 1 3 The, because this ew equatio must satisfy the Klei-Gordo equatio, Dirac thought that all that was left was to determie the ukow coefficiets α ad β. Now, whe we perform quatizatio o Eq. (4.1), we obtai the followig. 4 h ψ= h c + + ψ+ m cψ. (4) t x1 x x3 Extractig oly the operator from Eq. (5.3), ad makig a equatio by squarig both sides, we obtai h ψ= c h α1 + α + α3 t x1 x x 3 ihc α1 + α + α3 βmc x1 x x3 β h α α α β 4 ψ. (5) mc i c m c x1 x x3 Sice the left side of this equatio is the same as the Klei-Gordo equatio ad Eq. (5.4), the right side should fially be the same as the right side of Eq. (5.4). Next, expadig the right side of Eq. (5.5), we obtai the followig. h = h + + h + ψ c α1 α α 3 c ( αα 1 αα 1) t x1 x x3 x1 x i 6

7 h c + + ( αα αα ) h c ( αα αα ) x x3 x3 x1 ihmc + + ( αβ βα) ihmc ( αβ βα ) x1 x 3 i h mc ( αβ 3 + βα 3) + m c x β 4 ψ. 3 (6) I order to make Eqs. (5.4) ad (5.6) the same, the coefficiets 4 4 matrix satisfyig the followig coditios. α i ad β must be a αi = 1 αα i j + αα j i =, αβ i + βα i = β = 1 i, j = 1,,3( i j). (7) The solutio which satisfies these coditios ad is a clea combiatio is as follows. 1 i 1 i α1 = α = 1 i 1 i α3 = β= (8) Dirac, however, obtaied the followig for coefficiets α i ad β. α α 1 i 1 i = α = 1 i 1 i = β= (9) ψ from the Klei-Gordo equatio was a sigle compoet, but i order to develop a equatio that icludes this sort of matrix, the wave guide fuctio ψ itself would 7

8 eed to have four compoets. Namely, ψ ψ1 ψ. = ψ 3 ψ 4 The equatio obtaied i this paper, as i the Dirac equatio, also is a equatio i which the 4 coditios are etagled. However, the goal of this paper is ot to discuss the sigificace of these coditios, ad so that discussio will ot be take up here. 6. Coclusio A. The result we obtaied differs from Eistei s eergy-mometum relatioship. I macroscopic space, we obtai Eq. (1.1): E = c p + E. However, i the space iside a hydroge atom, we fid that Eq. (4.4) holds true: ( ) E + E + c p = E, ( = 1,, ). There exists a limitatio to the applicability of Eistei s eergy-mometum relatioship. B. I this paper we obtaied differet values (5.8) for the coefficiets of Dirac s equatio. However, these are ot iteded to disaffirm Dirac s equatio, but istead equatios i this paper with these discovered coefficiets are thought to be aother form of Dirac s equatio. Ackowledgemet The Appedix B was borrowed ad traslated from the Japaese laguage physics textbook of Dr. H. Ezawa s. I wish to express my gratitude to Dr. H. Ezawa. Appedix A Traditioally, Eq. (1.1), or Eistei s eergy-mometum relatioship, was thought to hold true eve withi a atom ad was icluded i quatum mechaics theory. This was show by quatizig Eq. (1.1) to derive the Klei-Gordo equatio from ad subsequetly derivig the Dirac equatio from this equatio. 8

9 Appedix B Let us review the eergy of a electro iside a hydroge atom. Let us suppose that the atomic ucleus is at rest because it is heavy, ad cosider the situatio where a electro (electric charge e, mass m) is orbitig at speed v alog a orbit (radius r) with the atomic ucleus as its ceter. A equatio describig the motio is as follows: mv e =. r 4πε r From this, we obtai: mv 1 e =. () 4πε r Meawhile, the potetial eergy of the electro is: e V( r) =. (3) 4πε r Sice the right side of Eq. (B.) is 1/ times the potetial eergy, Eq. (B.) idicates: mv = V( r). (4) Therefore, the total electro eergy: mv E = + V( r) (5a) mv = (5b) = K, (5c) is equal to the value whe expressed as kietic eergy. Also, the total eergy of the electro is equal to half its potetial eergy. V( r) E =. (6) Appedix C Gasiorowicz discusses the relativistic aalog of Schrödiger for a boud (scalar) electro iside a hydroge atom, which does iclude the rest mass eergy of the electro i a attractive, cetral potetial [4]. This equatio is E Ze 1 mc, c + 4πε c r = ψ ψ+ ψ h h h 9

10 which is the operator versio of Eq. (1.1) whe a potetial is icluded, ( ) E V = c p + E. () The solutio by solvig for this Eq. (C.1) did ot agree with the actual eergy level of the hydroge atom. The reaso proposed is that electros are 1/ spi particles ad do ot follow the Klei-Gordo equatio. However, as a remaiig problem, the left side of Eq. (C.) is as follows. E V = ( K + V) V (3a) = K. (3b) Thus, K >E, or (p /m) >( m c ), but this kid of iequality should ormally ot be possible. e Here, let us surmise that E of Eq. (C.) is defied ot as the E of Eq. (B.5c) but istead as: E = E K. (4) By substitutig this E ito Eq. (C.) ad cosiderig the relatio to Eq. (B.4), we obtai: ( ) E + K = c p + E (5). This equatio is idetical to Eistei s relatio. I the ed, total eergy E of Eq. (C.) is the eergy as defied by Eq. (C.4). E of Eq. (C.) icludes the electro s rest mass eergy ad is defied o a absolute scale. This is strog evidece to validate Eq. (4.3) as has bee ewly defied i this paper. Refereces [1] A.P.Frech, Special Relativity: THE M.I.T INTRODUCTORY PHYSICS SERIES, W.W.NORTON&COMPANY, New York. Lodo, 1 (1968). [] A.P.Frech, Special Relativity: THE M.I.T INTRODUCTORY PHYSICS SERIES, W.W.NORTON&COMPANY, New York. Lodo, (1968). [3] P.A.M.Dirac, DERECTIONS IN PHYSICS (Joh Wiley&Sos, Ic., New York). [4] S. Gasiorowicz, Quatum Physics, Wiley Iteratioal Editio, 144 (3). 1