Potential Energy of the Electron in a Hydrogen Atom and a Model of a Virtual Particle Pair Constituting the Vacuum

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1 Applid Physics Rsach; Vol 1, No 4; 18 ISSN ISSN Publishd by Caadia Ct of Scic ad ducatio Pottial gy of th lcto i a Hydog Atom ad a Modl of a Vitual Paticl Pai Costitutig th Vacuum 1 Chudaiji Buddhist Tmpl, Issaki, Japa Koshu Suto 1 Cospodc: Koshu Suto, Chudaiji Buddhist Tmpl, 5-4, Oot-Tow, Issaki, 37-48, Japa Tl: mail: koshu_suto19@mbiftycom Rcivd: July 1, 18 Accptd: July 5, 18 Oli Publishd: July 6, 18 doi:15539/apv14p93 URL: Abstact I a pviously publishd pap, th autho mad som mistaks i calculatig th pottial gy of th lcto i a hydog atom Thos mistaks occud du to applyig a pottial gy fomula with a ctai ag of applicatio i a gio wh it is ot applicabl Thfo, this pap cocts that o by divig a fomula fo pottial gy with o ag of applicatio Th pap also poposs a modl i which a vitual paticl pai pst i th vacuum gio isid a hydog atom simultaously has a photo with positiv gy ad a photo with gativ gy (I this pap, ths photos a calld dak photos) I th stat wh th lativistic gy is zo, th sum of th positiv gy ad gativ gy of th vitual paticl pai bcoms zo Accodig to this modl, this maks it possibl fo th paticls to las photos, ad captu gativ gy Kywods: Hydog Atom, Pottial gy, Modl of a Vitual Paticl Pai, Dak Photo, Dak Matt, Poto Radius 1 Itoductio O of th most impotat latioships i th Spcial Thoy of Rlativity (STR) is as follows: ( ) + c = ( ) p (1) H, is th lativistic gy of a objct o a paticl, ad is th st mass gy Cutly, isti s latioship (1) is usd to dscib th gy ad momtum of paticls i f spac, but fo xplaiig th bhavio of boud lctos isid atoms, opiio has shiftd to quatum haics as pstd by quatios such as th Diac s lativistic wav quatio Fo asos such as ths, th was o sach fo a latioship btw gy ad momtum applicabl to a lcto i th hydog atom Howv, th autho has vtud to tak up this poblm, ad divd th followig latioship (Suto, 11) ( ) p c = 1,, (), + =, H,, is th lativistic gy of th lcto, dscibd with a absolut scal Fom quatios (1) ad () it is vidt that, if a statioay lcto bgis to mov i f spac, o is icopoatd ito a atom, th th gy which svs as th dpatu poit is th st mass gy Cosid th cas wh a lcto cutly statioay i f spac is daw to a poto to fom a hydog atom At this tim, th st mass gy of th lcto dcass Th dcas i st mass gy of th lcto is xpssd as Δ If th gy of th photo lasd wh a lcto is daw ito a hydog atom is tak to b hν, ad th kitic gy acquid by th lcto is tak to b K, th th followig latioship holds Δ + hν + K = (3) Th autho pstd th followig quatio as a quatio idicatig th latioship btw th st mass gy ad pottial gy of th lcto i a hydog atom (Suto, 9) 93

2 apccstog Applid Physics Rsach Vol 1, No 4; 18 V () =Δ (4) Fom quatios (3) ad (4), it is vidt that th followig latioship holds btw pottial gy ad kitic gy 1 () V = K (5) Also, th pottial gy V() of th lcto is assumd to b wh th lcto is at st at a positio ifiitly fa fom th poto, ad thus it bcoms small tha that isid th atom, ad ca b dscibd as follows 1 V() = 4πε (6) Th is a low limit to pottial gy, ad th ag which gy ca assum is as follows H, if Δ < (7) V() is substitutd fo V i quatio (6), th th is = = (8) 4πε H, is th classical lcto adius Fom this, it is vidt that quatio (6) has th followig ag of applicatio (9) Howv, th autho also applid quatio (6) i th ag wh < Thus, i th followig sctio, a fomula fo pottial gy with o ag of applicatio is divd, ad that o is coctd Fomula fo Pottial gy of th lcto with No Rag of Applicatio Th lativistic gy of th lcto fomig a hydog atom ca b appoximatly dfid as follows = + (1a) = + V() + K (1b) 1 () = + V (1c) 1 = Δ (1d) quatio (1a) is a appoximatio is bcaus th total haical gy of a hydog atom divd by Boh is a appoximat valu (A igoous dfiitio of, is giv blow) H, th i quatio (1a) cospods to th dcas i th st mass gy of th lcto, ad cospods to th maiig pat of Th followig costait holds gadig th lativistic gy of th lcto du to quatio (1d) (h, th discussio is limitd to th odiay gy lvls of th lcto) 1 < (11) Th followig fomula ca also b divd fom quatio (1b) V() = m c K, K = = m c (1) quatio (1) is a fomula fo pottial gy with o ag of applicatio To dtmi th pottial gy i all gios withi a hydog atom, quatio (6) alo is ot sufficit, ad th suppot of quatio (1) is dd Icidtally, if quatio () is solvd fo gy, th followig solutios ca b divd (Suto, 14a, Suto, 14b) 1/, =±, =,1,, m c + α (13) H, α is th followig fi-stuctu costat 94

3 apccstog Applid Physics Rsach Vol 1, No 4; 18 α = (14) 4πεc If is substitutd fo i quatio (13),, = (15) H, quatio (13) is dividd ito th followig two quatios, by takig th positiv gy lvls amog th lativistic gy lvls of th lcto fomig a hydog atom to b +,, ad th gativ gy lvls to b, 1/ +, = 1,,, = + α (16) 1/, =, = 1,, + α (17) Icidtally, th vitual paticl pais costitutig th vacuum a fomd fom a vitual lcto ad a vitual posito As will b discussd blow, th gy wh = is thought to b ot th gy of th lcto, but th gy of a vitual lcto, ad thus it is xcludd h ( = is th gy of th vitual paticl pai Howv, th poblm big addssd h is th gy of th lcto, so h, = is gadd as th gy lvl of th vitual lcto) Wh quatio (16) is usd, th omal gy lvls of a hydog atom a as follows 1/ + α =, = 1+ 1, = 1,, (18) Now, if a Taylo xpasio is pfomd o th ight sid of q (18), 1/ 4 4 α α 3α α 3α = 1+ 1 m c Wh this is do, th quatios fo th gis is as follows (19) α m c, = 1,, Icidtally, i th classical quatum thoy of Boh, th gy lvls followig fomula (H th B i stads fo Boh ) B, () B, of a hydog atom a giv by th m 1 α B, 4πε = =, = 1,, (1) Fom this, it is vidt that Boh s gy quatio, quatio (1), is a appoximatio of quatio (18) Th followig compas gis wh = 1 Valu pdicatd by this pap quatio (18): 1 = V (a) Valu pdictd by Boh quatio (1): B,1 = V (b) B, = (c) 1 I quatio (1a), was dfid fom ad, but it is actually coct to dfi fom Icidtally, th lativistic gy of th lcto ca also b witt as follows ad 95

4 apccstog Applid Physics Rsach Vol 1, No 4; , = 4πε (3a) / = 1 (3b) Nxt, quatios (16) ad (3b) a joid with a quals sig That is, / = + α (4) Fom this, th followig quadatic quatio is obtaid If this quatio is solvd fo, + α + α α α 4 + = (5) 1/ α = 1+ 1± 1+ α (6) Wh th Taylo xpasio of quatio (6) is tak, th sult is as follows 4 α 3α ± + 4 (7) α 8 H, th gativ solutio of quatio (7), 4 B α a α + = + (8) Sic covgs to /4, /4 ca b gadd as th adius of th atomic uclus of a hydog atom (i, th poto) H, th thotical valu of th poto adius is: 4 15 = = m (9) Howv, if a attmpt is actually mad to masu th siz of th poto (atomic uclus), th gy of th poto chags Th siz of th poto dpds o th poto s gy, ad thus th masud valu dos ot match with quatio (9) (Radolf, 1; Suto, 14c) I additio, it is possibl to pdict that a difft masumt valu will b obtaid fom a xpimt usig a difft masumt mthod Nxt, wh th valu obtaid by sttig = i quatio (6) is tak to b, H, Stat with = is dfid as follows = (3) Stat : =, = (31) 3 Coctio of Pottial gy of th lcto i a Hydog Atom Th poits wh th autho mad a mistak i th valu of pottial gy of th lcto a 1 to 3 i th followig diagam (s Figu 1) (Suto, 17) Oigially, th pottial gy at 1 to 3 was foud fom quatio (6), but pottial gy i this gio must b foud fom quatio (1) That is, 1 V( ) = (3a) V( ) = 3 (3b) 3 V( ) = 4 (3c) Nxt, th gios i a hydog atom a classifid as follows at th lvl of classical thoy whil takig ito accout quatios (7) to (1) (s Tabl) 96

5 apccstog Applid Physics Rsach Vol 1, No 4; 18 Tabl 1 Rgios ad stats This is Figu 1 mad ito a tabl H, th valu of K was foud fom quatio (1) Rgios ad stats V( ) K Rgio A < + < < m c +, < V < () > K > Stat a Rgio B + < < < + < < K < Stat Rgio C < < 3 - < < < < K Stat c 3 Rgio D < < 4 3 < <, < V( ) < > K > Stat d 4 m c Rgio A is th gio wh th lcto fomig a hydog atom xists Howv, i Rgio B, th is o chag i th pottial gy of th paticl Thfo, what xists i this gio is ot chagd paticls Thus, this pap pdicts that Rgio B is a gio of a vitual paticl pai fomd fom a vitual lcto ad vitual posito Vitual paticl pais a th paticls costitutig th vacuum I this gio, th kitic gy of a vitual paticl pai dcass as th paticl pai appoachs th atomic uclus Howv, i th gios of th lcto wh + < ad /4 < < /3, kitic gy icass as th lcto appoachs th atomic uclus A vitual paticl pai with = xists i Stat Wh this vitual paticl pai absobs of gy, th vitual paticl pai tasitios ito Stat a (At this tim, th gy of th vitual lcto is 1/ th gy of th vitual paticl pai, ad thfo is ) / Also, this pap pdicts that this vitual paticl pai will spaat ito a vitual lcto ad vitual posito i Stat a Rgio C is a gio symmtical with Rgio B i tms of gy Th vitual paticl pais xistig i this gio hav a gativ gy (mass) Rgio D is symmtic with Rgio A i tms of gy lctos i this gio hav gativ gy (mass) i quatio (17) Th autho has alady poitd out that th systm fomd fom a poto ad a lcto with gativ gy is a cadidat fo dak matt, a typ of matt whos tu atu is ukow (Th autho calls lctos with this gativ gy dak lctos, ad photos with gativ gy dak photos) Wh Figu 1 is coctd basd o th abov, th sult is as follows (s Figu ) 97

6 apccstog Applid Physics Rsach Vol 1, No 4; 18 Figu 1 Rlatioship btw gis of th lcto ad vitual lcto pst i a hydog atom, ad thi positios Th gio wh th lcto fomig th hydog atom xists is < /3< < is th gio of th vitual paticl pai costitutig th vacuum, but th gy i this diagam idicats th gy of th vitual lcto (Th gy of th vitual paticl pai is twic th gy of th vitual lcto) Also, /4 < < /3 is th gio of th lcto with gativ gy (mass) This diagam is citd fom aoth pap, but th valus fo pottial gy at 1 to 3 a mistak, ad thus thy will b coctd i this pap Figu I this figu, pottial gy (vtical li) has b asd i th gio wh pottial gy dos ot xist ( /3< < ) Also, as th lcto i Rgio A, ad th dak lcto i Rgio D, appoach th atomic uclus, th kitic gy of th lcto icass Thus K i this gio is show with a dashd li 4 Discussio I th pvious sctio, th pottial gy valu of th lcto was coctd, ad thus th oigial pupos of this pap was achivd Howv, th a still a umb of poits that ca b discussd 1) How dos a vitual paticl pai with = acqui gativ gy? This pap xamis two itptatios Itptatio 1: A vitual paticl pai absobs a dak photo with gativ gy, ad lows its gy lvl Howv, a dak photo has v b obsvd i th atual wold Thfo, this itptatio caot b suppotd Thus, th pvious viw of vitual paticl pais with = is xamid That is, Pvious viw: A vitual paticl pai with = is i a stat wh st mass gy has b compltly cosumd, i, (to us a vulga xpssio) a akd stat uclothd by photo gy Th itptatio of this pap (Itptatio ): A vitual paticl with = simultaously has a photo with positiv gy ad a dak photo with gativ gy If h th positiv photo gy is tak to b P ( < P ) ad th gy of th dak photo is tak to b DP ( DP < ), th = ca b dfid as th stat wh th sum of ad is zo That is, P DP 98

7 apccstog Applid Physics Rsach Vol 1, No 4; 18 = + =, = (33), P DP P DP Icidtally, th dfiitio of th st mass gy of th lcto is is usd, this gy ca b dfid as follows = m c Howv, if th modl h = + = = m c (34) P DP P DP This pap caot pdict th gis of th photo ad dak photo blogig to th vitual paticl pai with = Howv, fo a dak lcto to attai Stat d, th vitual paticl pai with = must simultaously hav a photo ad a dak photo Also, accodig to this modl of th vitual paticl pai, th, i quatio (17) is ot th gy of th dak photo blogig to th dak lcto, cospods to th sum of th gy of th photo blogig to th dak lcto, ad th gy of th dak photo That is, = + =, < (35), P DP P DP P DP ) To stimat th umb of vitual paticl pais pst i th vacuum gio isid a hydog atom, lt us look at tiplt poductio Now, cosid th cas wh a icidt γ-ay has th gy cospodig to th mass of 4 lctos (44 MV) If this is discussd classically, th γ-ay ca cat a lcto ad posito a = / (s Figu 3) Figu 3 Itptatio of this pap gadig tiplt poductio This γ-ay will giv 1 MV of gy to th vitual paticls at = /, ad a lcto-posito pai will b catd ( 1) Wh this γ-ay appoachs clos to th atomic ucla, ad th lcto i th obital aoud th poto absobs this gy, th lcto will b xcitd ad appa i f spac ( ) If multipl vitual paticl pais xist i th = stat, th th is pottially a pobability that two lctos ad two positos a poducd i th pocss i 1 Howv, a phomo of this sot has ot b obsvd v if 1 MV of gy is cosumd i this pai catio, th γ-ay still has th gy of cospodig to th mass of lctos (1 MV) If th γ-ay givs gy to a lcto i th obital a th poto, th lcto will b xcitd ad appa i f spac As a sult, lctos ad 1 posito will appa i f spac 99

8 apccstog Applid Physics Rsach Vol 1, No 4; 18 Howv, if multipl vitual paticl pais xists i th, stat, th i pocss 1, th should also b a pobability of poducig two pais ( lctos ad positos) fom a gy of 44 MV ( 4 ) Howv, quaduplt poductio has v b actually obsvd Thus, this pap pdicts 1 vitual paticl pai i Stat Th is also a pobability that, asid fom a lcto, a sigl vitual lcto ad vitual posito a pst i Rgio A Takig ths poits ito cosidatio, th is a possibility that th pvious dfiitio of th hydog atom is too simpl, ad cosidatio may b cssay That is, Pvious viw: Hydog atom = 1 poto + 1 lcto Modl to b xamid: Hydog atom = 1 poto + 1 lcto + 1 vitual paticl pai (o 1 vitual lcto + 1 vitual posito) H, if o vitual paticl pai is addd, th th modl is applid wh th paticl pai is pst i Rgio B, ad if vitual paticls a addd, th modl is applid wh th vitual paticls a pst i Rgio A 3) If th gy absobd by th vitual paticl pai with = is i th ag < <, th th vitual lcto ad vitual posito spaatd i Stat a a pst tmpoaily i Rag A Now, what is th diffc btw th lcto fomig th hydog atom ad th vitual lcto? It is difficult to discimiat ths paticls fom th pspctiv of gy Howv, th vitual lcto ad vitual posito i Rgio A a likly ot compltly spaatd, ad i a stat of quatum taglmt Thfo, th lcto ad vitual lcto i Rgio A a ot i th sam stat Th is also thought to b a pobability that a vitual posito spaatd fom a vitual lcto at = will appoach th lcto of th hydog atom ad fom a vitual paticl pai If a w vitual paticl pai is fomd h, th maiig vitual lcto will th bhav as th lcto of th hydog atom If this modl is assumd to b coct, th lcto i th hydog atom dos ot dscib a cotiuous tajctoy, ad its motio is discotiuous Also, it is pdictd that th lcto will bhav as though it had movd to aoth locatio istataously (at sup lumial spd) 4) As is also vidt fom Figu, th positio occupid by th lcto ad dak lcto i th hydog atom, ad th gio of gy, a oly a small pat of th whol Th maiig majoity is th gio of th vitual paticl pai ad vitual paticls (vitual lcto ad vitual posito) If v th vitual paticl pai is icludd i th costituts of th hydog atom, th th will b a d to div th gy lvls of th vitual paticl pai Th gy lvls of th lcto ad dak lcto a disct, ad thus basd o commo ss, th gy lvls of th vitual paticl pai a also pdictd to b disct 5 Coclusio 1) I this pap, quatio (1) was usd to coct th valu fo pottial gy i a hydog atom, pviously foud icoctly by th autho As a sult, Figu 1 has b coctd as show i Figu ) Accodig to th modl poposd i this pap, a vitual paticl pai simultaously has a photo with positiv gy P ad a dak photo with gativ gy DP I this cas, th pvious gy is dfid as follows i) If th lativistic gy of a vitual paticl pai is zo, Pvious dfiitio:, = (36) Dfiitio i this pap:, = P + DP =, P = DP (37) ii) Rst mass gy of a lcto Pvious dfiitio: = m c (38) Dfiitio i this pap: = + = = m c (39) P DP P DP iii) gy lvls of a dak lcto with gativ gy Dfiitio i this pap:, = P + DP = P DP, P < DP (4) Icidtally, th xistc of dak photos caot b dictly dmostatd by xpimt, just lik vitual paticl pais Howv, if i th futu it is possibl to dmostat th xistc of gativ gy lvls, ad dak lctos i th hydog atom, th th xistc of dak photos will also b simultaously dmostatd 1

9 apccstog Applid Physics Rsach Vol 1, No 4; 18 Ackowldgmts I would lik to xpss my thaks to th staff at ACN Taslatio Svics fo thi taslatio assistac Also, I wish to xpss my gatitud to M H Shimada fo dawig figus Rfcs Radolf, P (1) Siz of th poto Natu, 466, Suto, K (9) Tu atu of pottial gy of a hydog atom Physics ssays, (), Suto, K (11) A gy-momtum latioship fo a boud lcto isid a hydog atom Physics ssays, 4(), Suto, K (14a) = gy lvl pst i th hydog atom Applid Physics Rsach, 6(5), Suto, K (14b) Pviously ukow ulta-low gy lvl of th hydog atom whos xistc ca b pdictd Applid Physics Rsach, 6(6), Suto, K (14c) Tu Factos Dtmiig th Ratio of Spac Cotactio ad Tim Dilatio Pdictd by th Spcial Thoy of Rlativity Physics ssays, 7(4), Suto, K (17) Pstatio of dak matt cadidats Applid Physics Rsach, 9(1), Copyights Copyight fo this aticl is taid by th autho(s), with fist publicatio ights gatd to th joual This is a op-accss aticl distibutd ud th tms ad coditios of th Cativ Commos Attibutio lics ( 11