Interaction of the Electromagnetic Radiation Quantum and Material Particle in a Vector - Potential Space

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1 Iteratioal Joural of High Eergy Physis 7; 4(4: doi:.648j.ijhep.744. ISSN: (Prit; ISSN: (Olie Methodology Artile Iteratio of the Eletromageti Radiatio Quatum ad Material Partile i a Vetor - Potetial Spae Adrey Nikolaevih Volobuev, *, Eugee Sergeevih Petrov, Paul Kostatiovih Kuzetsov Departmet of Medial ad Biologial Physis, Samara State Medial Uiversity, Samara, Russia Departmet of Eletri - Drive, Samara State Tehial Uiversity, Samara, Russia address: volobuev47@yadex.ru (A. N. Volobuev * Correspodig author To ite this artile: Adrey Nikolaevih Volobuev, Eugee Sergeevih Petrov, Paul Kostatiovih Kuzetsov. Iteratio of the Eletromageti Radiatio Quatum ad Material Partile i a Vetor - Potetial Spae. Iteratioal Joural of High Eergy Physis. Vol. 4, No. 4, 7, pp doi:.648j.ijhep.744. Reeived: September, 7; Aepted: September 5, 7; Published: Otober 9, 7 Abstrat: Iteratio of a photo ad material partile (eletro ad atom is osidered. It is show that this proess eeds to be desribed i the uiform referee system movig with light veloity i spae of a vetor - potetial. O the basis of the Noether s theorem it is show that i a vetor - potetial spae the volumetri desity of photo eergy, its veloity ad a rig urret desity of a material partile are oserved. O the basis of the Shrodiger s equatio solvig for a photo ad eletro ooperatig i vetor - potetial spae it is show the eletro durig iteratio should represet the quatum osillator with a disrete set eergies. Eletro flutuatios or Dira s eletro "jitter" are realized with a light veloity. The problem of a eletro mageti momet (spi ourree i a vetor - potetial spae is osidered. Coditios of the atom urrets quatizatio i vetor - potetial spae, ad also the Heiseberg s uertaity priiple i this spae are submitted. The Lamb s frequey shift i vetor - potetial spae is foud. The multiphoto system i a vetor - potetial spae is ivestigated. Keywords: A Photo, A Eletro, Dira s Eletro "Jitter", Noliear Shrodiger s Equatio, Quatum Osillator, Mageti Momet, Spi, Eletroi ad Atomi Rig Currets. Itrodutio At alulatio of a photo ad material partile iteratio there is a lie of problems. I partiular, iteratio of radiatio ad free eletros a take plae i the various ways. It a be oheret diret reradiatio of a photo o a free eletro or reradiatio of a photo with hage of a diretio of its movemet (Thomso s satterig, ot oheret satterig (Compto s effet that is typial for high eergies quatums: X-ray ad γ radiatio. The basi paradigm of quatum eletrodyamis osists that a photo or quatum a idivisible partile. Cosequee of it is two-stage harater of a photo ad eletro iteratio. First the photo is ompletely absorbed by a eletro, ad the the eletro radiates a photo. I spite of the fat that i reality absorptio of a photo by a eletro, obviously, takes plae, a alulatio of this proess is ioveiet. The law of a impulse preservatio at absorptio of a photo by motioless eletro looks like ħω mv where V there is eletro veloity after photo absorptio, m - its relativisti mass, ω - photo frequey, ħ - Plak's redued ostat, - light veloity i vauum. The law of eergy oservatio i relativisti form a be writte dow as: ħ ω m m m m m V m where m there is eletro rested mass. Havig divided the law of eergy oservatio o the law

2 37 Adrey Nikolaevih Volobuev et al.: Iteratio of the Eletromageti Radiatio Quatum ad Material Partile i a Vetor - Potetial Spae V V of a impulse preservatio we reeive, т.е. V, that is impossible, at least, i Eulidia spae. The reeived result is ot asual. The matter is that the photo exists i the referee system movig with a light veloity. To desribe iteratio of a photo ad eletro it is eessary that the eletro also it was examied i the referee system movig with a light veloity. I it the physial sese of the reeived result will osist. If the eletro is ot examied i the referee system movig with a light veloity that, for example, i proess of the Dira s equatio solvig by the method of perturbatio theory at iteratio of a photo ad eletro there arise poorly proved with the physial poit of view so-alled virtual eletro states []. Aordig to the speial theory of a relativity all partiles at the movemet with veloity V have redutio of the size uder the law [] l l V where l there is the size of a movig partile, l - the size of a partile rested i the give referee system. For a photo if it to osider as a partile [3] movig with a light veloity the legth i Eulidia spae is equal to zero l. But if to examie a photo i the iertial referee system movig with a light veloity i whih it atually is rested it is possible to reeive spatial display of a photo. It has bee made i work [4] where a vetor - potetial spae whih is propagated with a light veloity was used. I the referee system movig with a light veloity i a vetor - potetial spae it is possible to ivestigate various proesses. I partiular, i the give work proesses of a photo ad material partiles (eletro, atom iteratio, ourree of the eletro mageti momet are examied. However, this referee system has the restritios. First of all at it there a ot be a mass, ad a size i Eulidia oordiates. The mass i this referee system aspire to m m ifiity V, ad the size of ay material partile to zero. However, fields, urrets, eergy, frequey, et. i it a be preset, ad it is frequetly eough for the aalysis of some proesses.. Coservative Parameters i a Vetor - Potetial Spae First of all we shall fid out what parameters i a vetor - potetial spae are preserved, i.e. are ivariat. For this purpose we use Noether s theorem i this spae [5]. The volumetri desity of atio i a vetor potetial spae is equal s( qqɺ,, t ldt where q A there is geeralized (for Lagrage s equatio Лагранжа d l l [] the idepedet oordiate oeted to dt ɺ vetor - potetial A, l - Lagragia of systems a photo - eletro [4], t - time. The variatio of atio is size δs( qqɺ,, t δ ldt, ad aordig to a priiple of the least atio (i the ature this priiple is always realized δ l [6]. Let's fid a variatio of the Lagragia: l l l δl δq + δq + δt ɺ. ( ɺ We arry out ifiitesimal displaemet iitially time, ad the the geeralized oordiate. Usig Lagrage s equatio we shall replae the first term i (: ( δq d l l l d l l d l δl δq + δqɺ + δt δq + + δt dt dt dt qɺ qɺ qɺ qɺ d l l d l dl δq + δt δq + δt dt dt qɺ qɺ dt where δt δt there is some ifiitesimal a time displaemet. d ( δt Takig ito aout δ we have dt dl dl d ( δt d ( lδt δt δt + l, ad hee dl d ( lδt δ t. dt dt dt dt dt dt Thus: ( δ d l d l t d l δl δq + δq + lδt dt dt dt qɺ qɺ We arry out ifiitesimal displaemet of the geeralized oordiate: ( (3 δq δq δq + δt δq + qɺ δt (4 Usig (4 the formula (3 a be writte dow as: ( q qɺ d l dq δl δ δt + lδt dt ɺ dt Let's assume that hages of variatios δt δt ad δq δq results a variatio of atio (or Lagragia i zero δ l. Hee we shall reeive that the size (it sometimes ame a Noether s harge is equal: ( ɺ l l l Q δ δt + lδt l δt + δ ost q q q ɺ qɺ qɺ ɺ q Aordig to Noether s theorem the fators at variatios of time δ t ad the geeralized oordiate δq are oserved. (5 (6

3 Iteratioal Joural of High Eergy Physis 7; 4(4: l Aordig to [6] the size l w ɺ qɺ i.e. is equal to the volumetri desity of the photo eergy with the opposite sig. l qɺ The size i.e. is proportioal to the light ɺ 4π 4π veloity i spae of the geeralized oordiates [4]. Thus, the formula (6 a be opied as: Q wδt + δq ost (7 4π Hee, aordig to the Neother s theorem the volumetri desity of a photo eergy w ad its veloity i spae of the geeralized oordiates (or a vetor - potetial are oserved. Not stoppig o a dedutio we shall ote that Neother s theorem allows to fid ad other oservative sizes i a vetor - potetial spae. I partiular the desity of a rig urret j is oserved [7]. 3. Shrodiger s Equatios for a Photo ad Eletro i Spae of Vetor - Potetial Let's osider proess of a photo ad eletro iteratio i the referee system movig with light veloity i a vetor - potetial spae. Appliatio of Dira s equatio to eletro movemet has led, i partiular, to represetatio about so-alled eletro "jitter" [8, 9, ]. Dira i this oasio spoke There is the eletro whih is imagied to us slowly movig atually should make osillatory movemet of very big frequey ad small amplitude whih is summed to uiform movemet observable by us. As a result of this osillatory movemet the eletro veloity is always equaled the light veloity. []. Let's show that presee of eletro "jitter" with light veloity is a eessary oditio of the eletro ad photo iteratio opportuity. Uder the eletro "jitter" we mea ot its mehaial osillatios whih aot be desribed i a vetor - potetial spae, ad osillatios, first of all, its wave futio. I [4] oliear Shrodiger s equatio for wave futio of the photo propagatig i spae of vetor - potetial has bee foud: i ħ + π + l ħ (8 All sizes are writte dow i a vetor - potetial spae. As agaist Shrodiger s equatio i Eulidia spae the equatio (8 is relativisti. Therefore, i a vetor - potetial spae is ot eessity to write dow the separate relativisti equatio suh as Dira s equatio []. I spite of the fat that the equatio (8 is oliear this oliearity takes plae oly i the geeralized oordiate spae, i.e. vetor potetial spae. Noliearity of Shrodiger s equatio (8 is osequee of the geeralized oordiate from parameters oliear depedee, i partiular, from a mageti field [4]. I Eulidia oordiates the proess of eletromageti radiatio quatum propagatio has liear harater. Therefore, for example, i quatum-mehaial systems the liear priiple of superpositio is orret. Let's osider a photo ad eletro iteratio i spae of vetor - potetial. Let wave futio of a photo before iteratio is equal. The solutio of the equatio (8 for wave futio looks like [4]: ( q t с сq exp ħδ + exp exp i δ t 4π (9 ħ ħ where δ there is iitial photo frequey. After iteratio with eletro a wave futio of a photo to beome equal. It a be preseted as the sum of wave futio of ot oheret satterig photo ad some small perturbatio determied by iteratio of a photo ad eletro. We shall assume this iteratio is haraterized by the wave futio, so << : + ( We do ot postulate ay prelimiary properties of eletro. All its properties will be defied by a opportuity of its iteratio with a photo. Havig substituted ( i (8, we shall reeive: ( ( ħ π ħ + l + ( + ( i Let's alloate i ( the Shrodiger s equatio for a seodary photo: i π π iħ + ħ + ħ + ħ + + l + + l + + ( ( ( I oetio with that the seodary photo (as well as primary before iteratio after iteratio exists separately it is possible to write dow: π i ħ + + l ħ (3 i ħ + π ħ + l + ( + + l (4 The equatio (3 for a separate photo has the solutio [4]: ( q t с сq exp ħδ + exp exp i δt 4π (5 ħ ħ

4 39 Adrey Nikolaevih Volobuev et al.: Iteratio of the Eletromageti Radiatio Quatum ad Material Partile i a Vetor - Potetial Spae Where δ there is a frequey of a satterig photo i vetor potetial spae. This frequey a differ from frequey of a primary photo δ aordig to Compto s effet []. The equatio (4 reflets iteratio of the photo ad eletro i a vetor potetial spae. Takig ito aout << we shall trasform the Shrodiger s equatio (4 to a kid: π i ħ + + l ħ (6 Let's ote the Shrodiger s equatio (6 as agaist Shrodiger s equatio for a free photo (3 is liear. Wave futio haraterizes the eletro durig its iteratio with a photo. 4. "Jitter" of Eletro i a Vetor - Potetial Spae The amplitude of wave futio for a seodary photo aordig to (5 is equal: ( q t с exp ħδ + exp ħ Havig substituted (7 i (6 we shall fid: ( q t с iħ + πħ + ħδ + ħ (7 (8 Usig a stadard method, we shall exlude i (8 derivative o time havig preseted []: E iħ (9 Where Е there is full eletro eergy. The equatio (8 will be trasformed to a kid: ( q t с πħ + ħδ + + E ħ Usig a idepedet variable as trasform the equatio ( to a kid: ( t ξ q πħ ( we shall potetial. Thus the referee system also movig with a light veloity is used. The equatio ( allows draw a olusio, eletro ooperatig with a photo, should move with a light veloity. It ofirms Dira s olusio []. Solvig the equatio (, we shall fid harater of this movemet. Let's otie i ( depeds o time. the wave futio ( ξ Let's eter the followig desigatio: с λ δ + + E 8 π ħ ( The equatio ( will be writte dow as similar []: ( λ ξ + ξ (3 The solutio of the equatio (3 is well-kow []. It assumes that eletro ooperatig with a photo it is quatum osillator osillatig with a light veloity. The solutio exists oly at the whole positive values of umber : ξ A exp H ( ξ (4 where is,,,..., ad the size A - the fator determied by a oditio of ormalizatio, H ( ξ ( exp( ξ ( ξ d exp (5 dξ There are Hermit s polyomials. The oditio of ormalizatio has a stadard kid: + dξ (6 Substitutig the formula (4 i oditio (6 we shall fid []: + ( ( A exp ξ H ξ dξ A! π (7 Hee the fator looks a kid:, ad the solutio (4! π A с ξ + δ + + E ξ ħ ( All used physial parameters depedet o a variable ( q t ξ move with a light veloity i spae of vetor - πħ ξ exp H! π ( ξ (8

5 Iteratioal Joural of High Eergy Physis 7; 4(4: Figure. Wave futio for ad 3. O Figure the graph of eletro wave futio is show for ad 3. As a see from the graph the eletro wave futio ooperatig with a photo has osillatig harater. The amplitude of osillatios of wave futio with irease quikly falls. Let's fid values of eergies whih the eletro - osillator a have. For this purpose we shall fid a ow values of parameter λ. Substitutig the formula (8 i the differetial equatio (3 we have: d H dξ ( ξ dh ( ξ ( H ( ξ + λ ξ (9 dξ O the other had Hermit s polyomials (5 are solutios of the equatio []: d H dξ ( ξ dh ( ξ ( ξ ξ + H (3 dξ Comparig the equatios (9 ad (3 we reeive: λ + (3 Takig ito aout a desigatio ( we fid that full eergy of eletro - osillator a have oly disrete values: с E + ħ δ (3 At the lowermost power level at the eergy of eletro is equal: E с ħ δ (33 Usig the formula (33 it is possible to write dow (3 as: E E (34 The formula (34 represets a oditio for hage of eergy at trasitio eletro - osillator, i.e. at its Dira s "jitter", from oe state i aother i spae of vetor - potetial. Eergy of eletroi "jitter" is quatized. Aordig to the eergy oservatio law it is possible to write dow: E E ħδ ħδ ħ δ (35 The formula (35 allows alulate a differee of frequeies of primary ad sattered photos ad hee is aalogue of the formula for hage of frequey i Compto s effet i the vetor - potetial spae []. Due to the fat that Dira s "jitter" of eletro is supposed very big frequey ad small amplitude i the equatio (3 for umerial estimatios it is possible to aept λ >> ξ. I this ase the equatio (3 desribes lassial osillatory proess with the frequey proportioal λ, ad aordig to (8 the amplitude of wave futio of eletro "jitter" is equal ξ exp.! π Let's osider the ratio i.e. the ratio of the wave futios amplitudes of eletro "jitter" ad a photo. This ratio also haraterizes a validity of the assumptio <<. Proeedig from formulas (7 ad (8, with the aout (33, it is possible to fid: ~ exp E! π (36

6 4 Adrey Nikolaevih Volobuev et al.: Iteratio of the Eletromageti Radiatio Quatum ad Material Partile i a Vetor - Potetial Spae I [4] it is show the eergy of zero osillatios for a photo aordig to the equatio (3 it is equal. Assumig eergies of zero osillatios for a photo ad eletro durig their iteratio are equalized we have ~.! π Already for the seod eletro osillator power level the ratio is ~.7. At irease this ratio very quikly to ted to zero. Proeedig from the lead aalysis it is possible to assume also that Dira s "jitter" of eletro is oeted to quatum waves (8 arisig o a eletro surfae at its iteratio with a photo ad propagatig with a light veloity. Frots of these waves are oditioally show o Figure, urves. 5. The Mageti Momet of Eletro i a Vetor Potetial Spae urret j ad the geeralized oordiate q A. We assume that the eletro mageti momet has two diretios that meets θ ad θ 8 (j agaist vetor potetial А of a field, ad j o a vetor potetial of a field А, Figure. O Figure the spherial eletro form is give oditioally sie i spae of a vetor - potetial A the Eulidia oordiates are abset. Therefore to estimate the Eulidia veloity of a eletro surfae rotatio it is iorret. It is aepted a agle θ 8 (the substatiatio is lower i.e. the geeralized oordiate q it is direted agaist of a rig eletroi desity urret j (q ad A by determiatio are direted opposite eah other. The eletro mageti momet µ is always direted agaist its spi S ad hee o a diretio of a eletro mageti field stregth H rota. We shall ote that i spae of a vetor - potetial the stregth of mageti field Н (as a rotor o Eulidia oordiates ad eletro spi S a be show oly oditioally. Let's osider the physial reasos of ourree of the eletro mageti momet. Spi of eletro i.e. it mehaial harateristi - the momet of movemet quatity aot be desribed i spae of vetor - potetial. It is kow eergy of a mageti field ad a urret iteratio i lassial eletrodyamis is expressed by the formula [6]: U ja jq (37 where j there is a urret desity with whih the eletromageti field ooperates. This formula is similar to the formula W eφ for eergy of eletro with a harge е i a eletrostati field with potetial φ. The role of a harge i spae of vetor - potetial plays a urret desity j. The additioal iformatio about a eletro we shall itrodue assumig existee i it of ostat rig urrets. By these urrets it is determied the eletro mageti momet µ. As i spae of vetor - potetial there is o oept of mass i this spae it is possible to idetify eletro oly with desity of a quatum rig urret j, ad the formula (37 represets eletro eergy i a vetor - potetial spae. Let's trasform the equatio (6 usig represetatio (9: Figure. The sheme of the eletro vetor parameters. Thus the equatio (38 will be trasformed to a kid: + ( E ± jq πħ (4 For the solvig of the equatio (4 we shall itrodue a ew variable: η ( E ± jq π j ħ Passig to a variable η we shall fid: 3 (4 + ( E + l πħ (38 + η η (4 The equatio (38 is statioary, i.e. ot depedet o time. Let's itrodue ito the equatio (38 the eergy eletro mageti momet (or rig urrets supposig: l jq jqosθ ± jq (39 where θ there is a agle betwee a diretio of a eletro The solutio of the equatio (4 looks like []: where ( η ( η ( η Φ (43 A 3 u Φ os + uη du π 3 so-alled Airy s

7 Iteratioal Joural of High Eergy Physis 7; 4(4: futio. Normalizig wave futio ( η by a rule of the otiuous spetrum futios ormalizatio []: where δ ( η η + ( ( ( η η dq δ η η (44 there is delta - futio we fid: A 5 π j ħ 3 (45 Takig ito aout i a vetor - potetial spae full eletro eergy atually is eergy of its rig urrets we aept for alulatio E ± jq ± jq. O Figure 3 the graph of Airy s futio plotted at a sig or mius i the formula (4 i.e. at ( exp jq θ 8 is show. State exp ( jq or θ there is ustable sie at irease q a expoet quikly ireases. This state is ilikely beause a diretio of eletroi urret desity j ad a diretio of a vetor - potetial A (hee, ad the geeralized oordiate q are determied i statioary oditios by a diretio of a uiform vetor of mageti field H, Figure. Figure 3. The graph of Airy s futio determiig the wave futio of the eletro mageti momet. of As a see from the graph the wave futio ( η the mageti momet i a vetor - potetial spae has the osillatory harater. Apparetly it also reflets ourree of osillatios (Dira s "jitter" i a eletro surfae. Due to quatum waves o the eletro surfae o its surfae there are quatum urrets, ad hee the eletro mageti momet (ad also its spi i Eulidia spae. 6. Iteratio of a Photo ad Atom The formula (34 a be osidered as the first Bohr s oditio i spae of vetor - potetial for hage of eergy at trasitio of atom from oe state i aother. As agaist the Bohr s oditio for Eulidia spaes the eergies of atom states differ o a iteger. The seod Bohr s oditio a be reeived oly usig some additioal iformatio o atom. For a fidig of this oditio the formula (34 we shall write dow as: Ei (46 i E where Ei there is eergy of trasitio from i-th power level o i power level. The additioal iformatio o atom we shall eter assumig existee i it ostat rig urrets whih eergies i a vetor potetial spae look like (37. For i-th atom urret aordig to (37 we have: E j q (47 i i i Substitutig (47 i (46 we shall fid the itegrated sum: ji qi (48 i

8 43 Adrey Nikolaevih Volobuev et al.: Iteratio of the Eletromageti Radiatio Quatum ad Material Partile i a Vetor - Potetial Spae Passig to itegral o a losed urret yle (trajetory we shall fid the seod Bohr s oditio i Sommerfeld s form []: q de r e exp qdq π π ħ (5 jdq (49 Let's otie the role of a eletro impulse i a vetor - potetial spae plays the urret desity j. Meaig the give aalogy it is possible to write dow the Heiseberg s uertaity priiple i spae of vetor - potetial: j q (5 where j there is uertaity of a urret desity, q - uertaity of the geeralized oordiate (vetor- potetial. The atom eergy at the lowermost power level at quatum umber a be foud uder the formula (33 с E ħ δ. Near to atom there is a proess of self-atio of a vetor - potetial owig to what there is a uertaity of eergy E. Ourree of this uertaity i a vetor - potetial spae there is similarly to Lamb s effet i the Eulidia spae []. Let's estimate uertaity of eergy E. Eergy of proess of a mageti field self-atio i lassial eletrodyamis is determied the term a kid e A r ea i atom Hamiltoia, where e there is a m eletro harge, m - its mass, r e - so-alled lassial eletro radius []. Let's assume o a pulsatio of the vetor - potetial da arise ear to atom has effet the field with a vetor - potetial A. Cosequetly the eergy of iteratio of a field pulsatio ad the itself field looks like r e AdA. Passig i a vetor - potetial spae we have r qdq, where r e - i this ase e the ostat whih has bee writte dow i this spae. Ourree of a field pulsatio ear to atom it is a radom the proess iitiated by atom. Therefore, the elemet of eergy of self-atio de eeds to be writte dow as: de r qdq (5 e where the wave futio is determied by the formula (8 attributed to atom. Takig ito aout H [] i the oditioal time momet t we fid exp. π 4q π ħ Hee, the formula (5 looks like: The full eergy of self-atio of a field ear to atom is itegratio (5: q e exp e π ħ π ħ de r qdq r (53 Thus the full eergy of a field ear to atom at a zero power level is equal: с r e ( с E + de ħδ + ħ ħ δ + δ (54 where the size δ r e ħ there is Lamb s shift of frequey i a vetor - potetial spae. 7. Multiphoto System i a Vetor - Potetial Spae To fid the wave futio for multiphoto system i a vetor - potetial spae ioveietly sie it is eessary to kow a multisolito solutio of the Shrodiger s equatio (8. Therefore we shall be limited to fidig of the set photos eergy average value i a vetor - potetial spae. Average value of the system photos eergy a be alulated uder the formula similar []: ε ε + ε ε (55 where εi ħ δi there is eergy i-th photo i system, - probability of that photo has eergy ε i. The amplitude of a photo wave futio aordig to (7 is equal: i с ( q i exp i exp t ħδ + Bexp ( ħδ i (56 ħ ( q t с where B exp + exp. ħ We substitute the formula (56 i (55. Usig δi iδ where δ there is miimal frequey of the photo system we shall fid average value of the photo system eergy o oordiate q ost at the momet of time t:

9 Iteratioal Joural of High Eergy Physis 7; 4(4: B d ħδ ( exp iexp ħδ ħδ i δ δ i d ( δ ħ ħ ħ i i B exp ( ħδi exp ( ħiδ i i i ( ħiδ ( i d ( ε d ħδ exp ( iδ d ( δ ħ ħ i d ħδ l exp ( iδ d ( δ ħ ħ i exp Usig the formula of the geometrial progressio sum with a deomiator exp ( ħ δ we shall fid: i ( ħiδ ( ħδ exp exp exp ( ħδ ( ħδ exp (57 (58 Substitutig (58 i (57 we shall fid: d d ε ħδ + l exp ħ δ l exp ħδ d ( ( ( ( ( ħδi d ( ħδ i ( ħ δ ( ħδ ħδ ħδ ( ħ δ ( ħδ ( ħδ ( ħδ ( ħδ exp exp exp ħδ + exp exp exp exp (59 For the formula (59 beomes simpler: ħδ ħδ ε ħδ + exp exp ( ħδ ( ħδ (6 However there is also a essetial differee. I a vetor - potetial spae we do ot oet system of quatums with set of the quatum osillators []. Therefore there is o zero eergy of a photo i the basi oditio ad i the formula (6 is added omposed ħ δ - the miimal eergy of a photo i system. As show i [4] the eergy of a photo (iludig average iludes the eergy of vauum equal. For small eergies it is possible to aept exp( ħδ ħ δ. Hee ε. The lea average eergy of a photo is equal: ħδ εp exp ( ħδ If photos are abset ε p. 8. Colusio (6 To ivestigate iteratio of a photo ad a material partile, for example, eletro it is eessary to osider them i the uiform referee system movig with a light veloity. But the material partile aot be ivestigated i Eulidia spae i the referee system movig with a light veloity. This is opposed by the speial relativisti theory. It is eessary to pass to other spae where existee of a referee system of the ooperatig partiles movig with a light veloity is possible. Oe of suh spaes is the spae of a vetor - potetial. I this spae the volumetri desity of photo eergy, its veloity ad the desity of eletro ad atom rig urrets are exist ad preserved. Cosideratio of proess of a photo ad eletro iteratio i spae of a vetor - potetial shows: a. For a opportuity of a eletro ad photo iteratio the eletro should represet quatum osillator i.e. to "jitter" (as Dira with a disrete set of eergies. b. Dira s "jitter" of a eletro ours with a light veloity i vauum. It is foud the eletro eergy i vauum i spae of a vetor - potetial. The eletro mageti momet (ad hee its spi is determied by eletroi rig urrets. O the basis of depedee of the eletro mageti momet eergy from the geeralized oordiate ad desity of eletroi rig urrets the Shrodiger s equatio for wave futio of the eletro mageti momet (spi is reeived. The solutio of this equatio shows that this wave futio is oeted to Airy s futio, ad has essetially osillatory harater i a vetor potetial spae. The priiple of trasitio to model of a photo ad atom iteratio is show. Values of eergy whih the atom a have are foud. The oditio of a atom urrets quatizatio i a vetor - potetial spae as Sommerfeld s oditio is submitted. Also a priiple of the Heiseberg s uertaity i this spae is submitted. To that Lamb s shift of frequey equal i a vetor - potetial spae it is show. A average eergy of the photos i a vetor - potetial spae is foud. Thus it is show that a hypothesis about a photo as quatum osillator is superfluous.

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