The EmissionAbsorption of Energy analyzed by QuantumRelativity. Abstract


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1 The missionabsorption of nergy nlyzed by QuntumReltivity Alfred Bennun* & Néstor Ledesm** Abstrt The uslity horizon llows progressive quntifition, from n initil nk prtile, whih yields its energy s blk body emission within seonds. This primordil spetrum, llows subsequent inorportion of nk prtiles, interting oopertively by inresing photon number n λelongtion within the generted spetr. These two proesses, with individul rtes, whih ould not exeed, do so by integrting n overll exponentil rte, for the strethingout of infltionry spe. The Hwking formultion reltes blk body s rditing photons, t temperture inversely proportionl to its mss. Hene, the thermodynmis of ontour nd oupling for the energy to mss interonversion, llows differentiting the diretion of two symmetri opposite s bsorption vs emission. Hene, the nk vlues hve delimiting role. Aordingly, emissionbsorption ws nlyzed using the instein eqution, evluting the inrement of inertil mss s funtion of kineti energy with momentum p: p, but mximlly restrited for prtile by the nk limit: 4 T p + m γ m. This oneptuliztion, llows differentiting between ontinuum nd quntum proess, by pplying the dilttion oeffiient γ in reltionship to the wve funtion Ψ of prtile, within Shrödinger box (Sbox). Thus, induing prtile hnges by deresing the width or vlue, this llows simulting the inrement of inertil mss by inorportion of kineti energy. The renormliztion of the wve funtion, s dependent of the determintion of probbility density, shows tht the expettion vlues hve physil mening only for nonfrtionl vlues of the γprmeter. Its quntifition llows oniliting wve funtion nd quntum probbility density with restrited reltivity. This implies tht the bsorption beme quntized nd the inrement of inertil mssγ m, ould be evluted by quntum mehnis, whih prevents the γ prmeter to reh infinity. Hene, the generted wve funtion ollpses, when the veloity of the prtile pprohes or its mss rehes the nk limit. The imge sequene of the topogrphi projetion of the unidiretionl wve funtion, suggest tht within the prtile the inorported energy, produe orderly ptterns onforming energy onvetion ells of qusifrtl periodiity. This ptterns, in the diretion of bsorption shows inresing energy levels, deresing the ontour tht ould be relted by the reltionship tht deresing entropy inrese urvtures, until the nk limit. 1
2 If the sequene is exmined in reverse, or emission, lso shows proportionl orreltion between reltivisti mss or potentilenergy nd the urvture of spe. mission inresing entropy, eventully leds to deresing urvture of the prtile, whih is reveled by the tendeny of internl ontour forelines to beome prllel. Sle extrpoltion llows postulting, tht inresing entropy by deresing urvture, llows relting spe expnsion to the universe fltness. Introdution The Sbox, reltes oulomb wlls, generting resonne lous, the ltter expression llows ssigning osilltory vlues to probbility density. As well s desribing wve funtion s flututions of the vlues of energy density in reltionship to its lous, either when propgting s wve (λ) or being onfined s prtile (r λ/π) within the spetime (1,, 3, 4,, 6, 7, 8, 9). The obtined results demonstrte prllelism between the osillting properties of density of probbility nd frequeny. The lst one is restrited by spe to: /λv, this reltionship n lso be pplied to probbility density, whih is mnifesttion of probbility, relted to evolution of the spe dimensions. Prmeteriztion of Reltivisti mss s funtion of nk limit 4 The formul of restrited reltivity theory T p + m γ m desribe the term for bsorption of kineti energy p with linel momentum p, s the inertil mss γ m, 1/ were: dilttion oeffiient γ (1 v / ), v is the prtile veloity nd the veloity of light. Mthemtilly, if the vlue of v, beomes lose to, γ will tend to infinity. However, if the onept tht bsorption of kineti energy results in the inrement of inertil mss the theoretilly limit is the nk energy: η / (9), s result T. Reltivity s well s Quntum Mehnis, estblished the onept of mximum rehble limit for prtile, whih does not llow tht the vlue v ould equl to. If nk limit is epted it llows evluting the thermodynmi evolution of prtile. Results Repling in the instein formul totl energy T by the nk energy limit equivlent, it is obtin the expression: (6,7,8) η 4 ( m v) + ( m) γ m [1] The limit imposed to the formul, does not ontrdit the Reltivisti interprettion of the inrese of inertil mss nd the shortening of the prtile s length, in the diretion of its movement (6, 7, 8).
3 However, if the energy trnsferene does not only ffet one dimension, ould be implied hnge of ngulr frequeny given by the expression ω η, nd therefore shortening of its rdius (8, 10, 11). The eletron ould be experimentlly shown s prtile in rottion. letron mirosopy llows inresing resolution by ontrtion of the dimeter of the prtile. Inorportion of mss bsed on the dulity wveprtile In terms of the hypothesis of De Broglie (1) ll mss m, hs ssoite wvelength λ, by the eqution m v πη / λ m v h / λ. Consequently this reltionship introdue in the expression [1]. 1 st vlution η 4 ( m v) + ( m) γ m η πη v + λ v η Clering v it is obtined: 3 Fulfilling the ondition: λ 4 η > 0 m v η π λ π 3 v π 3 λ + η 4π η > λ > π l The formul shows the reltionship between dimeter of the prtile nd the orresponding wvelength. nd vlution 4 ( m v) + ( m) γ m m v + h v + λ v v Where: h λ It must fulfill the following thing: λ h > 0 h λ > h Numerilly m/s MeV λ > 1 MeV.s λ > m Reltion between liner moment nd resting mss 3
4 η p + 4 m γ m η p 4 4 p m γ m p m m p m γ 1 + γ η η η p m m p m + m 4 γ m 4 p 8 m p Kg m/s p 6. Kg m/s Figure 1. Reltion between mss nd moment. In the yxis, for eh vlue of resting mss, orresponds omplement vlue for the prtile s moment, on the xxis, to reh the nk s dimensions, with moment: p m p Kg p 6. Kg m/s. Drk line: physil vlues; brokenline: mthemtil vlues. Quntum prmeteriztion of the Reltivisti mss vrition by mens of the Shrödinger s box The reltivisti tretment n be homologted to the quntum one, by ssuming tht n eletron inside the Sbox, relte its sptil oordintes to the inrese of kineti energy. Thus, when pply fore ble to redimension the box, lso redimensions the eletron. Thus, trnsformtion of kineti energy into mss nd ontrtion width, by mens of the γoeffiient of l 0 reltivisti diltion (6) : m γ m0 l. γ Strting from this reltivisti onjeture, it ws quntumssyed the prtile s spetime lous s funtion of the bsorption of energy dimensioned by the γprmeter. Within the box the prtile hs the energy quntified in levels n, where width length of the n h box : n. Assuming, tht equls the dimeter of the prtile, then: 8m 4
5 n h n m 8 O/. Applying work in the xxis diretion whih nd the reltivisti onsidertions, it n h n h is obtined: n n γ O/ 8m0 O/ 8(γ m0 ) γ Hene, elertion dv/dt reltes the finl nd initil energy s follows: n f n i γ Prtile in box: n infinite potentil well (V(x)) si x < 0 ó x > [1] V ( x) 0 si 0 < x < Applying the Shrödinger eqution 1) xternl region to the well m [] ψ + ( ) ψ 0 whose trivil solution is ψ 0 η ) Internl region to the well d ψ m m d ψ m [3] + ψ ( x) 0 0 < x <, mking [4] k + ψ ( x) 0 dx η η dx η The solution is the expression: ψ ( x) A sen[ kx] + B os[ kx] As the eletron is defined within intervl 0 < x <, must fulfill the onditions of ontour: ψ ( 0) 0 ψ ( ) 0. Therefore, B 0 nd ψ ( x) A sen[ k ] 0. n π Aordingly, A 0, hene k n π, or: [] k. n π m From the expression [4] is obtined k, with n 1,, 3... the eletron nnot η n π η n h hve ny intermedite vlue of energy []. m 8m nπ i t η The wve funtion Ψ n ( x, A sen[ x] e with 0 x nd Ψn ( x, 0 in x < 0 or < x. * Renormlizing the wve funtion: Ψ n ( x, Ψn ( x, dx 1, / A / / nπ 1 beomes sen ( x) dx 1, then sen θ (1 osθ), whose solution equls: / A 1 A. Repling in [] Ψ n ( x, n π sen[ x] e i t η.
6 nπ Imposing time independene: [6] Ψ n ( x) sen[ x] ) b) Figure : ) Funtion of wve of the prtile. The eqution of Shrödinger is pplied so tht represents dimeter of the prtile between / nd /, nd renormlized mplitude /. b) The ssoite Probbility Density. The probbility density for the totl of energy of the prtile ws evluted between  / nd /, where it is observed tht the distribution is onentrted in the periphery or ontour. If we onsidered tht n idelized eletron ould be ontined in onedimensionl of infinite potentilbox, moving throughout box intervl: 0 x, ssuming tht the width length beme s smll s the dimeter of the eletron e. Ψ ( x, n O/ e nπ sen[ x] O/ \ In the initil ondition [1], n be supposed tht to vries s does the γprmeter, by the 0 reltivisti reltion. Reformulting by mens of this ide, the ondition [1] results: γ 0 si x < 0 ó x > γ si x < 0 ó x γ > 0 V ( x) Operting: V ( x γ ) 0 0 si 0 < x < 0 si 0 < x γ < 0 γ e Mking hnge of vrible x γ u, it is rehed the sme onlusion [6]: n π nπ Ψ n ( u, sen[ u] Tht is to sy: Ψ n ( x γ, sen[ x γ ] 0 0 6
7 The wve funtion dependene of prmeters x nd γ, ws desribed by the onstnt n. Nevertheless, this funtion desribes the evolution of the energy level, s if determined by the hnges of frequeny. Tht is to sy, without being ble to distinguish between the vlue of n nd the one of γ [fig. 3.)]. 1/ The Prmeter γ (1 v / ) is funtion of the veloity of the eletron in reltion to, whih uses tht its mthemti dominion vries between 1 <γ <, but does not reh the infinite vlue, by the restrition tht imposes the energy limit nk. The vrition of prmeter γ from 1 to very gret vlues genertes in its spet hrteristi frtl. ) b) ) d) e) f) 7
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