Joule-Lenz Energy of Quantum Electron Transitions Compared with the Electromagnetic Emission of Energy

Transcription

1 Joural of Modr Physics, 06, 7, Publishd Oli August 06 i SciRs Joul-Lz Ergy of Quatum Elctro Trasitios Compard with th Elctromagtic Emissio of Ergy Staisław Olszwski Istitut of Physical Chmistry, Polish Acadmy of Scics Kasprzaka, Warsaw, Polad Rcivd 4 July 06; accptd 9 August 06; publishd August 06 Copyright 06 by author ad Scitific Rsarch Publishig Ic This work is licsd udr th Crativ Commos Attributio Itratioal Lics (CC BY) Abstract I th first stp, th Joul-Lz dissipatio rgy spcifid for th lctro trasitios btw two ighbourig quatum lvls i th hydrog atom has b compard with th lctromagtic rgy of missio from a sigl lvl Both th lctric ad magtic vctors trig th Poitig vctor of th lctromagtic fild ar rfrrd to th o-lctro motio prformd alog a orbit i th atom I th xt stp, a similar compariso of missio rats is prformd for th harmoic oscillator Formally a full agrmt of th Joul-Lz ad lctromagtic xprssios for th rgy missio rats has b attaid Kywords Joul-Lz Ergy, Quatum Elctro Trasitios, ydrog Atom, Elctromagtic Ergy Emissio Itroductio Usually ay calculatio of th missio rat of rgy i th atom has as its backgroud a rathr complicatd statistical-ad-probabilistic thory This situatio sms to b ot chagd much sic th vry d of th itth ad bgiig of twtth ctury []-[3] I practic a idividual atomic systm has b vr cosidrd, but istad of it a smbl of th oscillatig atoms kow as th black body was xamid Rathr automatically th tmpratur paramtr importat for comparig th thortical rsults with xprimt has b ivolvd i such may-atomic calculatios Nxt th probabilistic approach to th missio itsity foud its justificatio, ad a rathr xtdd though complicatd applicatio, i quatum haics [4] [5] Mor rctly a approach to th tratmt of th rgy missio i a sigl atomic objct could b basd o ow to cit this papr: Olszwski, S (06) Joul-Lz Ergy of Quatum Elctro Trasitios Compard with th Elctromagtic Emissio of Ergy Joural of Modr Physics, 7,

2 S Olszwski th Joul-Lz law [6]-[0] For, wh th Bohr thory of th hydrog atom is tak as a xampl, ay atom has its lctro placd o a dfiit orbit which ca b approximatd by a circl Elctrically such a circular motio ca b rprstd by a currt havig a kow itsity For xampl, for th quatum stats ad + th currt itsity is rspctivly i = ad i+ =, () T T whr T ad T + ar th tim priods of th lctro circulatio about th proto uclus I th xt stp, th rgy diffrc btw lvls + ad, amly E = E E () provids us with th lctric pottial This lads to th lctric rsistac +, + E V = (3) V V V R = (4) i i i + whr th approximat rlatios i (4) hold i virtu of i i i + (5) valid for larg Th validity of (5) bcoms vidt if w apply Formula (7) i () For such larg w hav [] th rgy chag ( ) ( ) m m + m E = ; 3 = ( + ) + i th last stp of (6) th approximatio of larg is cosidrd Sic [] w obtai 3 3 π T =, 4 m ET π π R = = = = = ; i i m V E m h 3 4 this is a costat idpdt of Th sam valu of R ca b calculatd also for othr quatum systms tha th hydrog atom, s [6] [7] [0] A charactristic poit is that R is qual to a wll-kow rsult of xprimts do o th itgr quatum all ffct [] Th Joul-Lz law is rprstd by th wll-kow rlatio E = Ri (9) t whr t is th tim itrval cssary to produc th mittd rgy E I fact (9) implis that for i = i w hav or T π π π 3 4 π h h h E m m t = = = = = = T Ri m m m Morovr from (6) ad (0) w obtai m π E t = π h 3 4 m = = (6) (7) (8) (0) () 44

3 S Olszwski Thrfor th ratio (9) bcoms ( E) h t = () E 4 8 E m m = = 3 = 6 5 t h h π (3) Rsults similar to (0)-() ca b obtaid also for othr quatum systms tha th hydrog atom [6] [7] [0] Th pricipal aim of th papr is, i th first stp, to compar th ratio calculatd i (3) with th rat of rgy missio obtaid i trms of th lctromagtic thory Nxt, i ordr to compar th quatum missio with th classical missio rat, th proprtis of th harmoic oscillator missio ar also studid Filds Iducd by th Elctro Motio i th ydrog Atom Th lctric fild valu E actig o th lctro i th Bohr atom is wll kow: m E = E = = r Th last stp i (4) is attaid bcaus of th radius of th orbit which is [] (4) r = (5) m A lss-kow magtic fild omittd i th Bohr atomic modl [0] is iducd i th hydrog atom du to th circular lctro motio do with th frqucy Bcaus of th formula (s g [3]) th idtity btw (6) ad (7) combid with (7) givs π Ω = (6) T Ω =, (7) = = A charactristic poit is that wh xprssios for E ad = + c w obtai for th lctric compot of (9) E ar substitutd to th Lortz forc (8) F E [ v ], (9) = m r = = 6 4 m ad th sam valu is obtaid for th magtic compot of (9) m v = v = = c c c o coditio th vctor of th lctro vlocity havig th valu [] is ormal to v 3 6, (0) () = () 44

4 S Olszwski 3 Fild Valus Spcific for th O-Elctro Currt Prst i th Atom ad th Elctromagtic Rat of th Ergy Emissio Our aim is to costruct th Poytig vctor which provids us with th lctromagtic disps of rgy Th vctors E ad bcom slightly difffrt tha i Sctio bcaus thy rfr to th currt bhaviour of th lctro which is circulatig alog its orbit With th pottial V giv i (3) ad (6) ad qual to 3 m V =, (3) 3 th lctric fild o th orbit havig th lgth l = πr (4) attais th valu [6] E orbit πr π π 3 5 V m m m = = = This givs a lctric vctor dirctd alog th currt O th othr had th magtic fild dirctd ormally to th currt attais th valu [4] [6] 4 3 = i m m c orbit cr = Tcr = π = π This fild diffrs from that giv i (8) solly by th factor qual to π It should b otd that paramtr r trig (6) is th radius of th circular cross-sctio ara of th orbit assumd to b qual to th cross-sctio of th lctro microparticl cosidrd as a sphr [5] [7]: r (7) Th valu of th Poytig vctor maatig th rgy from th orbit is calculatd accordig to th formula [4] P c c S = Eorbit orbit Sorbit = Eorbit orbit Sorbit 4π 4π (8) c m m = π π = π π π m π whr (5) (6) Sorbit = πrπr = 4π rr (9) is th toroidal surfac of th orbit havig th lgth (4) ad th lgth of th cross-sctio circumfrc of th orbit is qual to π r (30) I ffct w obtai from (3), (6) ad (8) th rsult prcisly qual to Formula (3) calculatd from th Joul-Lz thory Sic (3) assumd th lctro trasitios solly btw th lvls +, (3) th idtity btw (3) ad (8) implis that th limitatio to trasitio (3) applis also to th lctromagtic rsult calculatd i (8) A problm may aris to what xtt th rgy rat (3), or (8), ca b radiatd as a lctromagtic wav A altativ bhaviour is that th rgy E is spt for a haical rarragmt of th lctro positio du to th trasitio procss A argumt for that is th prsc of th lctric forc E = π m 6 orbit 5 4 (3) 443

5 S Olszwski alog th orbit Th forc (3) multiplid by th orbit lgth calculatd i (4) givs which is prcisly th rgy le 6 4 m m = π = m π orbit E of th lctro trasitio obtaid i (6) (33) 4 Quatum ad Classical Emissio Rat Calculatd for th armoic Oscillator A atural tdcy is to compar th quatum rat of th rgy missio with th classical missio rat To this purpos th o-dimsioal harmoic oscillator has b chos as a suitabl objct of xamiatio Th classical rgy of th oscillator is E osc = ka, (34) a is th oscillator amplitud; m is th oscillator mass which togthr with th forc costat k rfrs to th circular frqucy of th oscillator k π ω = = ; (35) m T T is th oscillatio priod [8] Th quatum oscillator rgy is E = + ω ω (36) (th last stp holds for larg ) ad th chag of rgy du to trasitio btw th lvls + ad is E = ω (37) Accordig to th Joul-Lz approach to th quata [6]-[0] th missio rat btw th lvls + ad is so This givs ( E) ( ω) E = = t h h E ω hν = = = = ν =, t h h h T (38) (39) t = T, (40) bcaus th rfrc btw ω ad ν is ω = π ν (4) Th pottial V coctd with th rgy chag E is E ω V = = (4) If w ot that a maximal distac travlld by th lctro oscillator i o dirctio is l = a, (43) th lctric fild coctd with th oscillator paralll to its motio is V V ω E = = = (44) l a a Th lctric currt lt b cosidrd as rmaiig approximatly costat i cours of th oscillatio I this cas th magtic fild which is ormal to th currt [s (6)] is 444

6 S Olszwski i ν = = = = =, (45) cr Tcr Tc T sic th cross-sctio of th lctro currt is assumd to b idtical with th cross-sctio ara of th lctro microparticl, s (7) Th surfac ara of th sampl cotaiig th oscillator is o coditio th cotributio of th d aras of th sampl surfac qual to S = πrl = 4πra = 4π a, (46) πr = π 4π a (47) has b glctd bcaus (47) is a small umbr i compariso with S i (46) I cosquc, for th vctor ormal to vctor E th valu of th Poytig vctor bcoms 4π a P c c S ω 4π S = E = = ω = E 4π 4π a T 4πT T This is a rsult idtical with (38) o coditio Formula (39) is tak ito accout Accordig to th classical lctrodyamics [9] th missio rat of rgy from a classical oscillator is sic 4 4 de ω π a = p = 3 3 dt 3 c 3 T c p (48) (49) = a (50) is th dipol momt of th classical harmoic oscillator Formula (49) ca b compard with th quatum approach to th Joul-Lz missio rat of rgy [s (38)]: ( ω) E h h = = = t h T h T I th cas of vry small quatum systms th amplitud a i (49) ca b clos to its miimal lgth [0] ad th tim priod T ca approach its miimal siz [0] Th quality rquird btw (49) ad (5) lads to th rlatio a (5) (5) T (53) 4 π a h = 3 3 T c T Wh a ad T ar tak rspctivly from (5) ad (53), Formula (54) bcoms ( π) ( π) ( π) = = = h c 3 c c 3 c from which w hav th rlatio ( ) 3 c π 65 = = 3 α (54) (55) (56) 445

7 S Olszwski Th rsult obtaid i (56) diffrs by oly 0 prct from th rciprocal valu of th atomic costat qual to 37 5 Ratio of th Classical ad Quatum Emissio Rat Dfid by th Dampig Cofficit of th Classical Radiatio A attmpt of this Sctio is to dmostrat that th classical missio ca b cosidrd as a dampd quatum missio rat Th classical dampig cofficit of th oscillator is [9] γ 3 = ω (57) 3 O th othr had, th classical missio rat giv i (49) ca b modifid wh th amplitud a trig (49) is xprssd i trms of th oscillator rgy E [8]: E E a = k = ω mω = mω = (58) mω r, at th d of (58), th rgy E is rplacd by th approximat quatum formula for th oscillator rgy giv i (36) I ffct th classical missio rat i (49) bcoms 3 class ω η = ω a = ω = (59) c 3c mω 3c m Aothr trasformatio may cocr th quatum missio rat i (5): η quat As a rsult of (59) ad (60) w obtai th ratio E h π h ω = = = = t T T π ( π) class 3 η 4 ω ω 8π ω ω = : = = 4π = 4πγ = Tγ (6) quat η 3 c m π 3 c m 3 cmω ω which is proportioal to γ i (57) A multipl of th oscillatio tim priod T is th proportioality cofficit rprstig (6) i trms of γ Thrfor aothr way to writ (6) ca b η (60) class quat = Tγη (6) Lt us ot that E trig (36) ad (58) is proportioal to It is worth to ot that th Eisti cofficit A α of th missio probability ca b coupld with γ by th rlatio [0] so A hν = γ f hν (63) α A α, α = γ f (64), α Accordig to isbrg [0] [] w hav ( ) h h a, ν (, ) = ν (, ) = f, (65) πmω 8π m whr a is th quatum-thortical amplitud of th xpasio of th coordiat x x( t) = of a aharmoic oscillator; ω 0 is th circular frqucy of th harmoic oscillator For small pturbatio λ of th oscillator w hav [] ω = π ν, ω, (66) so Formula (65) givs 0 ( ) 0 hω hω h h = = f, (67) π m π m π m 8π m 0 ω0 ω0 446

8 S Olszwski or 4 = f (68), I ffct for α = tak i (63) w obtai from (64) ad (68): If γ is prstd, accordig to (6), i trms of th ratio of whr T is th oscillatio tim priod of th harmoic oscillator A = γ f, = 4 γ (69) class η ad quat η, w obtai class class 4 A η η = =, (70) quat quat T η T η 6 Rciprocal Valu of th Atomic Costat ad th Elctro Spi Th rciprocal valu α of th atomic costat (~37) approachd i (56) is importat i th tratmt of th lctro spi [0] [] [3] W show blow that th magtic fild itsity cssary to produc th lctro spi ca b obtaid approximatly as a rsult of a couplig of α with th radius r of th lctro microparticl, s (7) Accordig to th classical lctrodyamics [4] th magtic fild at a distat r from th ctr of th liar wir carryig a currt i is coupld with i ad r by th formula i = (7) cr If th currt i is flowig o a surfac of th coductor which is th lctro orbit, w ca assum that r = r which is both th radius of th lctro microparticl ad cross-sctio of th orbit Th fild bcoms i this cas [4] = = = Tcr c 4 3 m 3 3 π π whr th tim priod T of th lctro circulatio alog th orbit is tak from Formula (7) for = : 3 π T = T = 4 m Th ssc of th spi ffct is that th path of th spiig lctro circumvts th lctro orbit about α c = 37 tims durig th tim priod T idicatd i (73) I classical lctrodyamics this mas that th magtic fild producd i this way is α tims strogr tha that obtaid i (7): spi = c α = π = π, (7) (73) (74) (75) Th rsult i (75) diffrs solly by th factor of π from th magtic fild assumd to produc a spiig lctro particl i [0] [] [3]: 3 spi = A discrpacy btw (75) ad (76) ca b ascribd to som ucrtaity coctd with th calculatio of th radius r, s [4] 7 Coclusios Th aim of th papr was to gt mor isight ito a o-probabilistic dscriptio of th trasfr of rgy (76) 447

9 S Olszwski btw two quatum lvls A suitabl situatio for discussio is th cas wh th lvls ar ighbourig i thir mutual positio of th rgy stats Th th rgy chag ( E ) btw th lvls, ad th tim itrval t cssary to attai E, satisfy a vry simpl formula E t = h; (77) s [6]-[0] I th papr, Formula (77) fids its coutrparts supplid by th lctromagtic thory of missio Two physical objcts, amly th hydrog atom ad lctro harmoic oscillator, wr studid Th cas of th lctro oscillator allowd us to prform a mor dirct compariso of th quatum approach to th missio rat with th classical lctromagtic thory It occurs that th classical rat is qual to th quatum rat multiplid by th Bor dampig cofficit ad a itrval of tim, s (6) Rfrcs [] Plack, M (90) Acht Vorlsug ubr Thortisch Physik S irzl, Lipzig [] Eisti, A (97) Physikalisch Zitschrift, 8, [3] Bohr, N (967) O th Quatum Thory of Li Spctra I: Va dr Ward, BL, Ed, Sourcs of Quatum Mchaics, Dovr Publicatios, Nw York, [4] Bth, (933) Quathaik dr Ei- ud Zwi-Elktroproblm I: Gigr, ad Schl, K, Eds, adbuch dr Physik, Vol 4, Part, Sprigr, Brli, [5] Codo, EU ad Shortly, G (970) Th Thory of Atomic Spctra Cambridg Uivrsity Prss, Cambridg, UK [6] Olszwski, S (05) Joural of Modr Physics, 6, [7] Olszwski, S (06) Joural of Modr Physics, 7, [8] Olszwski, S (06) Joural of Modr Physics, 7, [9] Olszwski, S (06) Joural of Modr Physics, 7, [0] Olszwski, S (06) Rviws i Thortical Scic, 4, [] Sommrfld, A (93) Atombau ud Spktrallii 5th Editio, Vol, Viwg, Brauschwig [] MacDoald, A (989) Quatum all Effct A Prspctiv Kluwr, Milao [3] Slatr, JC (967) Quatum Thory of Molculs ad Solids Vol 3, McGraw-ill, Nw York [4] Lass, (950) Vctor ad Tsor Aalysis McGraw-ill, Nw York [5] Matvv, AN (964) Elctrodyamics ad th Thory of Rlativity Izd Wyzszaja Szkola, Moscow (I Russia) [6] Grir, W (998) Classical Elctrodyamics Sprigr, Nw York [7] Ladau, LD ad Lifshits, EM (969) Mchaics Elctrodyamics Izd Nauka, Moscow (I Russia) [8] Sommrfld, A (943) Mchaik Akadmisch Vrlagsgsllschaft, Lipzig [9] Bor, M (933) Optik Sprigr, Brli [0] Va dr Ward, BL (967) Itroductio I: Va dr Ward, BL, Ed, Sourcs of Quatum Mchaics, Dovr Publicatios, Nw York [] isbrg, W (95) Zitschrift fur Physik, 33, [] Olszwski, S (04) Joural of Modr Physics, 5, [3] Olszwski, S (04) Joural of Modr Physics, 5, [4] Olszwski, S (06) Joural of Modr Physics, 7,

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