Quantum Cattaneo wave equation for ultra-short laser pulses interaction with electron and nucleon gases

Transcription

1 Quanu Caano wav quaion for ulra-sor lasr pulss inracion wi lcron and nuclon gass JMarciak-Kozłowska 1 MKozłowski * 1 Insiu of Elcron cnology Warsaw Poland * Pysics Dparn Warsaw Univrsiy Warsaw Poland *Corrsponding auor -ail: iroslawkozlowski@asrpl Absrac In is papr quanu Caano wav quaion for ulra-sor lasr pulss inracion wi ar is obaind xplici forula for lcron and nuclon gass ar prsnd Ky words: quanu a ranspor Caano wav quaion ulasor lasr pulss

2 Inroducion ro orical poin of viw i ss qui obvious a diffusiv procsss can no ak plac wi infini vlociy insid ar sinc is fac would viola causaliy in spcial rlaiviy frawork Howvr sandard diffusion quaion is basd on ick s law [1] in cas of ass ranspor and on ourir s law [] in cas of a conducion So or qui coon consiuiv laws of aaical pysics sablis a siilar i-indpndn funcional rlaion bwn flux and spacial gradin of sa variabl is is cas of O s law in lcriciy [] of Darcy s law in fluid oion wiin porous dia [4] c Wn cobind wi adqua consrvaion principl (i: coninuiy quaion) s consiuiv laws lad o a pur parabolic aaical odl a prdics an infini spd of propagaion for ass or nrgy bing ranspord or s rasons infini spd paradox undrlis any aaical odls a ar frqunly usd in copuaional nginring and scinc I appnd o b ironical a firs coprnsiv drivaion for sandard diffusion quaion was givn by Einsin islf is issu is jus ignord in a nubr of applicaions in wic sandard linar parabolic odl is supposd o b accura noug for pracical purposs aloug sipl ida of ass or nrgy bing ranspord a infini spd is disurbing Howvr in so or applicaions i could b andaory o ak ino accoun wav naur of diffusiv procsss o prfor accura prdicions]or abov sad rasons a considrabl ffor as bn isorically dvod o rov infini spd paradox fro

3 sandard diffusion quaion asically wo kind of approacs av bn prsnd: on on sid so non-linar consiuiv laws av bn proposd uskping parabolic naur of odl [; on or so i-dpndn consiuiv laws av bn proposd wa lads o a nw class of yprbolic-yp wav odls as on firs proposdby Caano [6] Caano quaion is classical wav quaion for praur fild (x) In is papr w dvlop gnralizd quanu Caano quaion for a ranspor pnona inducd by aoscond ( s) lasr pulss- aoa ral wav pnona aoscond is sorr an all known rlaxaion is for ral procsss in ar In a cas duraion of injcd puls is so sor a basic assupion for applicaions of parabolic quaions is violadin papr w forula orical frawork for ulrasor lasr puls inracion wi ar quanu ral Caano and Proca quaions ar pillars of nw dscripion of aa procsss odl quaion Dynaical procsss ar coonly invsigad using lasr pup-prob xprins wi a pup puls xciing sys of inrs and a scond prob puls racking is poral voluion As i rsoluion aainabl in suc xprins dpnds on poral dfiniion of lasr puls puls coprssion o aoscond doain is a rcn proising dvlopn Afr sandards of i and spac wr dfind laws of classical pysics rlaing suc parars as disanc i vlociy praur ar assud o b indpndn of accuracy wi wic s parars can b asurd I sould b nod a is

4 assupion dos no nr xplicily ino forulaion of classical pysics I iplis a ogr wi assupion of xisnc of an objc and rally indpndnly of any asurns (in classical pysics) i was acily assud a r was a possibiliy of an unliid incras in accuracy of asurns aring in ind aoiciy of i i considring salls i priod Planck i abov san is obviously no ru Aoscond lasr pulss w ar a lii of lasr i rsoluion Wi aoscond lasr pulss blong o a nw ano World wr siz bcos coparabl o aoic dinsions wr ranspor pnona follow diffrn laws fro a in acro world is firs sag of iniaurizaion fro 10 - o 10-6 is ovr and nw on fro 10-6 o 10-9 jus bginning ano World is a quanu world wi all prdicabl and non-prdicabl (y) faurs In is paragrap w dvlop and solv quanu rlaivisic a ranspor quaion for ano World ranspor pnona wr xrnal forcs xis In onograp [ 7 ] nw rical frawork for sudy of ulra-sor lasr pulss inracion wi ar was dvlopd and yprbolic quaion for praur fild (x) was obaind: 1 1 ( v ) τ ( v ) 1 1 (1) wr dnos praur τ rlaxaion i for ral disurbanc of frionic sys and v is ri vlociy In wa follows w dvlop nw forulaion of HH considring dails of wo frionic syss: lcron gas in als and nuclon gas or lcron gas in als ri nrgy as for [1]

5 E / n (π ) () wr n dnos dnsiy and lcron ass Considring a n 1/ ~ a ~ () and a or radius on obains E ~ / n ~ a ~ α c (4) wr c lig vlociy and α 1/17 is fin-srucur consan for lcroagnic inracion or ri onu p w av p ~ ~ αc a (5) and for ri vlociy v p v ~ ~ α c (6) Considring forula (6) Eq (1) can b wrin as 1 1 α c c τ (7) As sn fro (7) HH quaion is a rlaivisic quaion sinc i aks ino accoun fini vlociy of lig or nuclon gas ri nrgy quals E / (9π ) (8) 8r 0

6 wr dnos nuclon ass and r0 wic dscribs rang of srong inracion is givn by r 0 (9) c π wrin π is pion ass ro forula (9) on obains for nuclon ri nrgy E π ~ c (10) In analogy o Eq (4) forula (10) can b wrin as E ~ c α s (11) π wr α 0 15 is fin-srucur consan for srong s inracions Analogously w obain nuclon ri onu p ~ r 0 ~ α c s (1) and nuclon ri vlociy p v ~ ~ α sc (1) and HH for nuclon gas can b wrin as 1 1 α s c c τ (14) In following procdur for discrizaion of praur ( r ) in o frion gas will b dvlopd irs of all w inroduc rducd d rogli wavlng

7 v v 1 1 c v c v s α α (15) and an fr pas and v τ v τ (16) In viw of forulas (15) and (16) w obain HHC for lcron and nuclon gass v (17) v (18) Equaions (17) and (18) ar yprbolic parial diffrnial quaions dapd wav quaion wic ar asr quaions for a propagaion in ri lcron and nuclon gass In following w will sudy quanu lii of a ranspor in frionic syss W dfin quanu a ranspor lii as follows: (19) In a cas Eqs (17) and (18) av for τ (0) τ (1) wr

8 τ ( ) v τ () ( ) v Equaions (0) and (1) dfin asr quaion for quanu wav a ranspor (QH) Having rlaxaion is τ and τ on can dfin pulsaions ω and or i ω 1 1 ω ( τ ) ω ( τ ) () ω ω ω v ( v ) ( v ) ω α c α s ( ) c v ( ) (4) forulas (4) dfin Planck-Einsin rlaion for a quana E and E E E ω ω ( v ) ( v ) a quanu wi nrgy E (5) ω can b nad aon in copl analogy o ponon agnon roon c or τ τ 0 Eqs (0) and (4) ar ourir quaions wi quanu diffusion cofficins D and D D D (6) D D (7) quanu diffusion cofficins D and D wr inroducd for firs i by E lson or fini τ and τ for Δ < τ Δ < τ Eqs (0) and (1) can b wrin as

9 ( v 1 ) (8) ( v 1 ) (9) Equaions (8) and (9) ar wav quaions for quanu a ranspor (QH) or Δ >τ on obains ourir quaions (6) and (7) In wa follows dinsionlss for of QH will b usd Inroducing rducd i and rducd lng x x ' / τ x ' (0) v τ on obains for QH (1) and for Q () () (4)

10 Conclusions In is papr quanu Caano quaion was proposd I was sown a for lasr puls wi i duraion τ quanu Caano quaion is quanu wav quaion wav vlociy is qual vαc wr α is fin srucur consan and c is lig vlociy Rfrncs [1] A ick Ubr diffusion Poggndorffs Annaln dr Pysik und Ci [] Jourir ori analyiqu d la calur Jacqus Gabay 18 [] GS O Di galvanisc K aaisc barbi H Riann 187 [4] SP uan orical Drivaionof Darcy slaw Aca Mcanica [5] A Einsin Zur ori dr rownscn wgung Annaln dr Pysik [6] MC Caano Sur un for d l quaion d la calur liinan l paradoxd un propagaion insanan Cops Rndus d L Acadi ds Scincs: Sris I-Maaics [7] MKozlowski J Marciak-Kozłowska ral procsss using aoscond lasr pulss Springr 006