Generalización fraccional de la ecuación de Schrodinger relacionada a la Mecánica Cuántica

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1 Revisa Tecocieíica URU Uivesidad Raael Udaea Faclad de Igeieía º Jlio - Diciee I: X / Depósio legal pp ZU3863 Geealizació accioal de la ecació de chodige elacioada a la Mecáica Cáica. L. Kalla R. K. aea ad Ravi aea 3 Vyas Isies o Highe dcaio Jodhp 34 Idia Depae o Maheaics & aisics Jai aai Vyas Uivesiy Jodhp-345 Idia 3 Depae o Civil gieeig Jai aai Vyas Uivesiy Jodhp -345 Idia Rese Reciido 8-4- Acepado 7-5- l ojeo de ese aajo es pesea a solció copacioal de a geealizació accioal idiesioal de la ecació de chodige elacioada a la Mecáica Cáica. l éodo iliza cojaee la asoada de d y la asoada de Foie. La solció es oeida e oa copacioal y ceada e éios de la ció de Miag-Lele y la ció H. l eslado picipal oeido aqí es geeal y a pai de ése se pede dedci ga úeo de casos especiales hasa ahoa dispesos e la lieaa. Adeás ése povee a eesió de eslado dado aeioee po Deah aea y Chaasia. l eslado picipal es peseado e oa de Teoea y se ecioa vaios casos especiales. Palaas clave: Fciò de Miag-Lele Fciò-H Tasoada d Tasoada de Laplace Deivada de Capo Facioal geealizaio o chödige eqaio elaed o Qa Mechaics Asac The ojec o his aicle is o pese he copaioal solio o a liea oe-diesioal acioal geealizaio o chödige eqaio occig i qa echaics. The ehod ollowed is ha o joi d aso ad Foie aso. The solio is deived i a closed ad copaioal o i es o he Miag- Lele cio ad he H-cio. The ai esl deived hee is geeal i ae ad capale o yieldig a lage e o special cases hiheo scaeed i lieae. I also povides a eesio o a esl give ealie y Deah aea e al. ad Chaasia e al. The ai esl is peseed i he o o a Theoe ad seveal special cases ae eioed. Key wods: Miag-Lele cio H-cio d aso Laplace aso Capo deivaive Maheaics jec Classiicaio : 6A33 44A 33C6 35J Coespodig aho shyaalla@yahoo.co 73

2 74 Geealizació accioal de la ecació de chodige elacioada a la Mecáica Cáica Revisa Tecocieíica URU º Jlio - Diciee Iodcio Facioal dieeial eqaios ae he geealizaios o he odiay dieeial eqaios o aiay ode eal o cople. Dig las wo decades oe iees is developed y vaios eseach woes i olaig acioal dieeial eqaios de o hei seless ad capailiy o odel ad solve cople syses. I his coecio oe ca ee o [ ad 9]. Facioal chödige eqaio is a daeal eqaio o qa echaics. This eqaio is discssed y Lasi [8 9 ad ] i a aep o ivesigae a geealizaio o Feya pah iegals o Bowia lie o Lévy lie qa echaical pahs. alie Feya ad His [] ecosced he chödige eqaio y aig se o he pah iegal appoach ad aig se o he well ow Gassia poailiy disiio. The chödige eqaio hs oaied coais space ad ie acioal deivaives. I a siila ae oe oais a ie acioal eqaio i o-macovia evolio is cosideed. I a ece pape ae [5] discssed ceai popeies o ie acioal chödige eqaio y epessig he chödige eqaio i es o acioal deivaives as diesioless ojecs. Tie acioal chödige eqaios ae also discssed y Deah [6] Bhai [3] ad Deah ad Bhai [8]. I a ece pape he ahos have ivesigaed he solio o he ollowig geealized liea oe diesioal acioal chödige eqaio o a ee paicle o ass deied y iħ / < < > o < > < < as 3 whee is he Capo acioal deivaive deied y 5 ad is he Lioville acioal space deivaive deied y is he wave cio h π ћ eg sec MeV sec. is he Plac cosa ad is a aiay cio. The aove deied chodige eqaio is he geealized ecely y aea e al. [35] y eployig he Hile acioal deivaive [6 p.3 eq. 5] isead o he Capo deivaive deied y 5. Poailiy sce o ie acioal chödige eqaio is ecely discssed y Toigh [36]. oe physical applicaios o acioal chödige eqaio ae ivesigaed y Go ad X [3] y deivig he solio o a ee paicle ad iiie sqae poeial well. This has oivaed he ahos o ivesigae he solio o a acioal geealizaio o chödige eqaio 6 i oe-diesio occig i qa echaics. Facioal eacio- disio eqaios ae eaed y Haold e al. [4] aea e al. [3 3 33] ad Hey ad Weae [5].. Maheaical peeqisies d aso is deied y [] G [;] e d d τ τ 4 ove he se o cios / τ A { M τ τ > < j j Me [ } 5

3 hya Kalla Ra Kishoe aea y Ravi aea Revisa Tecocieíica URU º Jlio - Diciee whee G is called he d aso o. I is clea ha i is a liea opeao. I is a sligh vaia o he well-ow Laplace aso. Fhe d aso peseves he i ad scale popeies which aes i a ideal ool o solvig seveal poles o physical & egieeig scieces wiho esoig o a ew eqecy doai. The elaios coecig he d aso ad Laplace aso deied y 6 ae give y he ollowig Theoe: Theoe. [ p. 5]. Le A wih he Laplace aso Fs. The he d aso G o is give y G F. 6 Coollay. Fo he d aso o is give y G ; Γ. R > R > Fhe we also have ;. R > R > 8 Γ Coollay. Le A. Res> ad F ad G ae he Laplace aso ad he d aso o he cio especively he F s G. 9 s s Lea. [ δ δ ω ; ] ω. whee R > R > ω < ad δ ω is he geealized Miag-Lele cio deied y Pahaa [8] i he o δ δ z z Γ! wih z δ Ci{ R R} >. The esl ca e easily poved y epadig he ioial cio ad iepeig he esl hs oaied y a appeal o he eqaio 8. I will e see ha his esl is diecly applicale i he deivaio o he solio o he acioal dieeial eqaio 6. oe. Whe δ edces o he Miag-Lele cio sdied y Wia [37] i he ollowig o z z Γ whee z C i{ R R } > Fo z C i{ R R} > edces o he Miag-Lele cio [ 4]: z z 3 Γ 7

4 76 whee. > R C z Lea. ; 4 whee a > R > R > R Poo: We have Fo aove i ollows ha ; valaig o he igh o aove eqaio wih he help o Lea. we aive a he desied esl 4. The e y e ivesio is jsiied i view o he esl [ ]. The ollowig acioal deivaive o ode > is iodced y Capo [4] i he o D c Γ d τ τ τ < Re >. 5 d d i. 6 whee d d is he h deivaive o ode o he cio wih espec o. Lea.3 The d aso o Capo deivaive deied y 5 is give y Γ d D c ; τ τ τ 7 ] [ } ; { d d h Geealizació accioal de la ecació de chodige elacioada a la Mecáica Cáica Revisa Tecocieíica URU º Jlio - Diciee 73-84

5 hya Kalla Ra Kishoe aea y Ravi aea Revisa Tecocieíica URU º Jlio - Diciee By he applicaio o he covolio heoe o he d aso [] he igh had side o he eqaio 7 ecoes [ ; ] [ ; ].. 8 Γ Applyig he d aso o liple dieeiaio we have G c D ; G 9 whee G [;]. The aove ola is sel i deivig he solio o dieeial ad iegal eqaios o acioal ode goveig ceai physical poles o eacio ad disio. I his coecio oe ca ee o he oogaphs wie y Podly [9] ao e al. [3] Kilas e al. [7] Mahai e al. [ ] ad Diehel [9 ]. oe. I hee is o cosio he he deivaive c D o sipliciy will e deoed y D The Lioville acioal deivaive o ode is deied i [3 ecio 4.] i he o y dy Γ y whee R > [ ] [ ] is he iege pa o. oe.3 The opeao deied y is also deoed y give i [3 p.59 A.] D.. Is Foie aso is F{ D ; } Ψ > whee Ψ is he Foie aso o. oe.4 Applicaios o acioal calcls i he solio o physical poles ca e od i he wos [7 6 6 ad 9]. p.]: The H-cio is deied y eas o a Melli- Baes ype iegal i he ollowig ae [ H whee i / p q z H p q z a Ap B [ ] a A... a p Ap p q z B... B H q q Ω Θ z π p q q i [ Π Γ Θ j Β j ][Π Γ a j A j ] j j [Π q Γ j Β j ][Π p Γa j A j ] j j d 3 4

6 78 Geealizació accioal de la ecació de chodige elacioada a la Mecáica Cáica Revisa Tecocieíica URU º Jlio - Diciee ad a epy podc is always iepeed as iy ; p q wih p q A i B j R a i j R o C i p ; j q sch ha A i j j i B a ; i ; j 5 whee we eploy he sal oaios: ; R - R ad C eig he cople e ield. 3. Uiied acioal geealizaio o chödige eqaio I his secio he solio o a liea oe-diesioal acioal chödige eqaio 6 is ivesigaed. The esl is peseed i he o o he ollowig: Theoe 3. Coside he oe-diesioal acioal geealizaio o he chödige eqaio o a ee paicle o ass deied y wih iiial codiios i D D D D ; < < < R 6 li > ; R 7 ± ± whee D D D ae he Capo acioal deivaives o odes > > > especively as deied y 5 D is he Lioville paial acioal deivaive o ode > deied y is he wave cio is he Plac cosa ad is a pescied cio. The de he aove codiios hee holds he ollowig ola o he solio o 6: π h πћ eg sec e i { MeV sec }d povided ha he seies ad iegals i 8 ae covege. a ih ; a 8 Poo: Applyig he d aso wih espec o he ie vaiale ad 4 ad sig he oday codiios we id ha s s. ih D 9

7 79 I we apply he Foie aso wih espec o he space vaiale ad apply he esl i yields. s s s a i a 3 olvig o i gives s 3 whee. a. To ive he eqaio 3 i is coveie o is ive he d aso ad ae ha he Foie aso. O aig he ivese d aso o he aove epessio wih he help o he esl 4 i is od ha 3 Fially he eqied solio 8 is oaied y aig he ivese Foie aso o he eqaio 3. I we se δ whee δ is he Diac dela cio Theoe 3. edces o he ollowig: Coollay 3. Coside he liea oe-diesioal acioal geealizaio o he chödige eqaio o a ee paicle o ass deied y 33 wih iiial codiios δ R > ± R ; li 33 whee D D D ae he Capo acioal deivaives o ode > > > especively ad deied y 5 D is he Lioville paial acioal deivaive o ode > deied y is he wave cio is he Plac cosa ad is a pescied cio. Ude he aove codiios hee holds he ollowig ola o he daeal solio o 33: [ ; i a a ] ] ; < < < D i D D D ; δ ± h πћ eg sec MeV sec δ hya Kalla Ra Kishoe aea y Ravi aea Revisa Tecocieíica URU º Jlio - Diciee 73-84

8 8 Geealizació accioal de la ecació de chodige elacioada a la Mecáica Cáica Revisa Tecocieíica URU º Jlio - Diciee π e i { i }d a ; a 34 povided ha he seies ad iegals i 34 ae covege. I we se he Theoe 3. yields he ollowig Coollay 3. Coside he liea oe-diesioal acioal geealizaio o he chödige eqaio o a ee paicle o ass deied y / / / i / 35 D D D D ; wih iiial codiios R li > ; R 36 ± ± whee / D he Capo acioal deivaives o ode deied y 5 / D is he Lioville paial acioal deivaive o ode deied y is he wave cio h πћ egs MeVs is he Pla cosa ad is he pescied cio he o he solio o 35 de he aove cosais hee holds he ollowig esl: e i π d / 37 whee a ih ; a ; h πћ egs MeVs povided ha he seies ad iegals i 37 ae covege. The ollowig esl de o Chaasia e al. [5] is oaied o he aove heoe o : Coollay 3.3 Coside he liea oe-diesioal acioal geealizaio o he chödige eqaio o a ee pacle o ass deied y i D D D ; < < < 38 wih iiial codiios R li > ; R 39 ± ±

9 hya Kalla Ra Kishoe aea y Ravi aea Revisa Tecocieíica URU º Jlio - Diciee whee D ad D ae he Capo acioal deivaives o ode > > especively ad deied y 5 D is he Lioville paial acioal deivaive o ode > deied y is he wave cio h πћ egs MeVs is he Plac cosa ad is a pescied cio he de he aove codiios hee holds he ollowig ola o he solio o 38: e i { π i a ; a }d 4 povided ha he seies ad iegals i 4 ae covege. I ode o pese he esls o he e Coollay we eed he ollowig: Lea 3. I R > R > a > 4 he hee holds he ola a ; 4 λ µ µ λ a a 4 whee λ ad µ ae he eal ad disic oos o he qadaic eqaio a. The ola 4 ca e esalished y ollowig he echiqe developed y aea e al. [33]. We have a λ a µ a 4 a λ µ λ µ The desied esl is oaied y aig he ivese d aso o oh sides o 4. ow i we se δ δ is eplaced y ad y i 8 ad se he esl 4 we oai he ollowig esl: Coollay 3.4 Coside he liea oe-diesioal liea acioal geealizaio o he chödige eqaio wih he iiial codiios < i a ; a 43 δ δ R li > 44 ± ± whee R δ is a Diac-dela cio whee ad ae he Capo acioal deivaives o ode ad especively deied y 5 is he wave cio

10 8 Geealizació accioal de la ecació de chodige elacioada a la Mecáica Cáica Revisa Tecocieíica URU º Jlio - Diciee The o he daeal solio o 43 de he aove cosais hee holds he ollowig ola: ep i { λ λ λ µ } d d π 4 45 whee λ ad µ ae he eal ad disic oos o he qadaic eqaio give y h πћ egs MeVs y y 46 λ 4 ad µ 4 47 whee i a a is he Miag - Lele cio deied y 3 ad povided ha he iegal i 47 is covege. Rea 4. A esl siila o Coollay 3.4 has ee give y Osighe ad Beghi [7] o he acioal elegaph eqaio. I is ieesig o oseve ha he Foie aso o he solio 45 o he eqaio 44 ca e epessed i he o λ µ a whee λ ad µ ae deied i 47 ad is he Miag -Lele cio deied i 3. I we se he he Theoe 3. gives ise o he ollowig Coollay 3.5 Coside he liea oe-diesioal acioal geealizaio o he chödige eqaio o a ee paicle o ass deied y wih iiial codiios i D D < 49 R li > ; ± 5 whee D is he Capo acioal deivaives o ode > deied y 5 D is he Lioville paial acioal deivaive o ode > deied y ad is he wave cio h πћ egs MeVs is he Plac cosa ad is a pescied cio. The de he aove codiios hee holds he ollowig ola o he solio o 48:

11 hya Kalla Ra Kishoe aea y Ravi aea Revisa Tecocieíica URU º Jlio - Diciee a e i d π G ζ dζ 5 whee he Gee cio G is give y G e i a d π a H 33 / 5 a y vie o a esl give y Haold e al. [4 p.686 eq.5 ]o evalaig he aove iegal; whee i a ad H. 3 3 is he H-cio deied i he eqaio 3. Fially i we he se δ we oai aohe esl give y aea e al. [35]. 4. Coclsio The ehod o joi d aso ad Foie aso is sed o solve a acioal geealizaio o he chodige eqaio. The solio is epessed i es o Miag-Lele cio ad he H-cio. eveal ow esls ollow as special cases o he ai esl esalished hee. Alhogh he d aso is close o he classical Laplace aso i ay e cosideed heoeical dal o i. Havig scale ad i pesevig popeies he d aso ay e sed o solve poles wiho esoig o a ew eqecy doai. 5. Reeeces. Belgace F.B.M. Kaaalli A. A. ad Kalla.L. Aalyic ivesigaios o he d aso ad applicaios o iegal podcio eqaios Maheaical Poles i gieeig Belgace F.B.M. Kaaalli A. A d aso daeal popeies ivesigaios ad applicaios Ie. J. Appl. Mah. och. Aal Bhai M. Facioal chödige wave ad acioal ceaiy piciple I. J. Coep. Mah. cieces o Capo M. lasiciá e Dissipazioe Zaichelli Bologa Chaasia. V.B.L. ad igh J Applicaio o d aso i chödige eqaio occig i qa echaics Appl. Mah. ci. 4 o Deah. L Facioal iegal ad acioal dieeial eqaios i lid Mechaics Fac. Calc. Appl. Aal Deah L. oliea Paial Dieeial qaios o cieiss ad giees ecod diio Bihäse Boso Basel Beli Deah L. ad Bhai M O acioal chödige ad Diac eqaios I. J. J. Pe Appl. Mah Diehel The Aalysis o Facioal Dieeial qaios. pige Beli.. Doesch G Aleig z Paiische Geach de Laplace - Tasoaio. Oldeeg Mich délyia. d. Highe Tascedeal Fcios Vol. 3 McGaw-Hill ew Yo. 955.

12 84 Geealizació accioal de la ecació de chodige elacioada a la Mecáica Cáica Revisa Tecocieíica URU º Jlio - Diciee Feya R.P. ad His A. R Qa Mechaics ad Pah iegals McGaw-Hill ew Yo Go X. ad X M oe physical applicaios o chödige eqaio J. Mah. Phys doi:. 63/ pages. 4. Haold H.J Mahai A.M. ad aea R.K.7. olio o eacio-disio eqaios i es o he H-cio Bll.Aso. oc.idia HeyB.I. ad Weae.L.. Facioal eacio-disio Physica A Hile R. edio Applicaios o Facioal Calcls i Physics Wold cieiic igapoe. 7. Kilas AA. ivasava H.M. ad Tjillo J.J Theoy ad Applicaios o Facioal Dieeial eqaios lsevie Aseda Lasi. Facals ad qa echaics Chaos o Lasi. Facioal qa echaics ad Levy pah iegals Physics Lees A Lasi Facioal chödige eqaio Physical Review axiv: qa-ph/698 v 4 J. MaiadiF Facioal Calcls ad Waves i Liea Viscoelacsiciy Wold cieiic igapoe.. Mahai A.M. aea R.K. ad Haold H.J. The H-cio: Theoy ad Applicaios pige ew Yo. 3. Mezle R. ad Klae J. The ado wal: A gide o aoalos disio: A acioal dyaics appoach. Phy. Rep Miag-Lele G.M la ovelle cio C.R. Acad. ci. Pais ae M Disied ode acioal s-disio Facals o Oldha K.B. ad paie J. The Facioal Calcls Theoy ad Appliaios o Dieeiaio ad Iegaio o Aiay Ode Acadeic Pessew Yo Osighe ad Beghi L. Tie- acioal elegaph eqaios ad elegaph pocesses wih owia ie Poa.Theoy Rela. Fields Pahaa T. R. A sigla iegal eqaio wih geealized Miag-Lele cio i he eel Yoohaa Mah. J Podly I. Facioal Dieeial qaios Acadeic Pess ew Yo ao.g. Kilas A.A. ad Maichev O.I Facioal Iegals ad Deivaives: Theoy ad Applicaios Godo ad Beach ciece Plishig Aseda aea R.K. Mahai A.M. ad Haold H.J. 6. Facioal eacio-disio eqaios Asophysics ad space sciece aea R.K. Mahai A.M. ad Haold H.J olio o geealized acioal eacio-disio eqaios Asophysics ad space sciece aea R.K. Mahai A.M. ad Haold H.J. Reacio disio syses ad oliea waves Asophysics ad space sciece aea R.K. aea R. ad Kalla.L Copaioal solio o a acioal geealizaio o he chödige eqaio occig i qa Mechaics Appl. Mah. Cop aea R.K. aea R. ad Kalla.L. olio o space-ie acioal chödige eqaio occig i qa Mechaics Fac. Calc. Appl. Aal. 3 o Toigh A. Poailiy sce o ie acioal chödige eqaio Aca Physica Poloica A 6 o Wia A Ue de daeal saz i de heoie de ioe. Aca Mah