Development of the Schrodinger equation for attosecond laser pulse interaction with Planck gas

Transcription

1 Develoment of the Shrodnger equaton for attoseond laser ulse nteraton wth Plank gas M. Kozlowsk 1 * J. Marak Kozlowska 1 Josef Plsudsk Warsaw Unversty, Insttute of Eletron Tehnology Abstrat The reaton of the new artles by the nteraton of the ultrarelatvst ons,from Large Hadron Collder( LHC), and attoseond laser ulse oen new ossbltes for laser hyssts ommunty.in ths aer we roose the hyerbol Shrödnger equaton (HSE) for gas of the lassal artles.e. artles wth mass= Plank mass We dsuss the nluson of the gravty to the HSE The soluton of the HSE for a artle n a box s obtaned. It s shown that for artles wth m greater than M the energy setrum s ndeendent of the mass of artle. Key words: attoseond laser ulses, Shrodnger equaton, Plank artles, thermal roesses * orresondng author: e-mal: mroslawkozlowsk@aster.l 1

2 1. Modfed Shrödnger Equaton When M. Plank made the frst quantum dsovery he noted an nterestng fat [1]. The seed of lght, Newton s gravty onstant and Plank s onstant learly reflet fundamental roertes of the world. From them t s ossble to derve the haraterst mass M P, length L P and tme T P wth aroxmate values [1] L P = m T P = s M P = 10-5 g. The great value of the Plank mass s stll the oen queston n hyss and osmology. In ths aer we formulate the desrton of the thermal roesses n Plank gas,.e. the gas of the artles wth mass=plank mass When the laser ulse nterats wth eletron or nuleon gases the rmordal Plank artles an modfy the thermal roesses ntted by laser ulse. Frst of all the Plank artles wll modfy the mathematal struture of the Shrodnger equaton.in aer by C Muller the roblem of the ostron ar reaton by ollson of the ultrarelatvst ons from Large Hadron Collder ( LHC) wth attoseond hotons was roosed. [] In ths aer we nvestgate the queston: how gravty an modfy the quantum mehans,.e. the nonrelatvst Shrödnger equaton (SE). We argue that SE wth relaxaton term desrbes roerly the quantum behavor of artle wth mass m < M P and ontans the art whh an be nterreted as the lot wave equaton. For m M P the soluton of the SE reresent the strngs wth mass M P. The thermal hstory of the system (heated gas ontaner, semondutor or Unverse) an be desrbed by the generalzed Fourer equaton [3,4] q( t) = t K( t t' ) T ( t' ) dt'. thermal hstory dffuson (1) In Eq. (1) q(t) s the densty of the energy flux, T s the temerature of the system and K(t t ') s the thermal memory of the system K ( t t' ) K ( t t' ) = ex, () where K s onstant, and denotes the relaxaton tme. As was shown n []

3 Kδ ( t t') dffuson K = onstant wave Kt ( t') = K ( t t') ex damed wave or hyerbol dffuson. The damed wave or hyerbol dffuson equaton an be wrtten as: T T + D 1 T = T. For 0, Eq. (3) s the Fourer thermal equaton T = DT T and D T s the thermal dffuson oeffent. The systems wth very short relaxaton tme have very short memory. On the other hand for (3) (4) Eq. (3) has the form of the thermal wave (undamed) equaton, or ballst thermal equaton. In the sold state hyss the ballst honons or eletrons are those for whh. The exerments wth ballst honons or eletrons demonstrate the exstene of the wave moton on the latte sale or on the eletron gas sale. T D = T T. For the systems wth very long memory Eq. (3) s tme symmetr equaton wth no arrow of tme, for the Eq. (5) does not hange the shae when In Eq. (3) we defne: υ = D T, t t. veloty of thermal wave roagaton and λ = υ, (7) where λ s the mean free ath of the heat arrers. Wth formula (6) equaton (3) an be wrtten as (5) (6) 1 T 1 T + = υ υ T From the mathematal ont of vew equaton:. (8) 1 T υ + 1 T D = T 3

4 s the hyerbol artal dfferental equaton (PDE). On the other hand Fourer equaton 1 T = T (9) D and Shrödnger equaton Ψ = (10) Ψ m are the arabol equatons. Formally wth substtutons t t, Ψ T. (11) Fourer equaton (9) an be wrtten as Ψ = D Ψ (1) and by omarson wth Shrödnger equaton one obtans and DT = (13) m D T =. m (14) Consderng that D T = υ (6) we obtan from (14) =. m υ h (15) Formula (15) desrbes the relaxaton tme for quantum thermal roesses. Startng wth Shrödnger equaton for artle wth mass m n otental V: (16) Ψ = Ψ+ V Ψ m and erformng the substtuton (11) one obtans T = T m VT (17) T = T m V T. (18) Equaton (18) s Fourer equaton (arabol PDE) for = 0. For 0 we obtan T T V T T = + + =, t m m υ (19) (0) 4

5 or 1 υ T m T Vm + + T = T. Wth the substtuton (11) equaton (19) an be wrtten as Ψ = VΨ Ψ. (1) m t The new term, relaxaton term () desrbes the nteraton of the artle wth mass m wth sae-tme.when the quantum artle s movng through the quantum vod t s nfluened by saterrng on the vrtual eletron=-ostron ars The relaxaton tme an be alulated as: 1 1 ( e ), (3) 1 = Plank where, for examle e- denotes the satterng of the artle m on the eletron-ostron 17 ar ( ~ 10 s) and the shortest relaxaton tme Plank s the Plank tme ( e Plank ~ s). From equaton (3) we onlude that Plank and equaton (1) an be wrtten as where Ψ = VΨ Ψ (4) m t, Plank Plank 1 1 G = 5 =. (5) M In formula (5) M s the mass Plank. Consderng Eq. (5), Eq. (4) an be wrtten as V m M Ψ = Ψ+ Ψ Ψ Ψ + Ψ. M M t (6) The last two terms n Eq. (6) an be defned as the Bohman lot wave M Ψ M = 0, (7).e. 5

6 1 Ψ = 0. (8) It s nterestng to observe that lot wave Ψ does not deend on the mass of the artle. Wth ostulate (8) we obtan from equaton (6) Ψ = Ψ+ V Ψ Ψ (9) m M and smultaneously M Ψ M = 0. (30) In the oerator form Eq. (1) an be wrtten as ˆ ˆ 1 ˆ E, m M E = + (31) where Ê and ˆ denote the oerators for energy and momentum of the artle wth mass m. Equaton (31) s the new dserson relaton for quantum artle wth mass m. From Eq. (1) one an onludes that Shrödnger quantum mehans s vald for artles wth mass m «M P. But lot wave exsts ndeendent of the mass of the artles. For artles wth mass m «M P Eq. (9) has the form Ψ = Ψ+ V Ψ m. (3) In the ase when m M Eq. (9) an be wrtten as Ψ = Ψ + VΨ, (33) M but onsderng Eq. (30) one obtans or Ψ = + VΨ (34) M M Ψ + VΨ = 0. (35). Gravty and Shrödnger Equaton 6

7 Classally, when the nertal mass m and the gravtatonal mass m g are equated the mass dros out of Newton s equaton of moton, mlyng that artles of dfferent mass wth the same ntal ondton follows the same trajetores. But n Shrödnger s equaton the masses do not anel. For examle n a unform gravtatonal feld [] Ψ = + mg gxψ (36) m x mlyng mass deendent dfferene n moton. In ths aragrah we nvestgate the moton of artle wth nertal mass m n the otental feld V. The otental V ontans all the ossble nteratons nludng the gravty. Ψ = VΨ Ψ (37) m t where the term, = (38) m desrbes the memory of the artle wth mass m. Above equaton for the wave funton Ψ s the loal equaton wth fnte nvarant seed, whh equals the lght seed n the vauum. Let us look for the soluton of the Eq. (6), V=0, n the form (for 1D) Ψ = Ψ( x t). (39) For 0,.e. for fnte Plank mass we obtan: µ Ψ ( x t) = ex ( x t) where the redued μ mass equals mm µ = m + M For m << M,.e. for all elementary artles one obtans μ = m and formula (40) desrbes the wave funton for free Shrödnger artles m Ψ( x t) = ex ( x t) (40) (41) (4) (43) For m >> M, μ = M M Ψ ( x t) = ex ( x t) (44) 7

8 From formula (44) we onlude that Ψ ( x t) s ndeendent of mass m. In the ase m < M from formulae (41) and (4) one obtans m µ = m 1 M m m m m Ψ( x t) = ex ( x t) ex x t M (45) In formula (45) we ut and obtan m k = m ω = (46) m ( kx ωt ) ( kx ωt ) M Ψ( x t) = e e (47) As an onluded from formula (47) the seond term deends on the gravty 1 m m G ex ( kx ωt) = ex ( kx ωt) M where G s the Newton gravty onstant. It s nterestng to observe that the new onstant, α G, (48) m G α G = (49) s the gravtatonal onstant. For m = m N nuleon mass 39 α G = (50) 3. The Partle n a Box Ths quantum mehanal system by a artle of mass m onfned n a onedmensonal box of length L. For the artle to be onfned wthn regon II, the otental energy outsde (regons I and III) s assumed to be nfnte. In order to understand further ths system, we need to formulate and solve the Shrödnger equatons (6). 8

9 Ψ 1 = Ψ+ VΨ Ψ+ Ψ. (51) m M M Consderng the lot wave equaton one obtans 1 Ψ Ψ = 0 (5) where Ψ = Ψ+ Ψ V, (53) µ mm µ = m + M. In regon II, V( x) = 0 In regons I and III, V( x) = Ψ+ VΨ= EΨ Ψ+ VΨ= EΨ µ µ Ψ+ 0 = EΨ Ψ+ Ψ= EΨ µ µ Ψ= EΨ Ψ= ( E ) Ψ µ µ For regons I and III (outsde the box), the soluton s straghtforward, the wavefunton Ψ s zero. For regon II (nsde the box), we need to fnd a funton that regenerates tself after takng ts seond dervatve. Ψ= EΨ µ µ E µ E Ψ= Ψ= k Ψ, where we defne k =. Perfet anddates would be the trgonometr sne and osne funtons. Ψ ( x) = Aos( kx) + B sn( kx). (56) To further refne the wavefunton, we need to mose boundary ondtons: At x=0, the wavefunton should be zero. Ψ (0) = Aos( k0) + Bsn( k0) = A 1+ B 0. (57) Equaton an only be true f A = 0: Ψ ( x) = B sn( kx). In addton, at x=l, the wavefunton should also be zero. (54) (55) 9

10 nπ Ψ (0) = 0 = B sn( kl). True when kl = nπ or k = : L nπ Ψ ( x) = Bsn( x), n= 1,,3,... L Gong bak to the Shrödnger's equaton, we an then formulate the energes µ E kl = nπ and k = µ E nπ µ E n π = = L L E n nπ nh h L 8 L = = sne =. µ µ π (58) (59) Thus, the alaton of the Shrödnger equaton to ths roblem results n the well known exressons for the wavefuntons and energes, namely: nπ x Ψ n = sn L L and nh En =. (60) 8µL From formula (60) we onlude that for heavy lassal artles,.e. for m >> M energy setrum of the artle n the box s ndeendent of the mass of artle E n nh =. (61) 8M L 4. Conlusons The thermal roesses ntated by nteraton of the attoseond laser ulse wth relatvst artles rodued by LHC an be desrbed by the hyerbol Shrodnger equatonthe hyerbol Shrödnger equaton when aled to the study of artle n a box offers new ture of the lassal extenson of the quantum mehans. The lassal artles,.e. artles wth m >> M have dstnt energy setrum whh s ndeendent of ts mass. 10

11 Referenes [1] M. Plank, The Theory of Heat Radaton, Dover Publshers, USA, [] C. Muller, Nonlnear Bethe- Hetler ar reaton wth attoseond laser ulses at LHC, arxv: [ he-h] [3] M. Kozlowsk, J. Marak Kozlowska, Thermal Proesses Usng Attoseond Laser Pulses, Srnger, USA, 006. [4] M.Kozlowsk, J Marak-Kozlowska, From Femto-to Attosene and beyond, Nova Sene ublshers, USA<