Quantum bouncer with dissipation

Transcription

1 ENSEÑANZA REVISTA MEXICANA DE FÍSICA E5 ) DICIEMBRE 006 Quatu boucer with dissipatio G. López G. Gozález Departaeto de Física de la Uiversidad de Guadalajara, Apartado Postal 4-137, Guadalajara, Jalisco, México. Recibido el 5 de abril de 005; aceptado el 17 de eero de 006 Effects o the spectra of the quatu boucer due to dissipatio are give whe a liear o quadratic dissipatio i the velocity of the particle is tae ito accout. Classical costats of otio Hailtoias are deduced for these systes their quatized eigevalues are estiated through perturbatio theory. Differeces were foud coparig the eigevalues of the costats of otio the eigevalues of the Hailtoias. The cases whe the dissipatio paraeters go to zero are copared with the odissipative cases. Keywords: Classical echaics; discrete systes; foralis. Los efectos e el espectro del rebotador cuático debido a la disipació so dados cuo ua disipació lieal o cuadrática e la velocidad de la partícula es toada e cueta. Costates de oviieto clásicas y Hailtoiaos se deduce para estos sisteas y sus eigevalores cuatizados so estiados ediate la teoría de perturbacioes. Se ecuetra diferecias coparo los eigevalores de las costates de oviieto y los eigevalores de los Hailtoiaos. Los casos cuo los paráetros de disipació se hace cero so coparados co los casos o disipativos. Descriptores: Mecáica clásica; sisteas discretos; foraliso. PACS: i; Ca 1. Itroductio Dissipative systes have bee oe of the ust subtle difficult topics to deal with i classical [1] quatu physics []. I geeral, costructig a cosistet Lagragia Hailtoia forulatio for a give dissipative syste ca be a big challege [3]. There are basically two approaches to studyig dissipative systes. The first oe tries to brig about dissipatio as a result of averagig over all the coordiates of the bath syste, where oe cosiders the whole syste as coposed of two parts, our origial coservative syste the bath syste, which iteracts with the coservative syste causes the dissipatio of eergy) i it [4]. This approach has its ow value will ot be followed or discussed here. The secod approach cosiders that the bath syste produces a average effect o our iitially coservative syste which is expressed as a additioal exteral velocity-depedet force actig o the coservative syste trasforig it ito a dissipative syste with this velocity-depedet force. The resultig classical dissipative syste thus cotais this pheoeological or theoretical) velocity-depedet force. The, the questio arises over its cosistet Lagragia Hailtoia foralis the cosequeces of its quatizatio. This approach, i additio, allows us to study test the Hailtoia approach for quatu echaics its cosistecy [5], is the approach we will follow i this paper. A syste which has attracted our attetio for dissipatio study through the above approach is the quatu boucer. The quatu boucer [6] is the quatizatio of the otio of a particle which is attracted by the costat gravity force, that is, close to surface of the earth. This particle hits a perfectly reflectig surface, producig the boucig effect. This syste, with a additioal dissipatio force, is particular iportace because of its potetial experietal realizatio. This dissipative syste has bee studied very little, util ow usig the first approach etioed above [7]. We will assue that the exteral velocity-depedet force has liear quadratic depedece with respect to the velocity. This approach gives us the opportuity to chec the ature of quatizatio usig the Hailtoia or costat of otio associated with the syste, that is, usig the usual quatizatio of the geeralized liear oetu or usig the quatizatio of the velocity. This cosideratio is particularly iterestig i dissipative systes, sice oe ca ot always fid a Hailtoia as a fuctio of the variable positio liear oetu [8]; that is, velocity v ca ot always be ow explicitly i ters of liear oetu p positio x of the particle through the relatio p = L/ v, where L is the Lagragia of the syste. This paper is orgaized as follows: we preset the classical study for the dissipative syste cosiderig the liear quadratic velocitydepedet force. The costat of otio, the Lagragia, the Hailtoia of the syste are derived, we give their expressios up to secod order i the dissipatio paraeter. We preset the odificatio for the eigevalues of the quatu boucer, whe this dissipatio is tae ito accout, for the above approxiated wea dissipatio) costat of otio Hailtoia usig quatu perturbatio theory. Fially, coclusios soe discussios of our results are give.. Classical liear dissipatio The otio of a particle of ass uder a costat gravitatioal force a liear dissipative force is described by the equatio d x = g v, 1) dt

2 QUANTUM BOUNCER WITH DISSIPATION 17 where x is the positio of the particle, g is the costat acceleratio due to the earth s gravity, is the paraeter which characterizes the dissipatio, v = dx/dt is the velocity of the particle. A costat of otio of the autooous syste 1) is a fuctio K = K x, v) satisfyig the equatio [9] v K x g + ) v K = 0. ) v The solutio of this equatio, such that li K = v / + gx 0 the usual total eergy for the o dissipative syste), is give by K = gv g ) l 1 + v ) + gx. 3) g The Lagragia associated with 1) ca be obtaied usig the ow expressio [5] K x, v) L = v v dv, 4) producig the followig Lagragia: L = gv l 1 + v ) g g ) + l 1 + v ) gx gv g. 5) Therefore, the geeralized liear oetu Hailtoia are give by p = g l 1 + v ) 6) g H = g ) ) ) p exp 1 g g p +gx. 7) At two orders i the dissipatio paraeter, oe has the costat of otio, the Lagragia, the geeralized liear oetu, Hailtoia give as K = 1 v + gx 3g v3 + L = 1 v gx 6g v3 + 4g v4, 8) 1g v4, 9) p = v g v + 3g v3, 10) H = p + gx + 6g p g p4. 11) Oe should ote that all these quatites go to the odissipative case whe the dissipative paraeter goes to zero. The costat of otio 3) or 8a) the Hailtoia 7) or 8d) brig about the dapig boucig effect o spaces x, v) x, p). The dissipative paraeter ca be deteried by easurig the velocity v o at the reflectig surface x = 0) the easurig its axiu displaceet x ax v = 0). By equalig the value of the costat of otio i both situatios, oe obtais the expressio gv o g ) l 1 + v ) o g where the paraeter ca be foud. 3. Classical quadratic dissipatio = gx ax, 1) I this case, the otio of the particle is described by the equatio d x = g γv v, 13) dt where γ represets a dissipatio costat which, of course, is differet fro the previous case. Proceedig i the sae way as we did for the liear case, the costat of otio, Lagragia, geeralized liear oetu, Hailtoia are give by K ± = 1 v exp ± γx L ± = 1 v exp ± γx p ± =v exp ± γx ) ± g exp ± γx ) ) 1, 14) γ ) g exp ± γx ) ) 1, 15) γ ), 16) H ± = p ± exp γx ) ± g exp ± γx ) ) 1, 17) γ where the upper sig correspods to the case v 0, the lower sig correspods to the case v < 0. These equatios were already give i Ref. 10, the odissipative case is obtaied whe the dissipative paraeter goes to zero. The dapig effect of the boucig particle i the space x, v) ca be traced i the followig way: startig with the iitial coditio x o = 0 v o > 0, for exaple, the costat of otio K + is deteried, K + = v o/. The, the axiu distace x ax v = 0) is calculated fro the expressio K + = g/γ)expγx ax /) 1), which helps to calculate the costat K, K = g/γ) exp γx ax /) 1 ). This K is ow used to calculate the velocity at the turig poit x = 0), v1 = K /. Cosiderig a perfectly reflexig surface, the velocity of the boucig particle Rev. Mex. Fís. E 5 ) 006)

3 18 G. LÓPEZ AND G. GONZÁLEZ for the ext cycle is v 1 = v 1 v 1 < v o ), the above cycle is reproduced agai, so o. Startig with the sae iitial coditios, the trajectories i this space are oe below the other at ay tie, as the dapig factor is greater. The dapig effect i the space x, p) ca be aalyzed siilarly through the Hailtoia approach. However, the trajectories startig with the sae iitial coditios i this space are ot below the other all the tie, as the dapig factor is greater. This strage effect is due to the chage i sig i 11d) with respect to 11a), produced by the positio velocity depedece of the expressio 11c). To deterie the costat γ through the costat of otio, oe ca start with the iitial coditios x o = 0, v o > 0) ca deterie the costat of otio K + = vo/. The, oe ca easure the axiu displaceet x ax v = 0) solve γ fro the equatio 1 v o = g exp γ γxax ) ) 1. 18) Up to secod order i the dissipatio paraeter, oe has, fro 11a) to 11d) the costat of otio, the Lagragia, the geeralized liear oetu, the Hailtoia give by K ± = 1 v + gx ± γ[v x + gx ] +γ [v x / + gx 3 /3], 19) L ± = 1 v gx ± γ[v x gx ] +γ [v x / gx 3 /3], 0) p ± = v ± γ[vx] + γ [vx /], 1) H ± = p + gx γ[p x/ gx ] +γ [p x / 3 + gx 3 /3]. ) 4. Quatizatio of the costat of otio Equatios 8a) 13a) ca be writte as Kx, v) = K o x, v) + V x, v), 3) where K o is the costat of otio without dissipatio K o x, v) = 1 v + gx, 4) V taes ito accout the dissipatio factors ) ) v 3 3g + v 4 4g V x, v) = γ [ [ v x + gx ] + γ v x ] + gx3 3 liear case) quadratic case) 5) The quatizatio of 14) ca be carried out through the associated Schrödiger s equatio of this costat of otio i Ψ t = K x, v)ψ, 16) where Ψ = Ψx, t) is the wave fuctio, is the Pla costat divided by π, K = Ko + V is a Heritia operator associated to 17), v is the velocity operator defied as v = i x. 6) Sice Eq. 16) represets a statioary proble, the usual propositio Ψx, t) = exp ie K t/ ) ψx) trasfors 16) to a eigevalue proble: K o + V )ψ = E K ψ. 7) Taig the operator V as a perturbatio of the costat of otio K o, oe ca calculate a approxiate solutio to the proble 18) through perturbatio theory. The solutio of the eigevalue proble K o ψ 0) = E 0) ψ 0) 8) is well ow [6], with ψ 0) beig the eigefuctio give by ψ 0) = Aiz z ) Ai z ), 9) where Ai Ai are the Airy fuctio its first differetiatio, z is its th-zero Ai z ) = 0) which ocurres for a egative arguet oly. z is the oralized variable z = x/l g with l g = / g ) 1/3, z is related to the eigevalue E 0 through the expressio z = E0). 30) gl g Up to secod order i perturbatio theory, the eigevalues of 18) are give i Dirac otatio [11]) as E K = E 0) + V + V E 0) 0), 31) Rev. Mex. Fís. E 5 ) 006)

4 where z = ψ 0). Usig the Heritia operators QUANTUM BOUNCER WITH DISSIPATION 19 v x = v x + vx v + x v )/3 v x = v x + vx v + x v + x v x + x vx v + vx vx)/6 for the associated expressios o 5b), usig the relatios x s = l s g z s d s /dx s = lg s d s /dz s for ay iteger s, oe has see appedix for a list of atrix eleets) E K = E 0) + l gz gl3 g γ 1gl gz 15 + γ 1/ + gl g /E 0) E 0) z3 105 E 0) 0) ) gl 3 g ) + 4g l 4 g a, liear) quadratic) 3) where a is a real uber give by a = 1 z z z ) + z z ) 3 z z ) 9. 33) Note that for the liear dissipative case, there is o real cotributio at a first approxiatio, for the quadratic dissipative case, the first order cotributio depeds o whether the particle is ovig up -) or dow -). Withi a full cycle, this first order correctio is cacelled out the secod order cotributio reais. Of course, for the approxiatio 3a) to be valid, oe ust assue that the secod ter which aes a restrictio o the possible value of the dissipative paraeter. Of course, whe the paraeters of dissipatio go to zero, oe gets the odissipative eigevalues. of this expressio ust be uch less tha E 0) 5. Quatizatio of the Hailtoia Equatios 8d) 13d) ca be writte as Hx, p) = H o x, p) + W x, p), 34) where H o is the Hailtoia without dissipatio, W has the dissipatio ters, H o x, p) = p + gx, 35) W x, p) = γ ) ) p 3 6g + p g [ ] [ ] p x gx + γ p x 3 + gx3 3 liear) quadratic) 36) It is ecessary to etio that the quatizatio of soe systes for quadratic dissipatio has bee solved by differet authors [10,1] but perfectly reflexig wall potetial, { for x < 0 Ṽ x) = gx for x 0. 37) Moreover, the solutio give i Ref. 10 is sigular whe the dissipatio paraeter goes to zero. Therefore, we thi it worth while to ae aalysis of the quatizatio for sall orders i the paraeter γ. For the usual Shrödiger quatizatio approach, oe has the statioary equatio i Ψ t = Ĥx, p)ψ, 38) where Ĥ is the Hailtoia operator associated to 4), p is the usual liear oetu operator p = i / x. Equatio 7) is trasfored to a eigevalue proble, Ĥψx)=E H ψx), through the propositio Ψx, t) = exp ie H t/ ) ψx), sice the Hailtoia Ĥ is give by Ĥ = Ĥo + Ŵ, where the solutio of the equatio Ĥ o ψ 0) = E 0) ψ 0) 39) Rev. Mex. Fís. E 5 ) 006)

5 130 G. LÓPEZ AND G. GONZÁLEZ is give by 0) 1). Perturbatio theory ca be used to deterie the approxiate values of the eigevalues E H expressio )). Usig the Heritia operators p x = p x + px p + x p )/3 p x = p x + px v + x p + x p x + x px p + px px)/6 for the associated expressios o 5b), oe gets E H = E 0) + l gz gl3 g ± γ 4gl gz 15 + γ 1/ + gl g /E 0) E 0) z3 105 E 0) 0) ) gl 3 g + 4g l 4 g ) a liear), quadratic) where a is give by 3b). Oe ust ote here too that, whe the paraeters of dissipatio go to zero, oe gets the usual eigevalues for the odissipative syste. As oe ca see fro 3) 9), there is a differece betwee the eigevalues associated with the costat of otio those associated with the Hailtoia. Their relative differece, δe = E H E K )/E 0), is give by 1 l g z 6 6 δe g 4 l g 1/ + gl g /E 0) 0) ) 9 z liear) E 0) 0) = 41) E 0) 40) ± γ 16gl gz 15, quadratic) 6. Coclusio The classical quatu proble of a particle boucig o a hard surface uder the ifluece of gravity subject to a liear quadratic velocity dissipative force were treated usig the costat of otio the Hailtoia of the syste. Expressios 3) 11a) give us the expected dapig behavior of the particle i the x, v) space, but the expressios 7) 11d) show a uexpected behavior i the x, p) space two trajectories i this space, for dissipative paraeters oe bigger tha the other, do ot follow oe uder the other all the tie). I additio, all our expressios are reduced to the odissipative case whe the paraeters of dissipatio go to zero. For the quatu case, we have aalyzed the eigevalues for the costat of otio Hailtoia up to the secod order i the dissipatio paraeter usig perturbatio theory. Relatio 30) tells us that that there is a differece whether the costat of otio or the Hailtoia is quatized, it suggests that aybe oe could see this differece experietally. Fially, oe ust observe that for the full liear case 7), it is possible to solve exactly the Shrödiger equatio i the oetu represetatio, this will be aalyzed i a future paper. Appedix We show here a list of soe atrix eleets fro Ref. 6 soe others calculated fro the sae referece a correctio of sig has bee ade to soe atrix eleets). Give the fuctios 0), oe has = δ z = 3 z z = 8 15 z A.1) z = 1)++1 z z ) A.) z = 4 1)++1 z z ) 4 A.3) Rev. Mex. Fís. E 5 ) 006)

6 QUANTUM BOUNCER WITH DISSIPATION 131 z 3 = z3 z 3 = 4z + z ) 1) ++1 z z ) 4 A.4) d dz = 0 d 1)+ = dz z z A.5) d dz = 1 3 z d3 dz 3 = 1 d4 dz 4 = 1 5 z d 1)+ = dz d3 dz 3 = z z ) A.6) ) ) + z z A.7) d4 dz 4 = z z ) + 4 z z z ) z z ) 4 1) + A.8) 1. R. Glauber V.I. Ma o, Sov. Phys. JEPT ) 450; V.V. Dodoov, Hadro J ) 173; G. López, M. Murgía, M. Sosa, Mod. Phys. Lett. B ) 65.. M. Razavy, Ca. J. Phys ) 037; S. Oubo, Phys. Rev. A ) 776; G. López, M. Murgía, M. Sosa, Mod. Phys. Lett. B ) M. Mijatovic, B. Veljaosi, D. Hajduovic, Hadroic J ) 107; G. López, A. Phys ) 37; G. López G. Gozález, IL Nuo. Ci. B, ) A.O. Caldeira A.T. Leggett, Physica A ) 587; W.G. Uruh W.H. Zure, Phys. Rev. D ) 1071; B.L. Hu, J.P. Paz, Y. Zhag, Phys. Rev. D ) 843; G.P. Bera, F. Borgoovi, G.V. López, V.I. Tsifriovich, Phys. Rev. A ) G. López, Rev. Mex. Fís ) J. Gea-Baacloche, A. J. Phys ) 776; D.M. Goodaso, A. J. Phys ) R. Oofrio L. Viola, quat-ph/ v1 1996). 8. G. López, Rev. Mex. Fís ) G. López, Partial Differetial Equatios of First Order Their Applicatios to Physics World Scietific, 1999) p F. Negro A. Tartaglia, Phys. Lett. A ); F. Negro A. Tartaglia, Phys. Rev. A ) P.A.M. Dirac, The Priciples of Quatu Mechaics Oxford Sciece Publicatios, 199). 1. J.S. Borges, L.N. Epele, H. Fachiotti, Phys. Rev. A ) 3101; C. Stuceo D.H. Kobe, Phys. Rev. A ) 3565; M. Mijatovic, B. Veljaosi, D. Hajduovic, Hadroic J ) 107; B.W. Huag, Z.V. Gu, S.W. Qia, Phys. Lett A ) 03; M. Razavy, Hadroic J., ) 406. Rev. Mex. Fís. E 5 ) 006)