Gaussian wave packet solution of the Schrodinger equation in the presence of a time-dependent linear potential. M. Maamache and Y.

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1 Gussin wve pcket solution of the Schrodinger eqution in the presence of time-dependent liner potentil M. Mmche nd Y. Sdi Lbortoire de Physique Quntique et Systèmes Dynmiques, Fculté des Sciences,Université Ferht Abbs de Sétif, Sétif 9, Algeri rxiv:85.49v [qunt-ph] 4 My 8 Abstrct We rgue tht the wy to get the generl solution of Schrodinger eqution in the presence of timedependent liner potentil bsed on the Lewis-Riesenfeld frmework is to use Hermitin liner invrint opertor. We demonstrte tht the liner invrint proposed in p nd q is n Hermitin opertor which hs the Gussin wve pcket s its eigenfunction. PACS: 3.65.Ge, 3.65.Fd The time evolution of quntum system subject to sptilly uniform, time-dependent force hs ttrcted considerble interest ltely. The exct propgtorfor this system hs long been known [], s hve set of exct solutions the Volkov solutions) [-3]. Recent work [4 3] focuses further on exct solutions nd their properties. First, Guedes [5] obtined, by mens of the invrint opertor method introduced by Lewis nd Riesenfeld L-R) [4], specil solution for the time-dependent liner potentil. The ide is tht ny opertor stisfying the quntum Liouville-von Neumnn eqution provides its eigenstte s solution of the time-dependent Schrodinger eqution up to time dependent phse fctor. Lter on, Feng [6] followed method bsed on sptiotemporl trnsformtions of the Schrodinger eqution to get the plne-wve-type nd the Airy-pcket solutions where the one in Ref. [5] constitutes prticulr cse which corresponds to the so-clled stnding prticle cse in liner potentil [6]. However, Bekkr et l. [8] pointed out tht the Airy-pcket solution is in fct only superposition of the plne-wve-type solution. In his Comment [7], Buer lined tht the solution found by Guedes [5] is simply specil cse of the Volkov solution, with zero wve vector k, to the time dependent Schrodinger eqution describing nonreltivistic chrged prticle moving in n electromgnetic field. Dunkel nd Trigger [] considered the initil minimum-uncertinty Gussin for sinusoidlly time-dependent liner potentil. Bowmn [] investigted the time evolution of the generl quntum stte for the time-dependent liner system, which ws shown to be tht of free-prticle stte, plus n overll motion rising from the clssicl force. Sng Pyo Kim [] hs shown tht chrged free prticle in constnt nd/or oscillting electric field hs bounded Gussin wve pcket, which is coherent stte of one-prmeter nd time-dependent ground stte for the free-prticle Hmiltonin. From the test-function method, Gengbio Lu et l.[3] constructed n exct n wve-pcket-trin n GWPT) solution, whose center moves ccelertively long the corresponding clssicl trjectory. Very recently, Lun et l. [9] hve reexmined the liner invrint proposed by Guedes [5] nd indicted tht besides the solutions described in Refs. [5-8], Gussin wve pcket eigenfunction solution cn nturlly be derived from the LR method if non-hermitin liner invrint is used, nd they rgue tht this solution ws ruled out before, becuse the uthors in Refs. [ 5-8] ssumed in dvnce the liner L-R invrint It) = At)p+Bt)x+Ct) be Hermitin opertor. The uthors [9] stted tht in Ref.[5, 8], the Gussin wve pcket eigenfunction solution ws ruled out, becuse the liner LR invrint is supposed to be Hermitin opertor, nd the B prmeter ws set to zero for hermiticity reson. In fct this is not the reson for setting B =, in Ref. [8], it ws ment to justify the B = imposed in Ref. [5]. We will show nd we will correct in the present pper, tht if B is different from zero B ), we still obtin Gussin wve pcket GWP) solutions. In fct, we could find ll the results using the Hermitin invrint opertor liner in p nd q with ny conditions imposed to get physiclly cceptble solutions s ws done in Lun et l. pper [9]. In the ltter pproch, the uthors obtined solutions lbelled by complex prmeters nd complex eigenvlues, this is mbiguous.

2 We recll tht ccording to the theory of Lewis nd Riesenfeld [4], n invrint is n opertor tht must necessrily stisfy three requirements:. It is Hermitin.. It stisfies the Von Neumnn eqution. 3. Its eigenvlues re rel nd time-independent we signl tht the Hermiticity of the invrint is one of the essentil conditions which mkes the eigenvlues of the invrint time-independent). Furthermore, ny invrint stisfying these three requirements leds to complete set of solutions of the corresponding Schrodinger eqution. So, conventionl solution is constructed s liner combintion of these solutions. The problem is to find the solutions of the Schrodinger eqution, for the Hmiltonin i h ψx,t) = H ψx,t) ) t H x,p,t) = m p Ft)x, ) where Ft) is time-dependent function. According to the theory of Lewis nd Riesenfeld [4], solution of the Schro dinger eqution with timedependent Hmiltonin is esily found if nontrivil Hermitin opertor It) exists nd stisfies the invrint eqution di = I t + [I,H] =. 3) i h Indeed,thisequtionisequivlenttosyingthtifϕ λ x,t)isneigenfunctionofit)withtime-independent eigenvlueλ, wecnfindsolutionoftheschrodingerequtionintheformψ λ x,t) = [iα λ t)]ϕ λ x,t) where α λ t) stisfies the eigenvlue eqution for the Schrodinger opertor, h α λ t)ϕ λ = i [ h t ] H ϕ λ. 4) It turns out tht the time-dependent invrint opertor tkes the liner form [9] It) = At)p+Bt)x+Ct). 5) The invrint eqution is stisfied if the time-dependent coefficients re such tht At) = A B m t, Bt) = B, 6) Ct) = C A where A, B, C re rbitrry rel constnts. Fτ)dτ + B m Fτ)τdτ, 7) The eigensttes of It) corresponding to time-independent eigenvlues re the solutions of the eqution It)ϕ λ x,t) = λ ϕ λ x,t). 8) It is esy to see tht the solutions of Eq. 8) re of the form { [ i λ Ct))x B x ] ϕ λ x,t) =. 9) h At) Substituting Eq. 9) into Eq. 4) nd ccomplishing the integrtion, we obtin

3 α λ t) = α λ ) = α λ ) { λ Cτ)) m haτ) + ib dτ maτ) { ) λ Cτ)) A m haτ) dτ iln At), ) note tht the logrithmic term goes downstirs s time-dependent normliztion fctor in ψ λ x,t). Therefore the physicl orthogonl wve functions ψ λ x,t) solutions of the Schrodinger eqution ) re given by It is esy to verify tht { A ψ λ x,t) = At) i λ Ct )) hmat ) + { i h [ λ Ct))x B x At) ]. ) ψ λ ψ λ = δλ λ ). ) Furethermore, the evolution of the generl Schrodinger stte cn be writen s + gλ)ψ λ x,t)dλ, 3) where gλ) is weight function which determines the stte of the system such tht Ψx,t) is squre integrble i.e. Ψx,t) dx is time-independent finite constnt. In Ref. [8], it hs been shown tht the choice iλ 3) gλ) = 3 leds to Airy function solutions. Here we shll show tht one obtins generl Gussin wve-pcket solution by choosing weight function s gussin too, gλ) = ha π π λ ), 5) where is positive rel constnt. Substituting Eq. ) nd 5) into Eq. 3) nd ccomplishing the integrtion, we obtin the normlized Gussin solution s 4) hat) π +i { i h ) hmat ) [ Ct)x B x ] At) { [ i Ct ) hmat ) Ct ) hmat ) + x hat) 4+i ) hmat ) ] 6) Let us now evlute the men vlue of x, p nd the quntum coordinte nd momentum fluctution in the stte Ψx,t). After some lgebr we find tht Ct ) x = Ψt) x Ψt) = At) mat ), 7) which is nothing but the clssicl position x c t), nd 3

4 is clssicl momentum p c t). The position uncertinty x = x x = hat) nd the momentum uncertinty p = Ψt) p Ψt) = Ct) At) + B Ct ) mat ) 8) t + ) ) hmat ), 9) p = = p p h x 4 + ) 4 mat ) h B At) t + hmat ), ) leds to the uncertinty reltion t p x = h mat ) h B At) t + hmat ) ) h. ) We cn rewrite 6) s follows hat) i ) t hmat ) { i π x { [ i Ct)x B x ] h At) i Ct ) hmat ) ) hmat ) 4 x [x x ]. ) Morever, the time-dependent probbility density ssocited with this Gussin wve pcket is Gussin for ll times { Ψx,t) x x ) = π x x, 3) weseetht xrepresentsthewihofthewvepcketttimet. Itislsoredilyverifietthetime-dependent probbility density is conserved Ψx,t) dx = 4) Equtions ) nd 3) describe Gussin wve pcket tht is centered t x = x whose wih xt) Ct ) nd mat ) vries with time. So, during time t, the pcket s center hs moved from x = to x = At) hmat ) ). The wve pcket its wih hs nded from x = ha to xt) = At) x A + therefore undergoes distortion; lthough it remins Gussin, its wih brodens with time wheres its height, π x, decreses with time. Further, it should be noted tht the wih of the Gussin pcket does not depend on the externl force Ft). Thus the shpe of the wve pcket is not chnged by the externl force. This mens tht the externl force Ft) cts uniformly in the wve pcket. Acknowledgments: 4

5 We wish to thnk Professor A. Lydi for his help. References [] R. Feynmn nd A. Hibbs, Quntum Mechnics nd Pth Integrls New York: McGrw-Hill), 965). [] W. Gordon, Z. Phys. 4, 7 96). [3] D. Volkov, Z. Phys. 94, 5 935). [4] A. Ru nd K. Unnikrishnn, Phys. Lett. A, ). [5] I. Guedes, Phys. Rev. A 63, 34 ). [6] M. Feng, Phys. Rev. A 64, 34 ). [7] J. Buer, Phys. Rev. A 65, 36 ). [8] H. Bekkr, F. Benmir, nd M. Mmche, Phys. Rev. A 68, 6 3). [9] Pi-Gng Lun nd Chi-Shung Tng, Phys. Rev. A 7, 4 5). [] J. Dunkel nd S. A. Trigger, Phys. Rev.A 7, 5 5). [] G.E. Bowmn, J. Phys. A. 39, 57 6). [] S. P. Kim, J. Koren Phys. Soc. 44, 464 6). [3] Gengbio Lu, Wenhu Hi nd Lihu Ci, Phys. Lett. A 357, 8 6). [4] H. R. Lewis, Jr. nd W. B. Reisenfeld, J. Mth. Phys., ). 5