Solution of the Schrödinger Equation for a Linear Potential using the Extended Baker-Campbell-Hausdorff Formula

Transcription

1 Appl. Mah. Inf. Sc. 9, No. 1, Appled Mahemacs & Informaon Scences An Inernaonal Journal hp://dx.do.org/ /ams/0901 Soluon of he Schrödnger Equaon for a Lnear Poenal usng he Exended Baker-Campbell-Hausdorff Formula Francsco Soo-Egubar and Hécor M. Moya-Cessa Insuo Naconal de Asrofísca, Ópca y Elecrónca, INAOE. Lus Enrque Erro 1, Sana María Tonanznla, San Andrés Cholula, Puebla, 780, Méxco Receved: Mar. 01, Revsed: Jun. 01, Acceped: Jun. 01 Publshed onlne: 1 Jan. 015 Absrac: The me-dependen Schrödnger equaon s solved for a lnear poenal usng operaonal mehods; n parcular, an exenson of he Baker-Campbell-Hausdorff formula s osed and used. Several nal condons are consdered. A closed form for he Wgner funcon s presened. The resuls can be exended o he propagaon of an elecromagnec feld n he paraxal approxmaon. Keywords: Schrödnger equaon, lnear poenal, Zassenhaus formula, Baker-Campbell-Hausdorff formula, Wgner funcon 1 Inroducon The man problem of non-relavsc quanum mechancs s he soluon of he Schrödnger equaon; several exac analycal, approxmaed analyc and numercal mehods have been nvened for hs purpose 1,,]. Beween he exac analycal mehods, we can coun he ones based on operaor echnques; such mehods are relavely new and can gve very smple soluons, o oherwse complcaed approaches, 5, 6, 7]. Tha s he case of he lnear poenal; he soluon of he Schrödnger equaon for he lnear poenal s very well known, bu has been negleced n he quanum mechancs books 8, 9]. The me dependen soluon s usually done n erms of he egenfuncons of he Hamlonan, he Ary funcons, as an negral over he connuous egenvalue, he energy; however, an lc soluon for a gven nal condon s no normally presened. In hs work, we use operaor echnques o fnd an exac analyc soluon of he me-dependen Schrödnger equaon wh a lnear poenal Vx = Fx. We solve he problem for several nal condons. To do ha, we use a very easy and drec generalzaon of he Baker-Campbell-Hausdorff formula, 7, 10], ha s derved from he Zassenhaus formula 11, 1, 10]. Ths exended Baker-Campbell-Hausdorff formula allows us o dsenangle he evoluon operaor, and o analyze he acon of hs operaor over nal condons adequaely wren. The lc soluon ha s found, allows us o wre a closed form for he Wgner funcon n erms of he Fourer ransform of he nal condon. The Wgner funcon can hen be calculaed for he nal condons ha we presen; as he resuls are very cumbersome, we presen only he case when he nal condon s an Ary funcon. The Schrödnger equaon for he lnear poenal mmcs exacly he paraxal equaon n wo dmensons 1, 1, 15]; so, our resuls works perfecly well for beam propagaon under he nfluence of a lnear graden refracve ndex 16]. The formal soluon of he Schrödnger equaon for a lnear poenal usng operaors The one dmensonal Schrödnger equaon for a lnear poenal Vx=Fx s ψx, ˆp = + F ˆx ψx,, 1 where ˆx and ˆp are wo Herman operaors.e., wo non-commung varables wh he commuaon relaon Correspondng auhor e-mal: fegubar@naoep.mx c 015 NSP

2 176 F. Soo-Egubar, H. M. Moya-Cessa: Soluon of he Schrödnger Equaon for... ˆx, ˆp] = 1,,]. We wll use a un sysem where h = 1 and he mass m=1. As he Hamlonan s me ndependen, equaon 1 can be negraed wh respec o me, and he formal soluon found as { } ˆp ψx,= + F x ψx,0, where ψx, 0 s he nal condon, namely, he sae of he sysem a me = 0. The exended Baker-Campbell-Hausdorff formula The Zassenhaus formula esablshes 11, 1, 10] ha e λ ˆX+Ŷ = e λ ˆX e λ Ŷ e λ C ˆX,Ŷ e λ C ˆX,Ŷ e λ C ˆX,Ŷ..., where ˆX and Ŷ are wo operaors, ha s, wo non-commung varables, and C ˆX,Ŷ = 1 ˆX,Ŷ ], formula. The lef orened verson of he Baker-Campbell-Hausdorff formula s e λ ˆX+Ŷ = e λ ˆX,Ŷ] e λ Ŷ e λ ˆX 9 and he lef orened exended Baker-Campbell-Hausdorff formula s e λ ˆX+Ŷ = e λ!ŷ, ˆX,Ŷ]]+ ˆX, ˆX,Ŷ]] e λ! ˆX,Ŷ] e λ Ŷ e λ ˆX. 10 Applcaon of he exended Baker-Campbell-Hausdorff formula o he Schrödnger equaon wh a lnear poenal We apply now resson 10 o he formal soluon. For ha, we choose λ =, ˆX = ˆp, and Ŷ = F x; we have ] ˆX,Ŷ = F ˆp, 11 ˆX, ˆX,Ŷ ]] = 0, Ŷ, ˆX,Ŷ ]] = F, 1 and Ŷ, Ŷ, ˆX,Ŷ ]]] = 0, 1 C ˆX,Ŷ = 1 Ŷ, ˆX,Ŷ ]] ˆX, ˆX,Ŷ ]], 5 C ˆX,Ŷ = 1 Ŷ, Ŷ, ]]] ˆX,Ŷ + Ŷ, ˆX, ˆX,Ŷ ]]] 8 1 ˆX, ˆX, ˆX,Ŷ ]]], 6 and so on. The general resson for C ˆX,Ŷ s very complcaed and we don need here See 11] and references here n. If Ŷ, ˆX,Ŷ ]] = 0 and ˆX, ˆX,Ŷ ]] = 0, C ˆX,Ŷ ] = 0, all he subsequen C s are also zero, and we go he well known Baker-Campbell-Hausdorff formula e λ ˆX+Ŷ = e λ ˆX e λ Ŷ e λ ˆX,Ŷ]. 7 If a leas one of he wo Ŷ, ˆX,Ŷ ]] or ˆX, ˆX,Ŷ ]] are Ŷ, Ŷ, dfferen from zero, and ]]] ˆX,Ŷ = 0, Ŷ, ˆX, ˆX,Ŷ ]]] = 0, ˆX, ˆX, ˆX,Ŷ ]]] = 0, only C and C wll be dfferen from zero, and all he ohers subsequen C s wll be zero, obanng he exended Baker-Campbell-Hausdorff formula e λ ˆX+Ŷ = e λ ˆX e λ Ŷ e λ! ˆX,Ŷ] e λ!ŷ, ˆX,Ŷ]]+ ˆX, ˆX,Ŷ]]. 8 There s also a lef orened verson of he Zassenhaus formula ha s as useful as he normal one, ha gves us lef orened versons of he Baker-Campbell-Hausdorff formula and of he exended Baker-Campbell-Hausdorff Ŷ, ˆX, ˆX,Ŷ ]]] = 0 1 ˆX, ˆX, ˆX,Ŷ ]]] = 0 15 so acually, we can apply he lef orened exended Baker-Campbell-Hausdorff formula 10. Usng 11 and 1, we fnd C = F ˆp and C = F, so F ψx,= F ˆp ˆp ψx,0. Fx 16 Usng he Hadamard lemma,7,10], s easy o show ha ˆp ψx,0 = ψx ˆp,0 and ha for an arbrary well behaved funcon f, we have F ˆp f x= f x+ F ; hus, ψx,= F F + 6x ] ψ x+ F 6 ˆp, Wrng he nal condon as a Fourer ransform Now, we wre he nal condon n erms of s Fourer ransform;.e., we wre ψx,0= dvyv xv, 18 c 015 NSP

3 Appl. Mah. Inf. Sc. 9, No. 1, / and we subsue n equaon 17, obanng ψx,= dvyv vf πvx ˆp]1; 19 he operaor n he las onenal can be dsenangled usng he Baker-Campbell-Hausdorff formula, 7, 10], and afer some rval algebra one ges F F + 6x ] ψx,= where dvyve vf 6 { ]} π x+ vv F, 0 Yv= dξ ψξ,0ξ v. 1 6 Some parcular nal condons 6.1 Inal condon: ψx, 0 = A kx We rea now he case when he nal sae of he sysem s a plane wave, namely ψx, = 0=e kx. I s a very easy exercse n hs case o fnd ha ψx,=a k x+ Fk F 6 Fx k ]. Ths soluon sasfes he Schrödnger equaon 1 and he nal condon ψx, = 0=e kx, so s he soluon. 6. Inal condon: ψx,0=ae ax b When he nal condon s a Gaussan funcon ψx,0=a ax b ], he Fourer ransform s also a Gaussan, and subsung n resson 0, we oban afer easy, bu cumbersome, calculaons where ψx,= A 1+a g 1 +g +g ], g 1 = 1ab + 1abF + af F, 1a and g = x ab+af F g = a, 5 ax 1 a. 6 Fg. 1: The soluon ψx,, when he nal condon s ψx,0=ae ax b, wh F = 1, A=1, a=1, and b=. Agan s no dffcul o show ha he nal condon s fulflled and ha equaon 1 s sasfed. In Fgure 1, we show he squared amplude as a funcon of he poson x and as a funcon of me, when F = 1, A=1, a=1, and b=. We observe ha he peak of ψx, goes o lower values of x; f he sgn of F s nvered, he moon of he peak s reversed, gong n ha case o he greaer values of x, as eced. 6. Inal condon: ψx, 0 = Aax In hs case, we ge Fv= a 1 8π v 0 ake us o ψx,= 1 dv π a v a F ] 6 + Fx F + xv+ a v v ha afer some rval algebra can be cas as, ha nsered n ], 7 ψx,= 1 F π a 6 + Fx a6 ] dv a v a +x a ] + F v. Usng he negral represenaon 17, 18] Ax= 1 π 8 u ] + xu du, 9 c 015 NSP

4 178 F. Soo-Egubar, H. M. Moya-Cessa: Soluon of he Schrödnger Equaon for... ha s vald for x real, we fnally arrve o { ψx,= a F a F 6x ]} 1 A ax 1 ] a a F 0 and h = a a c F + x ] c+. 5 The behavor of he soluon n hs case s presened n Fgure dsplays ψx, when F = 1, and a=1. As n Fg. : The soluon ψx,, when he nal condon s ψx,0=e cx Aax, wh F = 1, c=1, and a=1. Fg. : The soluon ψx,, when he nal condon s ψx,0=aax, wh F = 1 and a=1. he prevous case, he drecon of he ws of he maxmum of he probably dsrbuon s nvered when we change he sgn of F. 6. Inal condon: ψx,0=e cx Aax Frs, he Fourer ransform s calculaed usng he convoluon heorem; second, s subsued n 0; hrd, he negral represenaon 9 of he Ary funcon s used, and fnally 1 ψx,= h 1 +h +h ]Ah ] 1+c where h 1 = 1c a 6 a F1+c+F ] cf c+ 1c 1 h = x a + c F 6cF F c, h = cx 1 c, Fgure for F = 1, c=1, and a=1. The same observaon made n he prevous cases abou he behavor of ψx, wh respec o he sgn of F s vald. 7 Inal condon: ψx,0=j n ax When he nal condon s a Bessel funcon of he frs knd, we can no use drecly equaon 0 because we do no know he Fourer ransform, so we go back o resson 16 and we wre 18,19] J n ax= 1 π nτ axsnτ]dτ, 6 π o oban ψx,= F ˆp 1 π F ˆp Fx e nτ axsnτ]dτ. 7 Followng he same procedure ha ook us from equaon 16 o equaon 17, we arrve o ψx,= 1 π F ] 6 Fx { nτ af snτ 8 xasnτ a sn ]} τ dτ c 015 NSP

5 Appl. Mah. Inf. Sc. 9, No. 1, / From here, we make sn τ = 1 1 cosτ ] = 1 eτ e τ ], o ge ψx,= 1 π a a 8 eτ F ] 6 Fx { ]} F nτ + x asnτ ] a ] 8 e τ dτ, 9 we wre he las wo onenals nsde he negral n erms of her Taylor seres, ψx,= F 6 Fx a ] j,k=0 1 j!k! a 8 j+k 1 {n+j kτ π ]} F + x asnτ dτ. 0 Recallng 6, we oban ψx,= F 6 Fx a ] 1 a j+k J n+ j k a j!k! 8 j,k=0 ] x+ F. 1 Changng he ndex n he j sum o M = j k, ψx,= F 6 Fx a ] 1 a M+k k=0 M= km+ k!k! 8 ] J n+m+k k a x+ F ; he M sum can be exended from, snce we are addng zeros o he sum η! 1 = 0, when η s a negave neger, and usng ha a 1 J M = k a M+k, M+ k!k! 8 k=0 we oban he fnal resul ψx,= Fx+ F 6 + a ] M J n+m a x+ F M= ] J M a. The nal condon ψx,0=j n ax s realzed, and wh some work, can be verfed ha 1 s fulflled. Fgures, 5 and 6 show ψx, when F = 1, and a=1 for n=0, n=1, and n=7, respecvely. Fg. : The soluon ψx,, when he nal condon s ψx,0=j n ax, wh F = 1, n=0, and a=1. Fg. 5: The soluon ψx,, when he nal condon s ψx,0=j n ax, wh F = 1, n=1, and a=1. 8 The Wgner funcon Beng he Wgner funcon 0] one of he mos know quasprobably dsrbuon funcons 1, ], we wan o sudy nex. Usng formula 0, he Wgner funcon 0] Wx, p= 1 π dy ψ x+yψx ypy 5 c 015 NSP

6 180 F. Soo-Egubar, H. M. Moya-Cessa: Soluon of he Schrödnger Equaon for... Snce he Wgner funcon s only non-negave for Gaussan saes, ], presens large regons where s negave, as eced. Wh me he Wgner funcon moves bu keeps s nal form Fgure 8. Fg. 6: The soluon ψx,, when he nal condon s ψx,0=j n ax, wh F = 1, n=7, and a=1. can be calculaed n erms of he Fourer ransform of any reasonable nal condon as Wx, p= 1 { ] F } π π F+p F+p + πx dvy vy v p π + F π v { F F+p π F vf+p x ]}, 6 where Yv has he same meanng of he prevous secons. We presen lcly he case when he nal condon s an Ary funcon, ψx,0 = Aax; n hs case, Wx, p= 1 F+p a F + p x ] A 1/ a 7 In Fgure 7, we show he Wgner funcon a dfferen Fg. 8: The Wgner funcon when he nal condon s an Ary funcon wh a=1 and when F = 1 for = 1. 9 Conclusons Fg. 7: The Wgner funcon for he Ary funcon as nal condon wh F = 1 and a=1, for from 0 o 5 n 1 seps. mes from 0 o 5 n 1 seps when F = 1 and a = 1. The operaonal mehods are very easy o undersand, and n some condons, also very easy o apply. In hs work, we presen a drec generalzaon of he Baker-Campbell-Hausdorff formula from he Zassenhaus formula and we use o solve he me-dependen Schrödnger equaon for a lnear poenal for arbrary nal condon, whou solvng he correspondng saonary Schrödnger equaon. The Wgner funcon can be found and all physcally measurable quanes can be calculaed drecly from. As he paraxal equaon for a lnear GRIN medum s also 1, wh he adequae subsuons axal coordnae z subsues me, hese resuls can also be useful n opcs, n parcular, he Wgner funcon. c 015 NSP

7 Appl. Mah. Inf. Sc. 9, No. 1, / References 1] Alber Messah. Quanum Mechancs. Norh-Holland Publshng Company, ] Eugen Merzbacher. Quanum Mechancs. John Wley and Sons, Inc., hrd edon edon, ] Nouredne Zel. Quanum Mechancs. Conceps and Applcaons. John Wley and Sons, Inc., second edon edon, 009. ] Hécor M. Moya-Cessa and Francsco Soo Egubar. Dfferenal Equaons: An Operaonal Approach. Rnon Press, ] Hécor M. Moya-Cessa and Francsco Soo Egubar. Inroducon o Quanum Opcs. Rnon Press, 01. 6] P. C. García Qujas and L. M. Arévalo Agular. Facorzng he me evoluon operaor. PHYSICA SCRIPTA, 75:185 19, ] R. R. Pur. Mahemacal Mehods of Quanum Opcs. Sprnger Verlag, ] Sephen Gasorowcz. Quanum Physcs. John Wley and Sons, Inc., second edon edon, ] Lesle E. Ballenne. Quanum Mechancs: A Modern Developmen. World Scenfc Publshng Co. Pe. Ld., ] Bran C. Hall. Le Groups, Le Algebras, and Represenaons. An Elemenary Inroducon. Sprnger- Verlag, ] Mladen Nadnc Fernando Casas, Ander Muruab. Effcen compuaon of he zassenhaus formula. Compuer Physcs Communcaons, 18:86 91, 01. 1] W. Magnus. On he onenal soluon of dfferenal equaons for a lnear operaor. Communcaons on Pure and Appled Mahemacs, 7, 69 67, ] Geoffrey New. Inroducon o Nonlnear Opcs. Cambrdge Unversy Press, ] Joseph W. Goodman. Inroducon o Fourer Opcs. McGraw-Hll, second edon edon, ] C. G. Someda. Elecromagnec waves. Taylor & Francs, ] Eugene Hech. Opcs. Addson-Wesley, fourh edon edon, ] Wolfram funcon se. hp://funcons.wolfram.com/ 18] Frank W. J. Olver, edor. NIST Handbook of Mahemacal Funcons. NIST. 19] Wkpeda. hp://en.wkpeda.org/wk/bessel funcon 0] E. Wgner. Phys.Rev., 0:79 759, 19. 1] EJ Wenger, Nonlnear sequence ransformaons for he acceleraon of convergence and he summaon of dvergen seres, Compuer Physcs Repors, Elsever, ] JM Sage, S Sans, T Bergeman, D DeMlle, Opcal producon of ulracold polar molecules, Physcal revew leers, APS, 005. ] R. L. Hudson. Rep. Mah. Phys., 6, ] Francsco Soo-Egubar and Perre Clavere. When s he Wgner funcon of muldmensonal sysems nonnegave? Journal of Mahemacal Physcs,, Francsco Soo-Egubar s Tular Researcher a he Opcs Deparmen a he Naonal Insue of Asrophyscs, Opcs and Elecroncs INAOE n Puebla, Mexco. He receved hs PhD degree n Physcs a he Naonal Unversy of Mexco, n Mexco cy n 198. Hs research neres s n quanum opcs and n he foundaons of quanum mechancs. Hécor M. Moya-Cessa Hécor Manuel Moya-Cessa obaned hs PhD a Imperal College n 199 and snce hen he s a researcher/lecurer a Insuo Naconal de Asrofísca, Ópca y Elecrónca n Puebla, Mexco where he works on Quanum Opcs. He has publshed over 110 papers n nernaonal peer revewed journals. He s fellow of he Alexander von Humbold Foundaon and a Regular Assocae of he Inernaonal Cenre for Theorecal Physcs Trese, Ialy. c 015 NSP