Scattering States of l-wave Schrödinger Equation with Modified Rosen Morse Potential

Transcription

1 Commun. Thor. Phys Vol. 66, No., August, 06 Scattring Stats of l-wav Schrödingr Equation with Modifid Rosn Mors Potntial Wn-Li Chn í,, Yan-Wi Shi á, and Gao-Fng Wi Ôô, Gnral Education Cntr, Xi an Pihua Univrsity, Xi an 70065, China Shaanxi Ky Laboratory of Surfac Enginring and Rmanufacturing, School of Mchanical and Matrial Enginring, Xi an Univrsity, Xi an 70065, China Rcivd Fbruary 9, 06; rvisd manuscript rcivd Jun, 06 Abstract Within a Pkris-typ approximation to th cntrifugal trm, w xamin th approximatly analytical scattring stat solutions of th l-wav Schrödingr quation with th modifid Rosn Mors potntial. Th calculation formula of phas shifts is drivd, and th corrsponding bound stat nrgy lvls ar also obtaind from th pols of th scattring amplitud. PACS numbrs: w, Nk Ky words: modifid Rosn Mors potntial, Scattring stats, approximatly analytical solutions Introduction Th Mors and/or original Rosn Mors potntials, [ ] as th important molcular modl potntials, hav bn widly studid du to its validity in dscribing th vibration of actual diatomic molculs spcially for th lowst lctronic stats. [3] Nvrthlss, du to th Mors and/or original Rosn Mors potntials do not considr th innr-shll radii of two atoms, it is thrfor that thy ncountr invitably som problm in rproducing th xprimntal data for som molcul stats such as 6 Π u stat of 7 Li molcul and X 3 Π stat of SiC radical. [4 6] Vry rcntly, in ordr to fit th ffct of innr-shll radii of two atoms for diatomic molculs, [7] Zhang t al. [4] proposd basd on th original Rosn Mors potntial [] a modifid Rosn Mors potntial in th form of V r = D r rij +, r rij + whr paramtrs r ij = r KD /k, D is th dissociation nrgy, r is th quilibrium bond lngth, and k is th quilibrium harmonic vibrational forc constant, and th paramtr K is a dimnsional constant. [8] It is shown that th modifid Rosn Mors potntial is vry suprior to th Mors and original Rosn Mors potntials in fitting xprimntal data for som molcul stats. [4] Morovr, th vibrational nrgy lvls for 6 Π u stat of 7 Li molcul and X 3 Π stat of SiC radical with this potntial modl [5] ar good agrmnt with th xprimntal data dtrmind by th Rydbrg Klin Rss mthod. [9 ] Up to now, th bound stats of this potntial modl hav bn invstigatd within th non-rlativistic and/or rlativistic quantum mchanics. [ 4] Howvr, thr is no any rport on th scattring stats rgarding this potntial modl to our knowldg. Undoubtdly, th scattring stats spcially th phas shifts ar th important issu no mattr in quantum physics or nuclar physics. Thrfor, it is ncssary to study systmatically th scattring stats of this potntial spcially th analytical proprtis of scattring amplitud. Without considring th application of this potntial modl to a spcific molcular stat, w ar going to study th scattring stats of th modifid Rosn Mors potntial in this papr. On th othr hand, w ar going to study th proprtis of scattring stats to show th rlations btwn th nrgy lvls of scattring stats and thos of bound stats at th pols of scattring amplituds. This papr is organizd as follows. In Sc., w driv th scattring stat solutions of th modifid Rosn Mors potntial with cntrifugal trm. In Sc. 3, w discuss th analytical proprtis of scattring amplitud to find th discrt nrgy lvls from th pols of scattring amplitud. Th concluding rmarks ar givn in Sc. 4. Scattring Stats Th quantum bhaviour of a non-rlativistic particl in a tim-indpndnt potntial V r is dscribd by th stationary Schrödingr quation = µ = Ψr, θ, φ + V rψr, θ, φ = EΨr, θ, φ. Hr, th potntial is takn as th modifid Rosn Mors potntial, i.. Eq.. Taking Ψ nlm r, θ, φ = r u nl ry lm θ, φ, and substituting it into Eq., w Supportd by th National Natural Scinc Foundation of China undr Grant No. 4058, and Natural Scinc Basic Rsarch Plan in Shaanxi Provinc of China undr Grant No. 5JK093 physwlchn@63.com Corrsponding author, fgwi 000@63.com c 06 Chins Physical Socity and IOP Publishing Ltd

2 No. Communications in Thortical Physics 97 obtain th radial Schrödingr quation as follows, u ll + ] r+ [E D r r r ur = 0, 3 whr paramtr λ = rij. For scattring stats, at larg distanc th solutions of Eq. 3 ar asymptotically sinusoidal with phas shifts δ l that carry th scattring information. [5] Howvr, Eq. 3 cannot b solvd analytically du to th cntrifugal trm ll + /r. Thrfor, w hav to us a propr approximation to th cntrifugal trm. On th othr hand, it is noticd that this potntial has a minimum valu at r = r r 0. Thrfor, w can us th Pkris-typ [6 7] approximation to rplac th cntrifugal trm /r. To this nd, w introduc a variabl y = r r 0 /r 0 and xpand th cntrifugal trm in a powr sris around y = 0 r = r0 + = y r0 y + 3y + oy. 4 On th othr hand, w approximatly writ th cntrifugal trm as th following form r c 0 + c r + c r = c 0 + c r0+y + c r0+y. 5 Aftr xpanding th right hand of Eq. 5 in a powr sris around y = 0 and comparing it with Eq. 4, th cofficints c 0, c, and c can b xprssd as th following form Now w introduc a nw variabl [3 c 0 = r r + r r λ + r 3 + r ] r 4, c = r r [ λ3 + r + r 3 + r ] r 4, c = r r 3 [ r r 3 λ3 + r ] r 4. 6 x = r r r, x 7 and insrt Eq. 5 into Eq. 3, thn th radial Schrödingr quation 3 can b transformd into th following form { x x u x + x 3x + x u x + E c 0 ll + x D x [c ll + + r D ] λ + x [c ll + 4 r xd ] } ux = 0. 8 λ W now tak th trial wavfunction in th form of whr ux = x ik/ x η Fx, 9 c ll + + c l + c l η = λ + c 0 l + c 0 l Eλ + r D, 0 λ k = E c 0 l c 0 l D. Insrting Eq. 9 into Eq. 8 yilds th following hyprgomtric quation [ ik ] xxf x + + η + x + η F x k + ik + η η η + + c l + c l + r D λ Fx = 0, its gnral solution is nothing but th hyprgomtric function ã p b p x p Fx = F ã, b; c; x = c p p!, 3 whr x p is th Pochhammr symbol x p = p=0 Γx + p Γx, 4 and paramtrs ã, b and c ar dfind as follows ã = ik σ + η r, b = ik σ + η + r, c = + η, σ = 4c ll + + λ + 8 r D. 5 Now w can writ th analytical radial wav functions of arbitrary l-wav scattring stats for th modifid Rosn Mors potntial as [ ur = N r ] η ] [ r r ik/ r F ã, b; c; r r, 6

3 98 Communications in Thortical Physics Vol. 66 whr N is th normalizd constant. To dtrmin th normalizd constant N and scattring phas shifts δ l, w us th following rcurrnc formula of hyprgomtric functions [8] F ã, b; c; x = Γ cγ c ã b Γ c ãγ c b F ã, b; ã + b c + ; x + x c ã b Γ cγã + b c F c ã, c b; c ã b + ; x, 7 ΓãΓ b and notic th F a, b; c; 0 = and r / r λ r for r, thn w hav th following asymptotic bhavior for th hyprgomtric function F ã, b; r [ c; r Γ c Γ c ã b r Γ c ãγ c b ik/ ikr Γ c ã b ] Γ c ãγ c b [ = λ ik/ Γ c ã Γ c b Γ c ãγ c b λ ik/ + ikr Γ c ã b ] Γ c ãγ c b λ ik/. 8 By taking Γ c ã b Γ c ãγ c b = Γ c ã b Γ c ãγ c b iθ, λ ik/ = λ ik/ iθ, 9 and insrting it into Eq. 8, w hav F ã, b; r ik/ Γ c ã bλ c; r r λik/ Γ c Γ c ãγ c b ikr [ ikr+θ+θ + ikr+θ+θ ] = NΓ cλ ik/ ikr ik/ Γ c ã bλ Γ c ãγ c b sin kr + π + θ + θ. 0 Aftr insrting Eq. 0 into Eq. 6 and considring r / r and r / r λ r whn r, w can finally xprss th asymptotic form of th radial wav function 6 for r as follows ik/ Γ c ã bλ ur NΓ c r Γ c ãγ c b sin kr + π + θ + θ. Comparing Eq. with th gnral asymptotic solution [5] ur r sinkr /lπ + δ l, w gt th phas shifts and th normalizd constant as th following forms δ l = π l + + θ + θ = π l + + arg Γ c ã b arg Γ c ã argγ c b + argλ ik/, N = ik/ Γ c ãγ c bλ Γ c Γ c ã b. 3 Th substitution of paramtrs of ã, b and c into Eqs. and 3 allows us to xprss th phas shifts and normalizd constant in th following final forms δ l = π l + + θ + θ = π ik l + + argγ + argλ ik/ argγ + ik + η + 4c ll + + λ + 8 r D λ argγ + ik + η 4c ll + + λ + 8 r D, 4 λ N = λik/ Γ Γ + η + ik + η + 4c ll + + λ + 8 r D λ Γ ik Γ + ik + η 4c ll + + λ + 8 r D. 5 λ 3 Analytical Proprtis of Scattring Amplitud Now lt us study th analytical proprtis of th scattring amplitud. It is wll known that th discrt nrgy lvls of th bound stats corrspond to th singl pols of th scattring amplitud. [5] This suggsts that w discuss

4 No. Communications in Thortical Physics 99 th proprtis of Gamma function Γ + ik + η 4c ll + + λ + 8 r D λ basd on Eq. 4. From th dfinition of th Gamma function Γz = Γz + z = 6 Γz + zz + = Γz + 3 zz + z + =, 7 w know that z = 0,,,... ar th first ordr pols of th Γz. Thrfor, th first ordr pols of Γ + ik + η 4c ll + + λ + 8 r D λ ar situatd at + ik + η 4c ll + + λ + 8 r D = 0,,, 3,... = n r, n r = 0,,... 8 λ from which, w can obtain th corrsponding bound stat nrgy lvls as E nrl = { c 0ll + + D nr + + 4c ll r D / λ 4 [ll + c + c λ + r λ } D ] λ n r +, 9 + 4c ll r D / λ whr paramtrs c 0, c, and c ar givn by Eqs. 6. By comparing th abov nrgy lvls 9 with th nrgy quation 5 in Rf. [5] and obsrving th rlations of corrsponding paramtrs, it is asy to find that Eq. 9 is in ssnc th sam as Eq. 5 in Rf. [5]. That is to say, th pols of th S-matrix in th complx nrgy plan corrspond to bound stats for ral pols and scattring stats for complx pols in th lowr half of th nrgy plan. Without considring th application of this potntial modl to a spcific molcular stat, w now prsnt som numrical nrgy lvls calculatd from nrgy quation 9 and compar thm with thos obtaind by th MATHEMATICA packag programmd by Lucha and Schöbrl. [9] To this nd, w tak paramtrs of λ= and D =000 in nrgy quation 9 and th MATHEMAT- ICA packag to gt th numrical nrgy lvls as shown in Tabl. It can b found that our prsnt rsults ar in good agrmnt with thos from th MATHEMATICA packag, vn for larg potntial rang paramtr. Tabl Eignvalus E nrl as a function of for p, 3p, 3d, 4p, 4d, 4f, 5p, 5d, 5f, 5g, 6f and 6g stats for λ =, D = 000 and = µ =. Stats r =.6 r =.4 r = 3. Prsnt Lucha t al. Prsnt Lucha t al. Prsnt Lucha t al. p p d p

5 00 Communications in Thortical Physics Vol. 66 Tabl continud Stats r =.6 r =.4 r = 3. Prsnt Lucha t al. Prsnt Lucha t al. Prsnt Lucha t al. 4d f p d f g f g Conclusions In summary, within th framwork of non-rlativistic quantum thory, w hav studid th scattring stats of th modifid Rosn Mors potntial. By using a Pkris-typ approximation to th cntrifugal trm, th normalizd analytical wav functions ar obtaind and corrsponding calculation formula of phas shifts is drivd. Morovr, w hav discussd th analytical proprtis of scattring amplitud and found th discrt nrgy lvls of th bound stats from th pols of th scattring amplitud. Rfrncs [] P.M. Mors, Phys. Rv [] N. Rosn and P.M. Mors, Phys. Rv [3] G.F. Wi and W.L. Chn, Chin. Phys. B [4] G.D. Zhang, J.Y. Liu, L.H. Zhang, W. Zhou, and C.S. Jia, Phys. Rv. A [5] H.M. Tang, G.C. Liang, L.H. Zhang, F. Zhao, and C.S. Jia, Can. J. Chm [6] C.S. Jia, X.L. Png, and S. H, Bull, Koran Chm. Soc [7] A.A. Frost and B. Musulin, J. Am. Chm. Soc [8] C.S. Jia and Z.W. Shui, Eur. Phys. J. A [9] R. Rydbrg, Z. Phys [0] O. Klin, Z. Phys [] A.L.G. Rs, Proc. Phys. Soc [] C.S. Jia, L.H. Zhang, X.T. Hu, H.M. Tang, and G.C. Liang, J. Mol. Spctrosc [3] C.S. Jia, J.W. Dai, L.H. Zhang, J.Y. Liu, and G.D. Zhang, Chm. Phys. Ltt [4] C.S. Jia, J.W. Dai, L.H. Zhang, J.Y. Liu, and X.L. Png, Phys. Ltt. A [5] L.D. Landau and E.M. Lifshitz, Quantum Mchanics, Non-Rlativisitc Thory, 3rd d. Prgamon, Nw York 977. [6] C.L. Pkris, Phys. Rv [7] G.F. Wi and S.H. Dong, Phys. Ltt. B [8] I.S. Gradshtyn and I.M. Ryzhik, Tabls of Intgals, Sris, and Products, 5th d. Acadmic Prss, Nw York 994. [9] W. Lucha and F.F. Schöbrl, Int. J. Mod. Phys. C