Generating two-dimensional oscillator matrix elements sorted by angular momentum

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen do: / /38/43/010 Generatng two-dmensonal oscllator matrx elements sorted by angular momentum W Schwalm, M C Ntschke and P Res Physcs Department, Unversty of North Dakota, Grand Forks, ND, USA E-mal: wllam.schwalm@und.nodak.edu Receved 19 July 005, n fnal form 19 September 005 Publshed 1 October 005 Onlne at stacks.op.org/jphysa/38/9565 Abstract Generatng functons are found for two-dmensonal harmonc-oscllator ntegrals. These ntegrals are classfed by angular momentum, permttng ncluson of a constant magnetc feld. A generatng functon s obtaned for matrx elements of a Gaussan perturbaton, and as an example these are used to compute egenstates for a partcle n a wne bottle-bottom potental. Along wth ths specfc example a straghtforward method of generalzng the results s presented. PACS numbers: 0.10.Ab, w 1. Introducton The harmonc-oscllator paradgm n quantum mechancs has a range of applcatons. Due to ther convenent propertes, oscllator egenfunctons often serve as a bass set. For molecular or many-body calculatons, the Gaussan bass functons exhbt several advantages over alternatves such as the Slater-type orbtals Boys Although the Gaussans do not have the asymptotc forms expected of atomc wavefunctons, ether at large or at small rad, the greater effcency they afford for computng ntegrals more than compensates for the fact that a large number of Gaussans must be used. In partcular, the space of all Gaussan functons of the form φx,a = A exp x /a s closed under products, translatons and convolutons. The Fourer transforms are also Gaussans. Matrx elements of functons such as x n exp x /a, x n e kx, and so on are related smply enough to the Gaussan ntegrals so that they can be evaluated easly also Matthews and Walker So can the generalzatons to two or three dmensons. The oscllator egenfunctons share these advantages, and have others as well. For example, /05/ $ IOP Publshng Ltd Prnted n the UK 9565

2 9566 W Schwalm et al egenfunctons of a gven oscllator are naturally orthogonal to one another and can be manpulated easly wth the famlar ladder operators a and a. Even so, explct calculaton of a large number of matrx elements becomes cumbersome, whch motvates the use of generatng functons. The hypervral theorem has been used to obtan recurrence relatons for two-centre oscllator matrx elements n one dmenson Morales et al 1987, Morales Soon afterwards Rashd 1989 found generatng functons for two-centre oscllator ntegrals m fa,a n by transformng the recurrence relatons, whch he dd wthout the use of the hypervral theorem. Rashd derved recurrence relatons for two-centred matrx elements ψ g m x x of a, a ψ e x dx, n n one dmenson by purely algebrac methods. Ths was done for the cases fa,a = 1, x k, e ρx, e ρx. In the followng, we obtan generatng functons for the two-dmensonal case usng the Wesner method Wesner 1954, McBrde A complete set of ladder operators s obtaned by determnng smultaneous egenvectors of the angular momentum and Hamltonan operators. Thus, both energy and angular momentum are taken nto account by the resultng generatng functon, whch s of a rather smple form. To show the utlty of ths generatng functon, we calculate egenstates of a two-dmensonal oscllator wth a Gaussan perturbaton. The organzaton of ths paper s as follows. A generatng functon for two-dmensonal oscllator egenstates, sorted by angular momentum, s constructed n secton. Wth ths tool the case of an appled constant magnetc feld can also be treated. The results of secton are used to obtan a generatng functon for matrx elements of a Gaussan potental n secton 3. In addton to the generatng functon, an explct formula s gven for the general, angle-resolved Gaussan matrx element and specfc results are tabulated. The example of a hat potental s treated n secton 4. Fnally, n the last secton, we comment on generalzng to matrx elements of the form r k exp r /a.. Generatng functon for two-dmensonal oscllator Consder an sotropc harmonc oscllator wth mass m and force constant mα. The Hamltonan s H o = 1 m p + mα r. 1 The characterstc length scale s b = mα. When both angular momentum and energy are consdered, one fnds a complete set of ladder operators as presented n table 1. Each of these operators s chosen so that [L z,w] = λw and [H,w] = ɛw, 3 for partcular constants λ and ɛ, so that v rases both energy and angular momentum quantum numbers, whle u rases energy and lowers angular momentum. Four cases are shown. Also, u and v commute. The normalzed egenstate ψ n1,n x, y = u n 1 v n ψ 0,0 x, y 4 n1!n!

3 Oscllator matrx elements 9567 Table 1. Complete set of ladder operators. λ ɛ Operator 1 + α u = 1 mα α u = 1 mα 3 α v = 1 mα 4 + α v = 1 mα x +y + mα p x 1 mα p y x y mα p x 1 mα p y x y + mα p x + 1 mα p y x +y mα p x + 1 mα p y has egenvalues wth respect to both L z and H, namely L z ψ n1,n x, y = n n 1 ψ n1,n x, y, 5 Hψ n1,n x, y = αn 1 + n +1ψ n1,n x, y. The ground state ψ 0,0 annhlated by both u and v s ψ 0,0 x, y = 1 b π exp x + y. 6 b Thus, a convenent generatng functon would appear to be of the form Fx,y,s,t = 1 n1 n 1,n!n! sn 1 t n ψ n1,n x, y = 1 n n 1,n 1!n! sn 1 t n u n 1 v n ψ 0,0 x, y = expsu + tv ψ 0,0 x, y. 7 Equaton 7 reflects the fact that [u,v ] = 0. Startng from the u and v descrbed n table 1, Wesner s method Wesner 1955 s employed to obtan a generatng functon. The strategy s to fnd canoncal varables such that u and v are essentally dervatves and then to evaluate equaton 7 va the Taylor theorem. We note that the generatng functon defned n equaton 7 s also a coherent state, or a mutual egenstate of u and v. Characterstcs for the two frst-order partal dfferental equatons uψ 0,0 = 0 and vψ 0,0 = 0are or ξ = x +y b, ξ = x y b, 8 x = b ξ + ξ, y = b ξ ξ. 9 In these varables, u = 1 ξ v = 1 ξ. 10 ξ ξ The next step s to relate u and v each by gauge transformaton to a dervatve operator. Thus, suppose one has a functon µ = µξ, ξ such that u = 1 1 µ ξ µ. Then u = 1 ξ + 1 µ, µ ξ

4 9568 W Schwalm et al comparng ths to u n 10 1 µ µ ξ = ξ, so that the ntegratng factor µ s µ = e ξξ. Smlarly, one fnds the same ntegratng factor for v. Therefore, the ladder operators become u = e ξξ 1 e ξξ, v = e ξξ 1 e ξξ. 11 ξ ξ Equatons 11 generalze to u n 1 = e ξξ 1 n1 e ξξ, v n = e ξξ 1 n e ξξ. 1 ξ ξ Let Fξ,ξ,s,t= Fxξ,ξ, yξ, ξ, s, t. Insertng the expressons for u n 1 and v n one has Fξ,ξ 1,s,t= µξ, ξ exp s exp t µξ, ξ ψ ξ ξ 0,0 ξ, ξ, 13 where ψ 0,0 ξ, ξ = 1 b. 14 π e ξξ Ultmately, snce the exponentated operators n equaton 13 effect a shft n the arguments ξ and ξ, ths leads to the generatng functon Fs,t,ξ,ξ = ψ 0,0 ξ, ξ exp [ ξ s +ξ or n the orgnal Cartesan coordnates, [ x +ys Fs,t,x,y = ψ 0,0 x, y exp + b t st ], ] x yt e st. 15 b Expandng Fs,t,x,y n powers of s and t and collectng coeffcents, one can obtan any desred quantum state. One may note that the case of an appled magnetc feld can be treated wthout dffculty Landau and Lfshtz When we apply a feld B = B o ẑ, H = H o qb o mc L z + 1 m qbo mc r. 16 The presence of the feld breaks the chral symmetry of the oscllator and motvates the defntons ω b = qb o mc, 17 = α + ωb. 18 Thus, the characterstc length s now b = m. 19

5 Oscllator matrx elements 9569 Table. Complete set of ladder operators wth an appled feld. λ ɛ Operator 1 + ω b u = ω b u = ω b v = ω b v = 1 m m m m x +y + m p x 1 m p y x y m p x 1 m p y x y + m p x + 1 m p y x +y m p x + 1 m p y The normalzed egenstate ψ n1,n x, y s the same as n equaton 4, but wth parameters deformed to correspond to the transformed ladder operators defned n table. Therefore, L z ψ n1,n x, y = n n 1 ψ n1,n x, y, 0 Hψ n1,n x, y = ω b n 1 n + n 1 + n +1 ψ n1,n x, y. Thus, H s deformed contnuously from H o at ω b = 0 zero feld and the physcal propertes deform smoothly as well. The results obtaned prevously stll hold mutats mutands wth α replaced by. Now, however, the energy degeneracy between states wth rght- and left-handed angular momentum s splt by the feld as seen n the table. 3. Generatng functon for matrx elements of a Gaussan potental A generatng functon for the matrx elements of V where x,y V x,y =exp x + y a s Gs, t, p, q = Fx,y,s,t exp r /a Fx,y,p,q, where we have used an obvous notatonal shorthand. One has Gs, t, p, q = s n 1 t n p n 3 q n 4 n1!n!n 3!n 4! n 1,n V n 3,n 4. 3 By defnng the shape parameter ζ = a a + b, 4 and completng the square n the exponent, the ntegral mpled by equaton can be evaluated n a smple form: Gs, t, p, q = ζ exp [ζ 1st + pq + ζsp + tq]. 5 Equaton 5, whch gves Gs, t, p, q n a succnct form, s our prncpal result. Conservaton of angular momentum requres that n n 1 = n 4 n 3. In vew of equaton 5, we ntroduce an angular momentum quantum number l = n n 1. Thus, non-vanshng ntegrals are of the form n 1,n 1 + l V n 3,n 3 + l. 6 By expandng equaton 5, G = 1 j!k!g!h! ζ 1j+k ζ g+h+1 s j+g t j+h p k+g q k+h, 7 jkgh 1

6 9570 W Schwalm et al Table 3. Matrx elements m, m + l V n, n + l for selected l, m and n. l m n Gaussan matrx elements ζ ζ ζ ζ 3 ζ + ζ ζ 3 ζ + ζ ζ 4 5ζ 3 +3ζ ζ 6 0 6ζ 5 1ζ 4 +10ζ 3 4ζ + ζ ζ ζ 3 ζ ζ 4 4ζ 3 +ζ ζ 4 3ζ 3 + 3ζ ζ 5 4 6ζ ζ 3 6ζ ζ 6 4ζ 5 +4ζ 4 1ζ 3 +3ζ ζ ζ 4 3ζ ζ 5 6ζ 4 +3ζ ζ 5 6ζ 4 + 6ζ ζ 6 11 ζ 5 +9 ζ 4 3 ζ ζ 7 40ζ 6 +44ζ 5 4ζ 4 +6ζ 3 and comparng to equaton 3, one can assgn j + g = n 1, j + h = n 1 + l, k + g = n 3, k + h = n 3 + l. Therefore k = j n 1 + n 3, g = n 1 j, h = n 1 + l j, 8 and j remans to be summed. Thus, the general matrx element s n 1,n 1 + l V n 3,n 3 + l = n1!n 1 + l!n 3!n 3 + l! j!j n j 1 + n 3!n 1 j!n 1 + l j! ζ 1j n 1+n 3 ζ n1 j+l+1. 9 The range on j s determned by the denomnator of equaton 9, snce where the argument of any one of the factorals goes negatve, the denomnator becomes nfnte and so the coeffcent must vansh; thus 0 j l + n 1 and n 1 n 3 j n 1. Several matrx elements for a Gaussan bump are shown n table 3. Note that ζ contans the effect of an appled feld owng to ts dependence on b, snce when a feld s appled, b of equaton s replaced as ndcated n equaton Example of a hat potental Here, we apply the generatng functon developed n the prevous secton to a quadratc potental wth Gaussan perturbaton. As an example of a rotatonally symmetrc problem n two dmensons, consder a partcle of mass m subject to the potental Vr= 1 mα r + V o exp r /a. 30

7 Oscllator matrx elements 9571 Vx,y y x Fgure 1. Wne bottle-bottom potental. v Fgure. Shape for v o = 0, 4, 8, 1 and 16. Wth the proper choce of V o and a, the graph of Vrversus r resembles a hat or the bottom of a wne bottle. A qualtatve sketch of ths wne bottle-bottom potental s gven n fgure 1. Scalng to a dmensonless radal coordnate ρ = r/a yelds Wth Vr= 1 mα a ρ + v o exp ρ, 31 v o = V o mα a, so that v o s a dmensonless verson of the potental strength V o. The shape of the resultng dmensonless form vρ = ρ + v o exp ρ s shown n fgure for several values of v o. When v o > 1, the mnmum moves away from 0 to ρ = lnv o. The Hamltonan H s such that m + l, m H n + l, n =αn + l +1δ mn +αv m + l, m exp r /a n + l, n, 3 wth v = a v o /b. Fgure 3 shows the energy levels as a functon of v for the case ζ = 3/4 wth ζ as defned n equaton 4. Energes E are n unts of α. The energes shown n fgure 3 are found from nne separate varatonal calculatons, one for each angular quantum number from l = 0tol = 8 and each done n a bass of 16 l-projected oscllator functons. The energes for ±l are degenerate. For small v the levels splt lnearly, wth resdual ±l degeneracy, whle for large v one can expand n a harmonc

8 957 W Schwalm et al E v Fgure 3. Energy levels of the wne bottle-bottom n unts of α as a functon of central heght n the same unts. approxmaton about the effectve mnmum potental. So for 0 <ζ <1and v 1, E α lnv n ζ 1 ζ l α1+lnv +α 1 ζ ζ lnv. 33 Applyng a magnetc feld s straghtforward. In vew of the above dscusson, m + l, m H n + l, n = n + l +1δ mn + vα m + l, m exp r /a n + l, n. 34 So the matrx elements are deformed by an amount dependng on l, and n a rather smple way on ω o,.e. on the feld appled. The magnetc feld lfts the ±l degeneracy. 5. Generatng functon for matrx elements of r k exp r /a Wth the generatng functon developed as n the prevous sectons, t s straghtforward to calculate a number of related results. Here we gve just one llustraton. For ntegrals of the form m, m + l r k e r /a n, n + l 35 t suffces to evaluate the generatng functon Ks,t,p,q,σ = σ k k! s n 1 t n p n 3 q n 4 n1!n!n 3!n 4! n 1,n r k V n 3,n 4 = s n 1 t n p n 3 q n 4 n1!n!n 3!n 4! n 1,n exp σr r /a n 3,n One sees that ths expresson can be found from the functon Gs, t, p, q defned n equaton 3 by replacng 1/a n the exponent wth 1/a + σ. But ths s the same thng as replacng ζ n the result shown n equaton 5by whch gves Ks,t,p,q,σ = ζσ = ζ 1+ζσb 37 ζ ζσb +1 exp pq + st + st + pq + sp + tqζ. 38 ζσb +1

9 Oscllator matrx elements Summary By constructng smultaneous ladder operators of energy and angular momentum, as shown n table 1, t s not dffcult to derve the generatng functon Fx,y,s,t for the smultaneous egenfunctons of H and L z gven n equaton 15. Upon ntegratng the product of two copes of ths generatng functon wth the symmetrcal Gaussan exp r /a,wehave another generatng functon Gs, t, p, q n equaton 5 for the Gaussan matrx elements. A concse formula for matrx elements s gven n equaton 9. To llustrate the utlty of these, we have computed energy levels of a wne bottle-bottom potental as a functon of a shape parameter ζ as descrbed n secton 3. The smple form of Gs, t, p, q n equaton 5 makes t qute useful n calculatons wth an oscllator bass n problems wth cylndrcal symmetry. Related generatng functons such as Ks,t,p,q,σn secton 5 follow easly from the formula for Gs, t, p, q and are also useful. We have found t convenent to have smple Mathematca functons for obtanng the desred matrx elements drectly by evaluatng Taylor seres coeffcents. References Boys S F 1950 Proc. R. Soc Landau L and Lfshtz E 1981 Quantum Mechancs Oxford: Pergamon p 454 Matthews J and Walker R 1970 Mathematcal Methods of Physcs Boston, MA: Addson-Wesley p 58 McBrde E 1971 Obtanng Generatng Functons ed B Coleman New York: Sprnger p 5 Morales J 1987 Phys. Rev. A Morales J, Lopez-Bonlla J and Palma A 1987 J. Math. Phys Rashd M 1989 J. Math. Phys Wesner L 1955 Pac. J. Math