Motion in Spherically Symmetric Potentials

Transcription

1 Chapter 7 Motion in Sphericay Symmetric Potentias We escribe in this section the stationary boun states of quantum mechanica partices in sphericay symmetric potentias V (r), i.e., in potentias which are soey a function of r an are inepenent of the anges θ, φ. Four exampes wi be stuie. The first potentia V (r) = { 0 0 r R r > R (7.) confines a freey moving partice to a spherica box of raius R. The secon potentia is of the square we type { Vo 0 r R V (r) =. (7.) 0 r > R The thir potentia escribes an isotropic harmonic osciator. The fourth potentia V (r) = mω r (7.3) V (r) = Z e r (7.4) governs the motion of eectrons in hyrogen-type atoms. Potentia (7.4) is by far the most reevant of the four choices. It eas to the stationary eectronic states of the hyrogen atom (Z = ). The corresponing wave functions serve basis functions for muti-eectron systems in atoms, moecues, an crystas. The potentia (7.3) escribes the motion of a charge in a uniformey charge sphere an can be empoye to escribe the motion of protons an neutrons in atomic nucei. The potentias (7., 7.) serve as schematic escriptions of quantum partices, for exampe, in case of the so-cae bag moe of haronic matter. See, for exampe, Simpe Moes of Compex Nucei / The She Moe an the Interacting Boson Moe by I. Tami (Harwoo Acaemic Pubishers, Poststrasse, 7000 Chur, Switeran, 993) 83

2 84 Sphericay Symmetric Potentias 7. Raia Schröinger Equation A cassica partice moving in a potentia V (r) is governe by the Newtonian equation of motion m v = ê r r V (r). (7.5) In the case of an anguar inepenent potentia anguar momentum J = m r v is a constant of motion. In fact, the time variation of J can be written, using (7.5) an r = v, t J = v m v + r m v = 0 + r ê r ( r V (r)) = 0. (7.6) Since J k is aso equa to the poisson bracket {H, J k } where H is the Hamitonian, one can concue J k = {H, J k } = 0, k =,, 3. (7.7) The corresponence principe ictates then for the quantum mechanica Hamitonian operator Ĥ an anguar momentum operators J k Ĥ, J k = 0, k =,, 3. (7.8) This property can aso be proven reaiy empoying the expression of the kinetic energy operator c.f. (5.99) m = m r r r + J mr, (7.9) the expressions ( ) for J k as we as the commutation property (5.6) of J an J k. Accoringy, stationary states ψ E,,m ( r) can be chosen as simutaneous eigenstates of the Hamitonian operator Ĥ as we as of J an J 3, i.e., Ĥ ψ E,,m ( r) = E ψ E,,m ( r) (7.0) J ψ E,,m ( r) = ( + ) ψ E,,m ( r) (7.) J 3 ψ E,,m ( r) = m ψ E,,m ( r). (7.) In cassica mechanics one can expoit the conservation of anguar momentum to reuce the equation of motion to an equation governing soey the raia coorinate of the partice. For this purpose one concues first that the conservation of anguar momentum J impies a motion of the partice confine to a pane. Empoying in this pane the coorinates r, θ for the istance from the origin an for the anguar position, one can state The expression of the kinetic energy an conservation of energy yie J = m r θ. (7.3) p m = m ṙ + m r θ = m ṙ + J mr (7.4) m ṙ + J + V (r) = E. (7.5) mr

3 7.: Raia Schröinger Equation 85 This is a ifferentia equation which governs soey the raia coorinate. It can be sove by integration of { ṙ = ± E V (r) J }. (7.6) m mr Once, r(t) is etermine the anguar motion foows from (7.3), i.e., by integration of θ = J mr (t). (7.7) In anaogy to the cassica escription one can erivem, in the present case, for the wave function of a quantum mechanica partice a ifferentia equation which governs soey the r-epenence. Empoying the kinetic energy operator in the form (7.9) one can write the stationary Schröinger equation (7.0), using (7.), m r r r + J mr + V (r) E,m ψ E,,m ( r) = 0. (7.8) Aopting for ψ E,,m ( r) the functiona form ψ E,,m ( r) = v E,,m (r) Y m (θ, φ) (7.9) where Y m (θ, φ) are the anguar momentum eigenstates efine in Section 5.4, equations (7., 7.) are obeye an one obtains for (7.0) m r r r + ( + ) mr + V (r) E,m v E,,m (r) = 0. (7.0) Since this equation is inepenent of the quantum number m we rop the inex m on the raia wave function v E,,m (r) an E,m. One can write (7.0) in the form of the one-imensiona Schröinger equation m r + V eff (r) E φ E (r) = 0. (7.) where V eff (r) = V (r) + ( + ) mr (7.) φ E (r) = r v E,,m (r). (7.3) This emonstrates that the function r v E,,m (r) escribes the raia motion as a one-imensiona motion in the interva 0, governe by the effective potentia (7.) which is the origina potentia V (r) with an ae rotationa barrier potentia (+). This barrier, together with the origina mr potentia, can excue partices from the space with sma r vaues, but can aso trap partices in the atter space giving rise to strong scattering resonances (see Section??). Mutipying (7.0) by mr/ yies the so-cae raia Schröinger equation r ( + ) r U(r) κ r v κ, (r) = 0. (7.4)

4 86 Sphericay Symmetric Potentias where we efine U(r) = m V (r) (7.5) κ = m E (7.6) In case E < 0, κ assumes rea vaues. We repace in (7.0) the inex E by the equivaent inex κ. Bounary Conitions In orer to sove (7.0) one nees to specify proper bounary conitions. For r 0 one may assume that the term ( + )/r becomes arger than the potentia U(r). In this case the soution is governe my r ( + ) r an, accoringy, assumes the genera form or r v κ, (r) = 0, r 0 (7.7) r v κ, (r) A r + + B r (7.8) v κ, (r) A r + B r (7.9) Ony the first term is amissabe. This foows for > 0 from consieration of the integra which measures the tota partice ensity. The raia part of this integra is an, hence, the term Br (+) wou contribute 0 r r vκ, (r) (7.30) ɛ B r r (7.3) 0 which, for > 0 is not integrabe. For = 0 the contribution of Br (+) to the compete wave function is, using the expression (5.8) for Y 00, ψ E,,m ( r) B 4π r. (7.3) The tota kinetic energy resuting from this contribution is, accoring to a we-known resut in Cassica Eectromagnetism 3, m ψ E,,m ( r) 4π B δ(x )δ(x )δ(x 3 ). (7.33) A etaie iscussion of the proper bounary conitions, in particuar, at r = 0 is foun in the exceent monographs Quantum Mechanics I, II by A. Gaino an P. Pascua (Springer, Berin, 990) 3 We refer here to the fact that the function Φ( r) = /r is the soution of the Poisson equation = 4πδ(x)δ(y)δ(); see, for exampe, Cassica Eectroynamics, n E. by J.D. Jackson (John Wiey, New York, 975).

5 7.: Raia Schröinger Equation 87 Since there is no term in the stationary Schröinger equation which cou compensate this δ- function contribution we nee to postuate that the secon term in (7.9) is not permissibe. One can concue that the soution of the raia Schröinger equation must obey r v κ, (r) 0 for r 0. (7.34) The bounary conitions for r are governe by two terms in the raia Schröinger equation, namey, r κ r vκ, (r) = 0 for r. (7.35) We have assume here im r V (r) = 0 which is the convention for potentias. The soution of this equation is r v κ, (r) A e κr + B e +κr for r. (7.36) For boun states κ is rea an, hence, the secon contribution is not permissibe. We concue, therefore, that the asymptotic bounary conition for the soution of the raia Schröinger equation (7.0) is r v κ, (r) e κr for r. (7.37) Degeneracy of Energy Eigenvaues We have note above that the ifferentia operator appearing on the.h.s. of the in raia Schröinger equation (7.4) is inepenent of the anguar momentum quantum number m. This impies that the energy eigenvaues associate with stationary boun states of raiay symmetric potentias with ientica, but ifferent m quantum number, assume the same vaues. This behaviour is associate with the fact that any rotationa transformation of a stationary state eaves the energy of a stationary state unatere. This property hos since (7.8) impies Ĥ, exp( i ϑ J ) = 0. (7.38) Appying the rotationa transformation exp( i ϑ J ) to (7.0) yies then Ĥ exp( i ϑ J ) ψ E,,m ( r) = E exp( i ϑ J ) ψ E,,m ( r), (7.39) i.e., any rotationa transformation prouces energeticay egenerate stationary states. One might aso appy the operators J ± = J ± ij to (7.0) an obtain for < m < which, together with the ientities (5.7, 5.73), yies Ĥ J ± ψ E,,m ( r) = E J ± ψ E,,m ( r), (7.40) Ĥ ψ E,,m± ( r) = E ψ E,,m± ( r) (7.4) where E is the same eigenvaue as in (7.40). One expect, therefore, that the stationary states for sphericay symmetric potentias form groups of + energeticay egenerate states, so-cae mutipets where = 0,,,.... Foowing a convention from atomic spectroscopy, one refers to the mutipets with = 0,,, 3 as the s, p,, f-mutiptes, respectivey. In the remainer of this section we wi sove the raia Schröinger equation (7.0) for the potentias state in (7. 7.4). We seek to escribe boun states for the partices, i.e., states with E < 0. States with E > 0, which pay a key roe in scattering processes, wi be escribe in Section??.

6 88 Sphericay Symmetric Potentias 7. Free Partice Describe in Spherica Coorinates We consier first the case of a partice moving in a force-free space escribe by the potentia V (r) 0. (7.4) The stationary Schröinger equation for this potentia reas m E ψ E ( r) = 0. (7.43) Stationary States Expresse in Cartesian Coorinates expresse in ( ), is ψ( k r) = N e i k r where The genera soution of (7.43), as (7.44) E = k m 0. (7.45) The possibe energies can assume continuous vaues. N in (7.44) is some suitaby chosen normaiation constant; the reaer shou be aware that (7.44) oes not represent a ocaie partice an that the function is not square integrabe. One chooses N such that the orthonormaity property Ω 3 r ψ ( k r)ψ( k r) = δ( k k) (7.46) hos. The proper normaiation constant is N = (π) 3/. In case of a force-free motion momentum is conserve. In fact, the Hamitonian in the present case H o = m (7.47) commutes with the momentum operator ˆ p = ( /i) an, accoringy, the eigenfunctions of (7.45) can be chosen as simutaneous eigenfunction of the momentum operator. In fact, it hos ˆ p N e i k r = k N e i k r. (7.48) as one can erive using in (7.48) Cartesian coorinates, i.e., = 3, k r = k x + k x + k 3 x 3. (7.49) Stationary States Expresse in Spherica Coorinates Rather than specifying energy through k = k an the irection of the momentum through ˆk = k/ k one can expoit the fact that the anguar momentum operators J an J 3 given in (5.97) an in (5.9), respectivey, commute with H o as efine in (7.47). This atter property foows from (5.00) an (5.6). Accoringy, one can choose stationary states of the free partice which are eigenfunctions of (7.47) as we as eigenfunctions of J an J 3 escribe in Sect. 5.4.

7 7.: Free Partice 89 The corresponing stationary states, i.e., soutions of (7.43) are given by wave functions of the form ψ(k,, m r) = v k, (r) Y m (θ, φ) (7.50) where the raia wave functions obeys c.f. (7.0) m r r r + ( + ) mr E,m v k, (r) = 0. (7.5) Using (7.45) an mutipying (7.0) by mr/ yies the raia Schröinger equation r ( + ) r + k r v k, (r) = 0. (7.5) We want to etermine now the soutions v k, (r) of this equation. We first notice that the soution of (7.5) is actuay ony a function of kr, i.e., one can write v k, (r) = j (kr). In fact, one can reaiy show, introucing the new variabe = kr, that (7.5) is equivaent to ( + ) + j () = 0. (7.53) Accoring to the iscussion in Sect. 7. the reguar soution of this equation, at sma r, behaves ike j () for r 0. (7.54) There exists aso a so-cae irreguar soution of (7.53), enote by n () which behaves ike n () for r 0. (7.55) We wi iscuss further beow aso this soution, which near r = 0 is inamissabe in a quantum mechanica wave function, but amissabe for r 0. For arge vaues the soution of (7.53) is governe by + j () = 0 for r (7.56) the genera soution of which is j () sin( + α) for r (7.57) for some phase α. We note in passing that the functions g () = j (), n () obey the ifferentia equation equivaent to (7.53) + ( + ) + g () = 0. (7.58) Noting that sin( + α) can be written as an infinite power series in we attempt to express the soution of (7.53) for arbitary vaues in the form j () = f( ), f( ) = a n n. (7.59) n=0

8 90 Sphericay Symmetric Potentias The unknown expansion coefficients can be obtaine by inserting this series into (7.53). We have introuce here the assumption that the factor f in (7.59) epens on. This foows from + f() = + from which we can concue ( f() + ( + ) f + ( + ) f (7.60) + ( + ) Introucing the new variabe v = yies, using ) + f() = 0. (7.6) = v, = 4v v + v, (7.6) the ifferentia equation ( v v v + ) 4v f(v) = 0 (7.63) which is consistent with the functiona form in (7.59). The coefficients in the series expansion of f( ) can be obtaine from inserting n=0 a n n into (7.63) (v = ) ( a n n (n ) v n + ( + 3) a n v n + 4 ) a n v n = 0 (7.64) n Changing the summation inices for the first two terms in the sum yies ( a n+ n (n ) + ( + 3) a n + ) 4 a n v n = 0. (7.65) n In this expression each term v n must vanish iniviuay an, hence, One can reaiy erive a n+ = a =! ( + 3) a 0, a = 4 The common factor a 0 is arbitrary. Choosing (n + ) (n + + 3) a n (7.66)! ( + 3)( + 5) a 0. (7.67) a 0 = 3 5 ( + ). (7.68) the ensuing functions ( = 0,,,...) j () = 3 5 ( + )!( + 3) + ( )!( + 3)( + 5) + (7.69) are cae reguar spherica Besse functions.

9 7.: Free Partice 9 One can erive simiary for the soution (7.55) the series expansion ( = 0,,,...) n () = 3 5 ( ) + These functions are cae irreguar spherica Besse functions.!( ) + ( )!( )(3 ) + Exercise 7..: Demonstrate that (7.70) is a soution of (7.5) obeying (7.55).. (7.70) The Besse functions (7.69, 7.70) can be expresse through an infinite sum which we want to specify now. For this purpose we write (7.69) j () = ( ) 3 5 ( + ) + (( ) i (!( + 3 ) + (( i ) 4!( + 3 )( + 5 ) + (7.7) The factoria-type proucts 3 ( + ) can be expresse through the so-cae Gamma-function 4 efine through (7.7) Γ() = 0 t t e t. (7.73) This function has the foowing properties 5 Γ( + ) = Γ() (7.74) Γ(n + ) = n! for n N (7.75) Γ( ) = π (7.76) π Γ() Γ( ) = sin π. (7.77) from which one can euce reaiy Γ( + ) = π 3 ( + ). (7.78) One can write then j () = π ( ) n=0 ( i ) n n! Γ(n ). (7.79) 4 For further etais see Hanbook of Mathematica Functions by M. Abramowit an I.A. Stegun (Dover Pubications, New York) 5 The proof of ( ) is eementary; a erivation of (7.77) can be foun in Specia Functions of Mathematica Physics by A.F. Nikiforov an V.B. Uvarov, Birkhäuser, Boston, 988)

10 9 Sphericay Symmetric Potentias Simiary, one can express n () as given in (7.70) n () = π + Γ( + ) + ( i )!( ) + ( i ) 4!( )( 3 ) +. (7.80) Using (7.77) for = +, i.e., yies or Γ( + ) = ( ) n () = ( ) + π + n () = ( ) + π π Γ( ) (7.8) Γ( ) + ( i ) 4 +! Γ( )( )( 3 ) + ( ) + n=0 ( i )! Γ( )( ) ( i ) n. (7.8) n! Γ(n + ). (7.83) Linear inepenence of the Reguar an Irreguar spherica Besse Functions We want to emonstrate now that the soutions (7.69) an (7.69) of (7.53) are ineary inepenent. For this purpose we nee to emonstrate that the Wronskian W (j, n ) = j () n j () n (7.84) oes not vanish. Let f, f be soutions of (7.53), or equivaenty, of (7.58). Using f, one can emonstrate the ientity This equation is equivaent to the soution of which is = f, + ( + ) f, f, (7.85) W (f, f ) = W (f, f ) (7.86) n W = n (7.87) n W = c (7.88) for some constant c. For the case of f = j an f = n this constant can be etermine using the expansions (7.69, 7.70) keeping ony the eaing terms. One obtains c = an, hence, W (j, n ) =. (7.89) The Wronskian (7.89) oesn not vanish an, therefore, the reguar an irreguar Besse functions are ineary inepenent.

11 7.: Free Partice 93 Reationship to Besse functions The ifferentia equation (7.58) for the spherica Besse functions g () can be simpifie by seeking the corresponing equation for G + () efine through Using g () = G + (). (7.90) g () = g () = G + () G + () (7.9) G + () G + () G + () (7.9) one is ea to Besse s equation + ν + G ν () = 0 (7.93) where ν = +. The reguar soution of this equation is cae the reguar Besse function. Its power expansion, using the conventiona normaiation, is given by c.f. (7.79) J ν () = ( ) ν n=0 (i/) n n! Γ(ν + n + ). (7.94) One can show that J ν (), efine through (7.94), is aso a soution of (7.93). This foows from the fact that ν appears in (7.93) ony in the form ν. In the present case we consier soey the case ν = +. In this case J ν() is ineary inepenent of J ν () since the Wronskian W (J +, J ) = ( ) π (7.95) is non-vanishing. One can reate J + an J to the reguar an irreguar spherica Besse functions. Comparision with (7.79) an (7.83) shows j () = π J + () (7.96) n () = ( ) + π J (). (7.97) These reationships are empoye in case that since numerica agorithms provie the Besse functions J ν (), but not irecty the spherica Besse functions j () an n (). Exercise 7..: Demonstrate that expansion (7.94) is inee a reguar soution of (7.93). Aopt the proceures empoye for the function j (). Exercise 7..3: Prove the ientity (7.95).

12 94 Sphericay Symmetric Potentias Generating Function of Spherica Besse Functions The stationary Schröinger equation of free partices (7.43) has two soutions, namey, one given by (7.44, 7.45) an one given by (7.50). One can expan the former soution in terms of soutions (7.50). For exampe, in case of a free partice moving aong the x 3 -axis one expans e ik 3x 3 =,m a m j (kr) Y m (θ, φ). (7.98) The.h.s. can be written exp(ikr cos θ), i.e., the wave function oes not epen on φ. In this case the expansion on the r.h.s. of (7.98) oes not invove any non-vanishing m-vaues since Y m (θ, φ) for non-vanishing m has a non-trivia φ-epenence as escribe by (5.06). Since the spherica harmonics Y 0 (θ, φ), accoring to (5.78) are given in terms of Legenre poynomias P (cos θ) one can repace the expansion in (7.98) by e ik 3x 3 = b j (kr) P (cos θ). (7.99) =0 We want to etermine the expansion coefficients b. The orthogonaity properties (5.79) yie from (7.99) + cos θ e ikr cos θ P (cos θ) = b j (kr) +. (7.00) Defining x = cos θ, = kr, an using the Rorigues formua for Legenre poynomias (5.50) one obtains + x e ix P (x) = Integration by parts yies + x e ix! x (x ). (7.0) + x e ix P (x) =!! ix e + x (x ) ( x x eix + ) x (x ). (7.0) One can show x (x ) (x ) poynomia in x (7.03) an, hence, the surface term + vanishes. This hos for consecutive integrations by part an one can concue + x e ix P (x) = ( )! = (i)! + + x (x ) x eix x ( x ) e ix. (7.04)

13 7.: Free Partice 95 Comparision with (7.00) gives b j (kr) + = (i)! + x ( x ) e ix. (7.05) This expression aows one to etermine the expansion coefficients b. The ientity (7.05) must ho for a powers of, in particuar, for the eaing power x c.f. (7.69) b 3 5 ( + ) Empoying (5.7) one can write the r.h.s. + = (i)! + x ( x ). (7.06) or i! i ( + ) 3 5 ( + ) ()! 3 5 ( ) + + ()! 3 4 ( ) where the ast factor is equa to unity. Comparision with the.h.s. of (7.06) yies finay (7.07) (7.08) b = i ( + ) (7.09) or, after insertion into (7.99), e ikr cos θ = i ( + ) j (kr) P (cos θ). (7.0) =0 One refers to the.h.s. as the generating function of the spherica Besse functions. Integra Representation of Besse Functions Combining (7.05) an (7.09) resuts in the integra representation of j () j () = () +! + x ( x ) e ix. (7.) Empoying (7.96) one can express this, using ν = +, J ν () = πγ(ν + ) ( ν + x ( x ) ) ν e ix. (7.) We want to consier the expression G ν () = a ν ν f ν () (7.3) where we efine f ν () = C t ( t ) ν e it. (7.4)

14 96 Sphericay Symmetric Potentias Here C is an integration path in the compex pane with enpoints t, t. G ν (), for propery chosen enpoints t, t of the integration paths C, obeys Besse s equation (7.93) for arbitrary ν. To prove this we note f ν() = i t ( t ) ν t e it. (7.5) Integration by part yies f ν() = i ( t ) ν+ t e it ν + C t ν + In case that the enpoints of the integration path C satisfy ( t ) ν+ e it t one can write (7.6) or f ν() = From this we can concue ν + C f ν() = t ( t ) ν e it + ν + C t ( t ) ν+ e it. (7.6) t = 0 (7.7) C t ( t ) ν t e it (7.8) fν () + f ν (). (7.9) f ν () + ν + f ν() + f ν () = 0. (7.0) We note that equations (7.4, 7.0) impy aso the property (ν + ) f ν() + f ν+ () = 0. (7.) Exercise 7..4: Prove (7.). We can now emonstrate that G ν () efine in (7.3) obeys the Besse equation (7.93) as ong as the integration path in (7.3) satisfies (7.7). In fact, it hos for the erivatives of G ν () G ν() = ν a ν ν f ν () + a ν ν f ν() (7.) G () = ν(ν ) a ν ν f ν () + ν a ν ν f ν() + a ν ν f ν () (7.3) Insertion of these ientities into Besse s equation eas to a ifferentia equation for f ν () which is ientica to (7.0) such that we can concue that G ν () for proper integration paths is a soution of (7.93). We consier now the functions u (j) () = πγ(ν + ) ( ) ν x ( x ) ν e ix, j =,, 3, 4. (7.4) C j

15 7.: Free Partice 97 for the four integration paths in the compex pane parametrie as foows through a rea path ength s C (t = t = +) : t = s s C (t = t = + i ) : t = + is 0 s < C 3 (t = + i t = + i ) : t = + is s C 4 (t = + i t = ) : t = + is 0 s < (7.5) (7.6) (7.7) (7.8) C = C C C 3 C 4 is a cose path. Since the integran in (7.4) is anaytica in the part of the compex pane surroune by the path C we concue 4 u (j) () 0 (7.9) j= The integran in (7.4) vanishes aong the whoe path C 3 an, therefore, u (3) () 0. Comparision with (7.) shows u () () = J ν (). Accoringy, one can state J ν () = u () () + u (4) (). (7.30) We note that the enpoints of the integration paths C an C 4, for Re > 0 an ν R obey (7.7) an, hence, u () () an u () () are both soutions of Besse s equation (7.93). Foowing convention, we introuce the so-cae Hanke functions Accoring to (7.30) hos H () ν () = u () (), H () ν () = u (4) (). (7.3) For H ν () () one erives, using t = + is, t = i s, an the integra expression Simiary, one can erive J ν () = H ν () () + H ν () (). (7.3) ( t ) ν e it = ( t)( + t) ν e it = is( + is) ν e i e s = e i( πν/+π/4) ν s( + is/) ν e s, (7.33) H () ν () = e i( πν/ π/4) ν π Γ(ν + ) 0 s s( + is/) ν e s. (7.34) H ν () () = e i( πν/ π/4) ν π Γ(ν + ) s s( is/) ν e s. (7.35) 0 (7.34) an (7.35) are known as the Poisson integras of the Besse functions.

16 98 Sphericay Symmetric Potentias Asymptotic Behaviour of Besse Functions We want to obtain now expansions of the Hanke functions H (,) () in terms of such that + the expansions converge fast asymptoticay, i.e., converge fast for. We empoy for this purpose the Poisson integras (7.34) an (7.35) which rea for ν = + where The binomia formua yies H ( ) () = e ±i (+) π + π f ( ) () = f ( ) () = r=0 The formua for the Lapace transform of s n eas to an, hence, we obtain f ( ) () = H ( ) () = + r=0 0 )! f ( () (7.36) s s(s ± is/) e s. (7.37) ( ) ( ± i ) r s s +r e s. (7.38) r 0 ( + r)!! r!( r)! π e±i (+) π Besse Functions with Negative Inex ( ± i ) r ( ) +r+ (7.39) r=0 ( + r)! r!( r)! ( ± i ) r (7.40) Since ν enters the Besse equation (7.93) ony as ν, H ν () () as we as H () ν () are soutions of this equation. As a secon orer ifferentia equation the Besse equation has two ineary inepenent soutions. For such soutions g(), h() to be ineary inepenent, the Wronskian W (g, h) must be a non-vanishing function. For the Wronskian connecte with the Besse equation (7.93) hos the ientity W = W (7.4) the erivation of which foows the erivation on page 9 for the Wronskian of the raia Schröinger equation. The genera soution of (7.4) is W () = c. (7.4) In case of g() = H () () an h() = H () () one can ientify the constant c by using the + + eaing terms in the expansions (7.40). One obtains W () = 4i π, (7.43)

17 7.: Free Partice 99 i.e., H (,) () are, in fact, ineary inepenent. + One can then expan H () ν () = A H() () + B H () (). (7.44) + + The expansion coefficients A, B can be obtaine from the asymptotic expansion (7.40). the eaing terms yie e i(+ π ) = A e i( π π ) + B e i( π π ). (7.45) For This equation can ho ony for B = 0 an A = expi( + )π. We concue H () (+ )() = i ( ) H () (). (7.46) + Simary, one can show H () (+ )() = i ( ) H () (). (7.47) + Spherica Hanke Functions In anaogy to equations (7.90, 7.97) one efines the spherica Hanke functions h (,) () = π H(,) (). (7.48) + Foowing the arguments provie above (see page 93) the functions h (,) () are soutions of the raia Schröinger equation of free partices (7.53). Accoring to (7.96, 7.3, 7.48) hos for the reguar spherica Besse function j () = h() () + h () (). (7.49) We want to estabish aso the reationship between h (,) () an the irreguar spherica Besse function n () efine in (7.97). From (7.97, 7.3) foows n () = Accoring to (7.46, 7.47) this can be written π ( )+ H () (+ )() + H() (+ )() n () = Equations (7.49, 7.5) are equivaent to i h () () h () (). (7.50). (7.5) h () () = j () + i n () (7.5) h () () = j () i n (). (7.53)

18 00 Sphericay Symmetric Potentias Asymptotic Behaviour of Spherica Besse Functions We want to erive now the asymptotic behaviour of the spherica Besse functions h (,) (), j () an n (). From (7.40) an (7.48) one obtains reaiy h ( ) () = ( i)+ e ±i The eaing term in this expansion, at arge, is r=0 ( + r)! r!( r)! ( ± i ) r (7.54) h ( ) () = ( i)+ e ±i. (7.55) To etermine j () an n () we note that for R the spherica Hanke functions h () () an h () (), as given by (7.40, 7.48), are compex conjugates. Hence, it foows Using R : j () = Reh () (), n e () = Imh () (). (7.56) p () = Re i q () = i Im = r=0 { i r=0 ( + r)! r!( r)! ( + r)! r!( r)! ( ) i r = ( ) i r / r=0 ( + r)! (r)!( r)! / (+r+)! ( ) r r=0 (r+)!( r )! 4 0 = 0 ( ) r 4 (7.57) (7.58) one can erive then from (7.54) the ientities j () = cos ( + ) π n () = cos ( + ) π p () sin ( + ) π q () (7.59) cos( π/) q () + p (). (7.60) Empoying cos ( + ) π. = sin( π/) an sin ( + ) π. = cos( π/) resuts in the aternative expressions j () = n () = sin( π/) p () + sin( π/) q () The eaing terms in these expansions, at arge, are cos( π/) q () (7.6) cos( π/) p (). (7.6) j () = sin( π/) (7.63) n () = cos( π/). (7.64)

19 7.: Free Partice 0 Expressions for the Spherica besse Functions j () an n () The ientities (7.6, 7.6) aow one to provie expicit expressions for j () an n (). obtains for = 0,, j 0 () = sin n 0 () = cos j () = sin n () = cos j () = n () = cos sin ) ( 3 3 ( Recursion Formuas of Spherica Besse Functions The spherica Besse functions obey the recursion reationships One (7.65) (7.66) (7.67) (7.68) sin 3 cos (7.69) ) cos 3 sin (7.70) g + () = g () g () (7.7) g + () = + g () g () (7.7) where g () is either of the functions h (,) (), j () an n (). One can combine (7.7, 7.7) to obtain the recursion reationship ( ) ( ) g+ () g () g + () = A () g () (7.73) A () = (7.74) (+) + We want to prove these reationships. For this purpose we nee to emonstrate ony that the reationships ho for g () = h (,) (). From the inearity of the reationships ( ) an from (7.49, 7.5) foows then that the reatiosnhips ho aso for g () = j () an g () = n (). To emonstrate that (7.7) hos for g () = h (,) () we empoy (7.4, 7.3, 7.48) an express ( ν ) g () = h (,) π () = πγ(ν + ) x ( x ) ν e ix. (7.75) C,4 Using Γ( + ) =!, efining a = an empoying f ν () as efine in (7.4) we can write g () = a! f + (). (7.76)

20 0 Sphericay Symmetric Potentias The erivative of this expression is g () = g () + a! f (). (7.77) + Empoying (7.), i.e., yies, together with (7.76) f () = + + f + 3 (), (7.78) g () = g () g + () (7.79) from which foows (7.7). In orer to prove (7.7) we ifferentiate (7.79) g () = g () + g () g + (). (7.80) Since g () is a soution of the raia Schröinger equation (7.58) it hos g () = ( + ) g () + g () g (). (7.8) Using this ientity to repace the secon erivative in (7.80) yies g + () = g () ( + ) g () + + g (). (7.8) Repacing a first erivatives empoying (7.79) eas to (7.7). To prove (7.73, 7.74) we start from (7.7). The first component of (7.73), in fact, is equivaent to (7.7). The secon component of (7.73) is equivaent to (7.8). Exercise 7..5: Provie a etaie erivation of (7.7). Exercise 7..6: Empoy the recursion reationship (7.73, 7.74) to etermine (a) j (), j () from j 0 (), j 0 () using (7.65), an (b) n (), n () from n 0 (), n 0 () using (7.66).