Generalized Version of the Creation and Annihilation Operators for the Schro¨dinger Equation Santos, L. C. Na,∗, C.C. Barros Jra,∗ 7 1 aDepartamento de F´ısica - CFM - Universidade Federal de Santa Catarina, Florian´opolis - 0 SC - CP. 476 - CEP 88.040 - 900 - Brazil. 2 n a J 0 Abstract 3 A generalized version of the creation and annihilation operators is constructed ] h and the factorization of the Schr¨odinger equation is investigated. It is shown p - that the generalized version of factorization operators yield a factorization for h t the twelve different separable coordinates for the Schr¨odinger equation. a m Keywords: , factorization method, Schr¨odinger equation, operators [ 1 v 1. Introduction 1 8 5 The factorization method, introduced by Schr¨odinger [1] and Dirac [2] and 8 laterdevelopedbyInfeldandHull[3],isoneofthemethodsforsolvingquantum 0 . 1 mechanical problems. The idea is to consider a pair of first-order differential 0 7 5 equations which are equivalent to a given second-order differential equation. 1 Thecompletesetofnormalizedeigenfunctionscanbeobtainedbythesuccessive : v i application of the ladder operators on the eigenfunctions, which are the exact X solutions of the first order differential equation. r a Fromthe1970stotheearly1980,itwasacommonopinionthatthe method 10 was completely explored. However, Mielnik made an additional contribution to the traditional factorization method in 1984 [4]. In that work, he did not consider the particular, but the general solution to the Riccati type equation ∗Correspondingauthor Email addresses: [email protected] (Santos, L.C.N),[email protected] (C.C.BarrosJr) Preprint submitted toArxiv January 31, 2017 connected with the Infeld-Hull approach. Mielnik factorization is a powerful tool in the derivation of new Hamiltonians whose corresponding eigenproblem 15 is analytically solvable. On the other hand, the connection between the Infeld methodandsupersymmetricquantummechanics(SUSYQM)hasbeenexplored by many authors [5, 6, 7, 8]. For example, Witten noticed the possibility of arranging the second-order differential equations into isospectral pairs, the so- called supersymmetric partners. 20 In many works, the factorization method has been used as a tool to formu- late algebraic approaches to many non-relativistic quantum problems, the idea is build sets of one variable radial operators which are realizations for su(1,1) Lie algebra[9,10]. The separable coordinatesystems for the Schr¨odingerequa- tion are confocal quadric surfaces [11], and the potential is a function of the 25 coordinates [12]. In this paper, the possibility of factorization of the separated equations is investigated,anditisshownthatageneralizedversionofthecreationandanni- hilationoperatorscanbeconstructed. Indeed,theseoperatorsyieldafactoriza- tionfor the twelve differentseparablecoordinatesfor the Schr¨odingerequation. Inthe Infeldmethod, the originalformofthe second-orderdifferentialequation d dψ p +q(θ)ψ+λρ(θ)ψ =0, (1) dθ dθ (cid:18) (cid:19) is transformed in the form d2y +r(x,m)y+λy =0, (2) dx2 wherem=1,2,3...andp,ρarepositivefunctions. Thetransformationconnect- ing these equations is y =(pρ)1/2ψ, dx=(ρ/p)1/2dθ. (3) In this work, we propose to apply a factorization method for the original form of the separated Schr¨odinger equation. In this way, our approach yield fac- torization operators for the twelve different separable coordinate systems. An 2 interesting feature of this work is that the original Hilbert space of theory is 30 sustained. So, this paper will show the following contents: In section II, we enumerate thecoordinatessystemswhichwillallowseparationoftheSchr¨odingerequation. InsectionIII, ageneralizedversionofthecreationandannihilationoperatorsis proposed. InsectionIVandV, we apply the method to the radialsecond-order 35 Schr¨odinger equation. 2. Separable coordinate systems for the wave equation Inthissection,westudytheseparablecoordinatesystemsfortheSchr¨odinger equation, for more details and information see [13]. We consider the standard differential equation for the scalar field 2ψ+k2ψ =0, (4) ∇ 1 where ▽2 is the Laplaceoperator. When k is a function of the coordinates,we 1 obtain the Schr¨odinger equation. The rectangular coordinates x, y, z and the curvilinear coordinates ξ , ξ , ξ are related by the scale factors 1 2 3 2 2 2 ∂x ∂y ∂z h = + + , (5) n s(cid:18)∂ξn(cid:19) (cid:18)∂ξn(cid:19) (cid:18)∂ξn(cid:19) where n=1,2,3. The Laplacian can be expressed in its generalized form 1 ∂ h h h ∂ψ 2ψ = 1 2 3 , (6) ∇ h h h ∂ξ h2 ∂ξ n 1 2 3 n (cid:20) n n(cid:21) X thus we can rewrite the equation (4) into 1 ∂ h h h ∂ψ 1 2 3 +k2ψ =0. (7) h h h ∂ξ h2 ∂ξ 1 n 1 2 3 n (cid:20) n n(cid:21) X In order to obtain the separated equations, we introduce the St¨ackel deter- 3 minant Φ Φ Φ 11 12 13 S =|Φmn|=(cid:12)(cid:12)(cid:12) Φ21 Φ22 Φ23 (cid:12)(cid:12)(cid:12) (cid:12) (cid:12) (cid:12) Φ31 Φ32 Φ33 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) =Φ Φ Φ(cid:12) +Φ Φ Φ +Φ(cid:12) Φ Φ 11 22 (cid:12)33 12 23 31 (cid:12) 13 21 32 Φ Φ Φ Φ Φ Φ Φ Φ Φ , (8) 13 22 31 11 23 32 12 21 33 − − − where Φ are functions of ξ alone. mn n If the separated equations for the three-dimensional case are 1 d dX f (ξ ) m + Φ (ξ )k2X =0, (9) f (ξ )dξ m m dξ mn m n m m m m (cid:20) m (cid:21) n X we can relate the equations (7) and (9) by the Robertson condition [13] h h h 1 2 3 =f (ξ )f (ξ )f (ξ ), 1 1 2 1 3 1 S which limits the kinds of coordinates systems which will allow separation. The equation(9) is supposedto be a separatedequation,thereforethe functions f , n 40 Φn1, Φn2, and Φn3 must all be functions of ξn alone. 3. Creation and annihilation operators In this section, we propose a generalized versionof the creation and annihi- lation operators. For the Schr¨odinger equation the constant k must have the 1 3 formk2 =ε v (ξ )[13],wherev (ξ )= 2mV (ξ ),substituting itintothe 1 − n n n n ~2 n n n equation (9) gPives 1 d dX m f (ξ ) + m m f (ξ )dξ dξ m m m (cid:20) m (cid:21) 3 Φ (ξ ) ε v (ξ ) X + m1 m n n m " − # n X Φ (ξ )k2X +Φ (ξ )k2X =0. (10) m2 m 2 m m3 m 3 m 4 Introducing k′ =ε, k =k′, k =k′ into (10) we obtain 1 2 2 2 2 1 d dX m f (ξ ) m m f (ξ )dξ dξ m m m (cid:20) m (cid:21) 3 + k′2Φ (ξ )X v X m nm m n− n n m6=n X =k′2Φ X . (11) n nn n These are the separated equations for the three-dimensional space. Now we proposetoapplyafactorizationmethodfortheseparatedSchr¨odingerequation (11). The idea is to define two ladder operators d 1 df m A = + R(ξ ), (12) m dξ 2f dξ − m m m d 1 df A+ = + m +R(ξ ), (13) m dξ 2f dξ m m m and to show that these operators yield a factorization of the equation (11). Indeed, if we multiply Aby A+, we find 2 1 d dX 1 df AA+ = f m m f dξ m dξ − 4f2 dξ m m (cid:18) m (cid:19) m (cid:18) m(cid:19) 1 d2f + m +R′ R2, (14) 2f dξ2 − m m and comparing with equation (11) we find a Riccati type equation R′ R2 =ǫ+Γ(ξ ), (15) n − where ǫ is a constant and 1 df 2 1 d2f 3 Γ(ξ )= m m + k′2Φ v . (16) n 4f2 dξ − 2f dξ2 m nm− n m (cid:18) m(cid:19) m m m6=n X The occurrence of the Ricatti equation in the factorization of second-order di- fferentialequationsisatypicalphenomenon. Specifically,the factorizationope- rators convert the equation (11) into product of A and A+ with a extra con- 45 dition, a Riccati type equation. The explicit solution of this type of equation, in ge-neral, is not Known [14]. In the following, we will obtain two particular solutions of the Riccati equation (15) in a spherical coordinate system for the Coulomb and isotropic oscillator potentials. 5 4. Application to hydrogen atom WewanttoshowhowtosolvetheradialSchr¨odingerequationwithCoulomb potential. Weapplythemethodtotheradialsecond-orderdifferentialequation, inthis case,the sphericalcoordinatesaredenotedbyξ =r, ξ =θ, ξ =φand 1 2 3 the f functions are f =r2, f =1 cos2θ, f = 1 cos2φ . Therefore the n 1 2 3 − − s matrix is given by p 1 1 0 r2 S = 0 1 1 . (17) cos2θ−1 (cos2θ−1)2  0 0 cos21φ−1    The separation constants are then required to be of the form k2 =ε, k2 = l(l+1), k3 =m. 1 2 − 3 So the radial Schr¨odinger equation with potential v = K/r is n − d2X 2dX l(l+1) K 1 1 + + = εX , (18) dr2 r dr − r2 r − 1 and the Ricatti equation is l(l+1) K R′ R2 =ǫ + , (19) − − r2 r the particular solution for this equation is given by l K R= , r − 2l where ǫ = K2. The creation and annihilation operators in (12) and (13) can −4l2 be written in the form d 1 l K A+ = + + , l dr r r − 2l d 1 l K A = + + . l dr r − r 2l The commutator of A and A+ is r dependent l l 2l A+,A =A+A A A+ = . (20) l l l l− l l r2 (cid:2) (cid:3) 6 Indeed, the A and A+ are creation and annihilation operators. We can prove l l this directly, the first step is to consider the product between the operators A+A and the radial wave function X , i.e, l l n,l−1 K2 A+A X = H X l l n,l−1 l−1− 4l2 n,l−1 (cid:18) (cid:19) K2 = ε X , (21) n,l−1− 4l2 n,l−1 (cid:18) (cid:19) in a similar way K2 A A+X = H X l l n,l l− 4l2 n,l (cid:18) (cid:19) K2 = ε X . (22) n,l− 4l2 n,l (cid:18) (cid:19) A direct calculation shows that these operators satisfy 2l [H ,A ]= A (23) l l r2 l and 2l H ,A+ = A+, (24) l−1 l −r2 l so the action of the A and H(cid:2) on the s(cid:3)tates X is l l n.l 2l H AX =A +H X l l n.l l r2 l n.l (cid:18) (cid:19) =ε A X , (25) n,l−1 l n.l this result imply X A X n.l−1 l n.l ∝ or X =cA X , (26) n.l−1 l n.l where c is a constant. Equivalently X =cA+X . (27) n.l l n.l−1 These results imply that the action of the operators A and A+ on the states l l X and X is to change the quantum number l. In order to determinate n.l n.l−1 c, we apply A to the left-hand side of eq. (27) l A X =cA A+X (28) l n.l l l n.l−1 7 Therefore, using equation (26) we find 1 c= . (29) ε K2 n,l− 4l2 q (cid:0) (cid:1) 5. Radial harmonic oscillator 50 Solutions of equation (11) are limited to a small set of potentials and the radialharmonicoscillatorisoneofthefewquantumsystemswhereanexactand analyticalsolutionisknown. Thetime-independentSchr¨odingerradialequation for the isotropic oscillator v =kr2 reads n d2 2 d l(l+1) + kr2 X = εX , (30) dr2 rdr − r2 − 1 − 1 (cid:20) (cid:21) for this case, the Riccati equation is l(l+1) R′ R2 =ǫ kr2 (31) − − r2 − the particular solution of equation (31) is given by l R= √kr, ǫ=√k(2l 1). (32) r − − This solution R leads to factorizing operators d 1 l A = + +√kr, (33) l dr r − r and d 1 l A+ = + + √kr. (34) l dr r r − Hence, we get the following commutation rules 2l A+,A = +2√k l l r (cid:2) (cid:3) and [H ,A ]=A A+,A l l l l l =A (cid:2) 2l +2(cid:3)√k , (35) l r (cid:18) (cid:19) 8 H ,A+ =A+ A,A+ l−1 l l l l (cid:2) (cid:3)= A+(cid:2) 2l +(cid:3)2√k , (36) − l r (cid:18) (cid:19) thus, the action of the A and H on the states X is l l n.l H A X =ε A X . (37) l l n,l n,l−1 l n,l Equivalently, the action of the A+ and H on the states X leads to l l−1 n.l−1 H A+X =ε A+X , (38) l−1 l n.l−1 n,l l n,l−1 thus, A+ andA are the raisingandloweringoperatorsfor the isotropicoscilla- l l tor. It follows that X =cA X , X =cA+X , (39) nl−1 l nl nl l nl−1 where 1 c= . ε +√k(2l 1) nl − q 6. Summary and conclusions In this paper, we have determined the separable coordinate systems for the Schr¨odingerequation. Fromtheseresults,factorizationoperators forthetwelve different separable coordinates have been provided. We have determined the 55 Schr¨odinger equation in the presence of a Coulomb potential and a radial har- monic oscillator potential. We have shown that, a new set of generalized cre- ation and annihilation operators has been introduced. By using the apparatus developed in this work, we believe that other potentials in different coordinate systems can be solved. 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