Etats cohérents et intelligents généralisés pour des systèmes quantiques exactement solubles

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(1)UNIVERSITE MOHAMMED V - AGDAL FACULTE DES SCIENCES RABAT Numéro d’ordre : 2294. THESE DE DOCTORAT Présentée par : Abdelhamid EL KINANI. Discipline : Physique Spécialité : Physique-Mathématique ETATS COHERENTS ET INTELLIGENTS GENERALISES POUR DES SYSTEMES QUANTIQUES EXACTEMENT SOLUBLES Soutenue le 1 Février 2006 à 10 heure, devant le Jury : Président F. GUEDIRA. : Professeur à la faculté des sciences de Rabat. Examinateurs A. INTISSAR. : Professeur. à la faculté des sciences de Rabat. E. H. SAIDI. : Professeur à la faculté des sciences de Rabat. M. DAOUD. : Professeur à la faculté des sciences Ibn Zohr Agadir. M. B. SEDRA. : Professeur à la faculté des sciences de Kenitra. M. ZAKKARI. : Professeur à la faculté des sciences et techniques de Settat. Faculté des sciences, 4 Avenue Ibn Battouta B.P. 1014, RP, Rabat-Maroc Tél +212 (0) 37 77 18 34/35/38. Fax : +212 (0) 37 77 42 61, http://www.fsr.ac.ma.

(2) Je dédie ce travail à :. - La mémoire de ma mère - Mon père - Mes frères - Mes sœurs - Mes amis - Celle que je cherche encore... -Tous ceux qui me sont chères.

(3) Remerciements Les travaux présentés dans cette thèse ont été effectués au sein du laboratoire de PhysiqueThéorique à la Faculté des sciences de Rabat. Cette thèse a été dirigée par M. Mohammed Daoud, Professeur à la Faculté des Sciences Ibn Zohr d’Agadir; il m’est très difficile de lui exprimer en quelques mots ma profonde gratitude. J’ai eu l’honneur d’être son étudiant et le grand plaisir d’accomplir ce travail sous sa direction. J’ai toujours bénéficié de son aide précieuse, de sa haute compétence et patience. Qu’il trouve ici l’expression de ma haute considération pour m’avoir orienté et avoir suivi mes premiers pas d’un œil bienveillant. Je ne remercierai jamais assez le Professeur Mohammed Baghdadi pour ses conseils, ses encouragements ainsi que pour ses séminaires qui m’ont été d’une énorme utilité. Qu’il me soit permis aussi de remercier Messieurs : -. Abdellah El Hassouni, Professeur à la Faculté des Sciences de Rabat, pour ses cours et ses conseils aussi pour m’avoir accepté dans l’UFR de Physique théoriques.. -. Yassine Hassouni, Professeur à la Faculté des Sciences de Rabat pour m’avoir encadrer durant la préparation de mon DESA.. Je tiens à remercier vivement Mme. Fouzia Guedira Professeur à la Faculté des Sciences de Rabat pour avoir accepter de présider le jury de cette thèse. Je remercie également : -. M. Ahmed Intissar, Professeur à la Faculté des Sciences de Rabat,. -. M. El Hassan Saidi, Professeur à la Faculté des Sciences de Rabat,. -. M. My Brahim Sedra, Professeur à la Faculté des Sciences de Kenitra,. -. M. Mohammed Zakkari, Professeur à la Faculté des Sciences et Techniques de Settat,. du grand honneur qu’ils m’ont fait en acceptant d’être membres de jury. Mes remerciements vont également à M. Jean-Pierre Antoine, Professeur de physique mathématique à l’université Catholique de Louvain, pour avoir accepter de faire un rapport sur nos travaux et ce malgré ses multiples occupations. Je tiens aussi à remercier tous les membres du laboratoire pour leurs remarques et discussions scientifiques très utiles que nous avons entamé lors des séminaires de notre groupe de recherche qui m’ont permis de mener à bien cette thèse. Enfin je remercie tous mes amis pour leur aide précieuse et pour l’affection dont ils ont fait preuve..

(4) Numéro d’ordre : 2294. Titre de la thèse : «Etats cohérents et intelligents généralisés pour des systèmes quantiques exactement solubles» Prénom, Nom : Abdelhamid EL KINANI Résumé : Ce travail traite de la construction des états cohérents et intelligents pour des systèmes quantiques exactement solubles. Dans ce sens, nous développons deux approches différentes conduisant aux états cohérents de type Gazeau-Klauder et Klauder-Perelomov. Les méthodes et les techniques de la mécanique quantique supersymétrique nous ont permis de définir les opérateurs de création et d’annihilation associés à un système quantique exactement soluble. Ces opérateurs sont indispensables pour la génération des états cohérents. Nous avons également étudié les représentations analytiques de Fock-Bargmann correspondantes à chaque classe d’états cohérents obtenue. Nous montrons que ces réalisations analytiques permettent d’obtenir de façon aisée les états intelligents pour un système quantique donné. Ces derniers états minimisent la relation d’incertitude de Robertson-Schrödinger. En guise d’illustrations des résultats obtenus dans ce mémoire, nous nous sommes intéressé aux systèmes quantiques plongés dans des potentiels de type Pöschl-Teller, Morse, puits infinis et les oscillateurs non-linéaires.. Mots-clefs (5): Etats cohérents, états intelligents, relation d’incertitude de Robertson-Schrödinger, représentation analytique de Fock-Bargmann, systèmes quantiques exactement solubles, mécanique quantique supersymétrique..

(5) Sommaire Introduction et motivations.....................................................................................................................1 Chapitre I : Etats cohérents des algèbres symplectiques..................................................................8 I. 1. Introduction.............................................................................................................................................8 I. 2. Etats cohérents de l’algèbre sp(1,R)........................................................................................................9 I. 2. 1. Algèbre sp(1,R)...................................................................................................................................9 I. 2. 2. Etats cohérents de Barut-Girardello…………..................................................................................10 I. 2. 2. 1. Représentation analytique..............................................................................................................11 I. 2. 3. Etats cohérents de Klauder-Perelomov..............................................................................................12 I. 2. 3. 1. Théorie générale............................................................................................................................12 I. 2. 3. 2. Etats cohérents de Klauder-Perelomov pour l’algèbre sp(1,R).....................................................14 I. 2. 3. 3. Représentation analytique..............................................................................................................15 I. 3. Etats cohérents des algèbres symplectiques sp(N,R).............................................................................16 I. 3. 1. Algèbre symplectiques sp(N,R).........................................................................................................16 I. 3. 2. Etats cohérents de Barut-Girardello pour les algèbres sp(N,R).........................................................16 I. 4. Conclusion.............................................................................................................................................19. Chapitre II: Etats cohérents des superalgèbres de Lie orthosymplectique osp(1/2) et osp(2/2)..20 II 1. Introduction...........................................................................................................................................20 II. 2. Généralités sur les superalgèbres de Lie..............................................................................................21 II. 2. 1. Algèbres graduées............................................................................................................................21 II. 2. 2. Superalgèbres de Lie........................................................................................................................22 II. 3. Etats cohérents des superalgèbres de Lie orthosymplectiques............................................................24 II. 3. 1. Les Superalgèbres de Lie orthosymplectiques................................................................................24 II. 3. 2. Superétats cohérents associés à la superalgèbre osp(1/2)...............................................................25 II. 3. 2. 1. Stucture algèbrique de la superalgèbre osp(1/2).........................................................................25 II. 3. 2. 2. Représentations de la superalgèbre osp(1/2)...............................................................................25 II. 3. 2. 3. Superétats cohérents de Klauder-Perelomov pour osp(1/2)........................................................26 II. 3. 3. Superétats cohérents associés à la Superalgèbre osp(2/2)...............................................................28 II. 3. 3. 1. La Superalgèbre de Lie orthosymplectique osp(2/2)..................................................................28 II. 3. 3. 2. Superétats cohérents de Klauder-Perelomov pour osp(2/2)........................................................30 II. 4. Conclusion...........................................................................................................................................32. Chapitre III : La théorie générale des états intelligents.................................................................................................................................................33 III. 1. Introduction.......................................................................................................................................33 III. 2. La relation d’incertitude de Robertson-Schrödinger.........................................................................34 III. 2. 1. Cas de deux observables................................................................................................................34 II.I 2. 2. Cas de n observables......................................................................................................................36 III. 2. 2. 1. Matrice des dispersions..............................................................................................................36 III. 2. 2. 2. L’inégalité de Robertson-Schrödinger.......................................................................................36 III. 3. Diagonalisation de la matrice des dispersions...................................................................................38 III. 4. La théorie générale des états intelligents...........................................................................................40 III. 4. 1. Minimisation de La relation d’incertitude de Robertson-Schrödinger..........................................40 III. 4. 2. Expressions explicites des états intelligents..................................................................................43 III. 4. 2. 1. Cas de l’algèbre de Lie su(1,1)..................................................................................................43 III. 4. 2. 2. Cas de l’algèbre de Lie su(2).....................................................................................................45.

(6) III. 5. Conclusion.........................................................................................................................................48. Chapitre IV : Systèmes quantiques arbitraires états cohérents et intelligents........................49 IV. 1. Introduction.......................................................................................................................................49 IV. 2. Systèmes quantiques arbitraires........................................................................................................50 IV. 2. 1. Opérateurs de création et d’annihilation.......................................................................................50 IV. 3. Etats cohérents de Gazeau-Klauder...................................................................................................55 IV. 3. 1. Etats propres de l’opérateur d’annihilation...................................................................................55 IV. 3. 2. La représentation de Fock-Bargmann...........................................................................................57 IV. 4. Etats cohérents de Klauder-Perelomov.............................................................................................59 IV. 5. Etats intelligents généralisées............................................................................................................61 IV. 6. Applications.......................................................................................................................................65 IV. 6. 1. Le potentiel de Pöschl-Teller.........................................................................................................66 IV. 6. 1. 1. Etats cohérents de Gazeau-Klauder...........................................................................................68 IV. 6. 1. 2. Etats cohérents de Klauder-Perelomov.....................................................................................70 IV. 6. 1. 3. Etats intelligents généralisés.....................................................................................................73 IV. 6. 2. Le potentiel de Morse...................................................................................................................77 IV. 6. 2. 1. Etats cohérents de Klauder-Perelomov.....................................................................................80 IV. 7. Conclusion.........................................................................................................................................81. Contributions................................................................................................................................83 Article 1 : Coherent states à la Klauder–Perelomov for the Pöschl–Teller potentials..................84 A. H. El Kinani and M. Daoud, Phys. Lett. A 283, 291-299 (2001). Article 2 : Generalized intelligent states for an arbitrary quantum system.……………………..90 A. H. El Kinani and M. Daoud, J. Phys. A : Math. Gen. 34, 5373-5387 (2001). Article 3 : Generalized intelligent states for nonlinear oscillators…………….…………….…..99 A. H. El Kinani and M. Daoud, IJMPB, Vol 15, 2465-2483 (2001). Article 4 : Generalized coherent and intelligent states for exact solvable quantum systems.….110 A. H. El Kinani and M. Daoud, J. Math. Phys. Vol 43, 714-733 (2002).. Article 5 : Coherent and generalized intelligent states for infinite square well potential and non-linear oscillators……………………………………………...…….……….…….........121 A. H. El Kinani and M. Daoud, IJMPB, Vol 16, 3915-3937 (2002).. Conclusion générale....................................................................................................................133 Références....................................................................................................................................135.

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(91) Article 1.

(92) 21 May 2001. Physics Letters A 283 (2001) 291–299 www.elsevier.nl/locate/pla. Coherent states à la Klauder–Perelomov for the Pöschl–Teller potentials A.H. El Kinani a,∗ , M. Daoud b a LPT, Physics Department, Faculty of Sciences, University Mohammed V, P.O. Box 1014, Rabat, Morocco b LPMC, Physics Department, Faculty of Sciences, University Ibn Zohr, P.O. Box 28/S, Agadir, Morocco. Received 28 March 2001; received in revised form 10 April 2001; accepted 11 April 2001 Communicated by V.M. Agranovich. Abstract In this Letter we present a scheme for constructing the coherent states of Klauder–Perelomov’s type for a particle which is trapped in Pöschl–Teller potentials.  2001 Elsevier Science B.V. All rights reserved.. 1. Introduction Coherent states are one of the important concepts in physics today (for review, see Refs. [1–3]). The original coherent states based on the Heisenberg–Weyl group has been extended for a number of Lie groups with square integrable representations, and they have many applications in quantum mechanics. In particular, they are used as bases of coherent states path integrals [4] or dynamical wavepackets for describing the quantum systems in semiclassical approximations [5]. Many definitions of coherent states exist. The first one defines the usual coherent states as the eigenstates of the annihilation operator a − for each individual oscillator mode of the electromagnetic field: a − |z = z|z.. (1). Here [a − , a + ] = 1 ((a − )† = a + ) and z is a complex constant with conjugate z¯ . The unit normalized states |z are given by |z = e−|z|. 2 /2. ∞ zn √ |n, n! n=0. (2). where |n is an element of the Fock space H ≡ {|n, n 0}. A second definition of coherent states for oscillators assumes the existence of a unitary “displacement” operator D(z) = exp za + − z¯ a − , (3) * Corresponding author.. E-mail address: ham_kin@hotmail.com (A.H. El Kinani). 0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 2 4 9 - 3.

(93) 292. A.H. El Kinani, M. Daoud / Physics Letters A 283 (2001) 291–299. whose action on the ground states |0 give the coherent states (2) parametrized by z. The unitarity of D(z) ensures the correct normalization of |z. In view of the canonical commutation relation [a − , a + ] = 1, the second definition is equivalent to the first one. A third definition is based on the Heisenberg uncertainty relation with the position x and momentum p given, as usual, by 1 x = √ a− + a+ 2. i and p = √ a + − a − , 2. (4). the coherent states defined above have the minimum-uncertainty value xp = 1/2 and maintain this relation in time (temporal stability of coherent states). Coherent states have two important properties. First, they are not orthogonal to each other. Second, they provide a resolution of the identity, i.e., they form an overcomplete set of states. Following the first approach, a formalism, in which coherent states are defined as eigenstates of a lowering operator, has been proposed by Gazeau and Klauder for an arbitrary quantum mechanical system [6] (see also Refs. [7–9]). Recently, we gave a complete classification of states minimizing the Robertson–Schrödinger uncertainty relation for an exactly solvable quantum system [10,11]. We obtained the so-called “generalized intelligent states”. This approach follows the definition of coherent states related to the optimization of Heisenberg relation of familiar harmonic oscillator coherent states. But, up to now, a formalism defining the coherent states for an arbitrary quantum system as the action of some displacement operator on a reference state (coherent states of Klauder– Perelomov kind) has not been considered in the literature as far as we know. Hence, the main purpose of this Letter is the construction of the coherent states of Klauder–Perelomov’s type for a quantum mechanical system evolving in the Pöschl–Teller potentials. This Letter is organized as follows. In Section 2, we review the factorization of the Pöschl–Teller Hamiltonian and we introduce creation and annihilation operators which will be useful for our purpose. Section 3 is devoted to the definition of the set of coherent states of Klauder–Perelomov’s type for the quantum system under consideration. In Section 4, we discuss and compare the three definitions of coherent states. Other properties such as stability in time, overcompleteness, and resolution to the unity are discussed. Concluding remarks are summarized in the last section.. 2. Creation and annihilation operators We start by recalling the eigenvalues and eigenvectors of a particle trapped in the Pöschl–Teller potentials of trigonometric type. The corresponding Hamiltonian is given by [12] H =−. d2 + Vκ,κ (x), dx 2. (5). where Vκ,κ (x) is the family of Pöschl–Teller potentials indexed continuously by the parameters κ > 1 and κ > 1: κ(κ−1) κ (κ −1) (κ+κ )2 1 Vκ,κ (x) = 4a 2 sin2 (x/2a) + cos2 (x/2a) − 4a 2 , 0 < x < πa, (6) ∞, x 0, x πa. It is also called, sometimes, the Pöschl–Teller of first type. Clearly, this is a smooth approximation for κ, κ → 1+ of the infinite square-well potentials over the interval [0, πa]. The Pöschl–Teller potentials is closely related to several other potentials which are widely used in molecular and solid state physics like (i) the symmetric Pöschl– Teller potentials well (κ = κ 1), (ii) the Scarf potentials 1/2 κ < 1 [13], (iii) the modified Pöschl–Teller potentials which can be obtained by replacing the trigonometric functions by their hyperbolic counterparts [12,14], (iv) the Rosen–Morse potential which is the symmetric modified Pöschl–Teller potentials [15]. Many interesting.

(94) A.H. El Kinani, M. Daoud / Physics Letters A 283 (2001) 291–299. 293. properties of the Pöschl–Teller potentials was recently reexamined at classical as well as at quantum levels, in a nice paper by Antoine et al. [8]. The Hamiltonian H can be written in the following factorized form: + − H = aκ,κ aκ,κ ,. (7). − + where the annihilation aκ,κ and the creation aκ,κ operators are given by. d + Wκ,κ (x), dx in terms of the superpotentials Wκ,κ (x): . x 1 x κ cot − κ tan . Wκ,κ (x) = 2a 2a 2a ± aκ,κ =∓. The normalized eigenstates ψn (x) and the corresponding eigenvalues en are given by κ κ −1/2 x x x (κ−1/2,κ −1/2) Pn ψn (x) = cn (κ, κ ) cos sin cos , 2a 2a a. (8). (9). (10). where cn (κ, κ ) is a normalization factor that can be evaluated, and en = n(n + κ + κ ),. n = 0, 1, 2, . . . .. (11). (α,β) ’s Pn. stand for the Jacobi polynomials. In Eq. (10), the We define the actions of the creation and annihilation operators on the eigenstates |ψn as follows:. + (n + 1)(n + 1 + κ + κ )e−iα(2n+1+κ+κ ) |ψn+1 , aκ,κ |ψn =. − aκ,κ n(n + κ + κ )eiα(2n−1+κ+κ ) |ψn−1 , |ψn =. (12) (13). where the real parameter α plays an important role in the temporal stability of coherent states. This remark will + − be clarified in the sequel of this Letter. From Eqs. (12) and (13), one can verify that the operators aκ,κ and aκ,κ satisfy the following commutation relation: − + aκ,κ aκ,κ = Gκ,κ (N), (14) where Gκ,κ (N) ≡ G = 2N + κ + κ + 1,. (15). and the operator N is defined by his action on states |ψn as N|ψn = n|ψn . We note that N. + − = aκ,κ aκ,κ . (16) = H.. 3. Coherent states of Klauder–Perelomov’s type In view of the second definition of coherent states (CS) for the standard harmonic oscillator, we define the CS of Klauder–Perelomov’s type for a particle submitted to Pöschl–Teller potentials as follows: + − |z, α = exp zaκ,κ (17) z ∈ C. − zaκ,κ |ψ0 , So, we have to compute the action of the unitary operator + − D(z) = exp zaκ,κ − zaκ,κ . (18).

(95) 294. A.H. El Kinani, M. Daoud / Physics Letters A 283 (2001) 291–299. on the ground state |ψ0 of the system under consideration. Using the action of the annihilation and the creation operators on the Hilbert space H = {|ψn , n = 0, 1, 2, . . .} (Eqs. (12) and (13)), one can show that the state |z, α can be written, in a compact form, as |z, α =. +∞ . an |z| zn e−iαen |ψn .. (19). n=0. The quantities an (|z|) in (19) are defined by

(96) (n + 1)(n + κ + κ + 1) cn |z| , an |z| = (κ + κ + 1) and the coefficients cn (|z|) are given by n+1 i +1 ij−1 +1 +∞ 1 (−|z|2 )j ei1 ei2 . . . eij . cn |z| = (n + 2j )! j =0. i1 =1. i2 =1. (20). (21). ij =1. Setting π(n + 1, j ) =. n+1 . ei1. i1 =1. i 1 +1. ij−1 +1. ei2 . . .. i2 =1. . eij. and π(n + 1, 0) = 1,. (22). ij =1. one can verify that the π ’s satisfy the following relation: π(n + 1, j ) − π(n, j ) = (n + 1)(n + κ + κ + 1)π(n + 2, j − 1).. (23). Using this recurrence formula, it is not difficult to show that the cn (|z|)’s satisfy the following differential equation: |z|. dcn (|z|) = cn−1 |z| − ncn |z| − (n + 1)(n + 1 + κ + κ )|z|2 cn+1 |z| . d|z|. (24). Hence, solving this equation, we can explicitly obtain the Pöschl–Teller coherent states of Klauder–Perelomov’s type. The solutions of the differential equation (24) for κ + κ ∈ N \ {0, 1, 2} are of the form cn |z| =. 1 −(1/2)(κ+κ +1) ßm,n+(1/2)(κ+κ +1) cosh 2|z| , n n!|z|. (25). because the Jacobi functions ß satisfy d l ßm,n−l cosh 2|z| = nßlm,n−1−l cosh 2|z| − (2l + n)ßlm,n+1−l cosh 2|z| , d|z|. (26). where l = −(1/2)(κ + κ + 1), and m is free integer or half-integer parameter. It is clear that the differential equation (24) admits several solutions depending on m. However, an admissible solution in our case is obtained by noting that D(z = 0) = 1 (unit operator). Recall that the Jacobi functions are defined by [16] ßlm,n cosh(2x) = (l + n + 1)(l − n + 1)(cosh x)2l (tanh x)n−m ×. +∞ s=max(0,m−n). (tanh x)2s ((s + 1))−1 ((l − n − s + 1))−1 , (n − m + s + 1)(l + m + 1). (27). where l is any complex number, and m, n are both integer or both half-integers. These functions play an important role in the representation theory for the group QU(2) of unimodular quasi-unitary matrices. Then, using.

(97) A.H. El Kinani, M. Daoud / Physics Letters A 283 (2001) 291–299. 295. definition (27), we obtain the unique solution compatible with the condition D(z = 0) = 1, cn |z| =. 1 −(1/2)(κ+κ +1) ß(1/2)(κ+κ +1),n+(1/2)(κ+κ +1) cosh 2|z| , n n!|z|. (28). which can be written also as. 1 −(κ+κ +1) tanh |z| n . cn |z| = cosh |z| n! |z|. (29). The coherent states of Klauder–Perelomov’s type take now the form . 1/2 +∞ (1/2)(κ+κ +1) z tanh |z| n (n + 1 + κ + κ ) |z, α = 1 − tanh |z| e−iαn(n+κ+κ ) |ψn . |z| (n + 1)(1 + κ + κ ). (30). n=0. Finally, setting ζ = z tanh |z|/|z|, we obtain +∞ (1/2)(κ+κ +1) |ζ , α ≡ |z, α = 1 − |ζ |2 ζn n=0. . (n + 1 + κ + κ ) (n + 1)(1 + κ + κ ). 1/2. . e−iαn(n+κ+κ ) |ψn .. The parameter ζ is restricted by |ζ | < 1. The identity resolution is |ζ , αζ , α| dµ(ζ ) = IH ,. (31). (32). where the measure is given by dµ(ζ ) =. d 2ζ κ + κ . π (1 − |ζ |2 )2. (33). There are two main consequence arising from the former result. First, one can express any coherent state |ζ , α in terms of the others: |ζ , α = |ζ , αζ , α|ζ , α dµ(ζ ). (34) The kernel ζ , α|ζ , α is easy to evaluate from (31): (1/2)(κ+κ +1) (1/2)(κ+κ +1) ζ , α|ζ , α = 1 − |ζ |2 1 − |ζ |2 ×. +∞ n ζζ n=0. (n + 1 + κ + κ ) e−i(α+α )n(n+κ+κ ) . (n + 1)(1 + κ + κ ). (35). The coherent states are normalized (ζ , α|ζ , α = 1), but they are not orthogonal to each other. Second, an arbitrary element state of the Hilbert space H, let us call it |f , can be written in terms of the coherent states, |f = f ζ , ζ |ζ , α dµ(ζ ), (36) where the analytic function +∞ (1/2)(κ+κ +1) n f ζ , ζ = 1 − |ζ |2 ζ n=0. determines completely the state |f ∈ H.. . (n + 1 + κ + κ ) (n + 1)(1 + κ + κ ). 1/2. . e−iαn(n+κ+κ ) ψn |f . (37).

(98) 296. A.H. El Kinani, M. Daoud / Physics Letters A 283 (2001) 291–299. Let us now consider the dynamical evolution of the coherent states. More precisely, we have U (t)|ζ , α = e−iH t |ζ , α = |ζ , α + t.. (38). The coherent states are stable temporally.. 4. Other kinds of coherent states It is well known that, and as we mentioned in the introduction, there are three different group-theoretic approaches to coherent states [1–3]. These approaches follow three possible definitions of the familiar Glauber coherent states [17] of a harmonic oscillator. In a similar way, we have developed (Section 2) the formalism in which coherent states, for Pöschl–Teller potentials, are generated by the action of the displacement operator on the ground state. In a second approach, developed recently by Gazeau and Klauder [6,7], one deals with eigenstates of the annihilation operator. This second definition follows the Barut–Girardello construction of the su(1, 1) coherent states [18]. The third definition of coherent states is related to the optimization of Robertson–Schrödinger (R–S) uncertainty relation [19,20]. States that minimize (R–S) uncertainty relation are called intelligent states [10,11,21–24]. These different definitions of the coherent states for an arbitrary quantum system coincide only in the special case of the Weyl–Heisenberg algebra that is the dynamical symmetry of a quantized harmonic oscillator. For other quantum mechanical systems, like a particle in the Pöschl–Teller potentials, these different approaches lead to distinct states. Relation between various sets of coherent states should be studied by developing a formalism that provides a unified description of these different states for an arbitrary quantum system. This matter remains an open problem. In this section, for review purposes, we give the main properties of Gazeau–Klauder coherent states [6] and intelligent states (states minimizing the R–S uncertainty relation) [10], for the Pöschl–Teller potentials in order to compare various sets of coherent states obtained following the standard three definitions. 4.1. Gazeau–Klauder coherent states − The so-called Gazeau–Klauder coherent states [6,7], defined as eigenstates of the annihilation operator aκ,κ , − aκ,κ |z, α = z|z, α,. z ∈ C, α ∈ R,. (39). are given by [8] (see also [10]) +∞ . . zn e−iαn(n+κ+κ ) |ψn √ |z, α = N |z| (n + 1)(n + κ + κ + 1) n=0 . (40). for a particle trapped in the Pöschl–Teller potentials of trigonometric type (κ, κ > 1). The normalization constant N (|z|) is given by 2 N |z| =. . |z|κ+κ , Iκ+κ 2|z|. (41). where Iκ+κ (2|z|) is the modified Bessel function of the first kind. The resolution of unity is explicitly given by |z, αz, α| dµ(z) = IH ,. (42). where the measure can be computed by using the inverse Mellin transform [25] (for more details, see [10,11]): dµ(z) =. 2 Iκ+κ (2r)Kκ+κ (2r)r dr dφ, π. z = reiφ .. (43).

(99) A.H. El Kinani, M. Daoud / Physics Letters A 283 (2001) 291–299. 297. The Gazeau–Klauder coherent states are continuous in the labelling z and α. They form an overcomplete family of states (identity resolution) and satisfy the identity action z, α|H |z, α = |z|2 .. (44). One can verify that the states |z, α (Eq. (40)) are stable temporally, i.e., e−iH t |z, α = |z, α + t.. (45). Clearly, the Gazeau–Klauder coherent states (40) are different from the ones constructed à la Klauder–Perelomov (31). 4.2. States minimizing the Robertson–Schrödinger uncertainty relation + − Using the creation and annihilation (aκ,κ and aκ,κ ) operators, we introduce two Hermitian operators. 1 + − W ≡ Wκ,κ = √ aκ,κ + aκ,κ , 2. i + − P ≡ Pκ,κ = √ aκ,κ − aκ,κ , 2. (46). which satisfy the commutation relation [W, P ] = iGκ,κ ≡ iG.. (47). It is well known that the variances (W )2 and (P )2 obey the Robertson–Schrödinger uncertainty relation 1 G2 + F 2 , 4 where the operator F is defined by F = W − W , P − P . (W )2 (P )2 . (48). (49). The symbol { , } stands for the anticommutator (for more details, see Refs. [10,11]). The states minimizing the Robertson–Schrödinger uncertainty relation satisfy the eigenvalues equation √ (W + iλP )|z, λ, α = z 2|z, λ, α, (50) which can be written also as + − (1 − λ)aκ,κ + (1 + λ)aκ,κ |z, λ, α = 2z|z, λ, α.. (51). The parameter λ ∈ C is called, sometimes, the squeezing parameter. In the states satisfying (50), we have the following relations: (W )2 = |λ|∆,. (P )2 =. 1 ∆, |λ|. (52). with. 1 (53) G2 + F 2 . 2 A complete classification of the solutions (generalized intelligent states) of (50) was obtained in [10] for an arbitrary exactly solvable quantum system. Here, we are interested by the situation where |λ| = 1. In this case, we have ∆=. (W )2 = (P )2 ,. (54).

(100) 298. A.H. El Kinani, M. Daoud / Physics Letters A 283 (2001) 291–299. and the states satisfying (50) with |λ| = 1 are called generalized coherent states. They are given by [10] (up to the normalization constant) |z, λ, α = U (z, λ)|ψ0 ,. (55). where the operator U (z, λ), providing the state |z, λ, α by acting on the ground state |ψ0 , is given by n . +∞ λ − 1 1 + 2 2z 1 + a + a U (z, λ) = . 1 + λ H κ,κ λ + 1 H κ,κ. (56). n=0. Coherence and squeezing of the generalized intelligent states was discussed in [10]. Here also it is clear that the generalized coherent states (|λ| = 1) obtained by minimizing the Robertson–Schrödinger uncertainty relation are different from two sets of coherent states discussed before (Klauder–Perelomov and Gazeau–Klauder ones). The generalized coherent states |z, λ, α (Eq. (55)), obtained by minimizing the Robertson–Schrödinger uncertainty relation, generalize the Gazeau–Klauder ones. Indeed, the latters are obtained by simply setting λ = 1. − The states |z, λ = 1, α ≡ Eq. (51) are eigenstates of the annihilation operator aκ,κ . In this case (λ = 1), we get 1 (W )2 = (P )2 = G, 2 where the average value of G is. (57). G = z, λ = 1, α Gκ,κ (N)z, λ = 1, α = (1 + κ + κ ) +. (58). G 1 + κ + κ ,. (59). 2 2|z|2 0 F1 (2 + κ + κ , |z| ) (1 + κ + κ ) 0 F1 (1 + κ + κ , |z|2 ) in terms of the confluent hypergeometric function 0 F1 (a, x) [16]. Note that. which traduces the fact that the dispersions (W )2. and (P )2. are greater than 1/2, unlike the case of the harmonic. oscillator in which we have (W )2 = (P )2 = 1/2. 5. Concluding remarks In this Letter, we have defined coherent states of Klauder–Perelomov’s type of a particle trapped in Pöschl–Teller potentials. They present an important differences from the ones derived à la Gazeau–Klauder and by minimization of the Robertson–Schrödinger uncertainty relation. To close this Letter, let us mention that the results obtained here can be extended to introduce coherent states of Klauder–Perelomov’s type for the infinite square well potential. We should also quote Nieto and Simmons [26], who have considered the Pöschl–Teller potentials as examples of their construction of coherent states for confining one dimensional potentials. More recently, the classical limit of coherent states for Pöschl–Teller potentials was investigated [27]. The states constructed in [26,27] have a totally different meaning from the ones discussed in this Letter. In fact, here we deal with coherent states of Klauder– Perelomov’s kind. Moreover, the so-called Gazeau–Klauder coherent states and ones minimizing the Robertson– Schrödinger uncertainty relation are more general than ones constructed in [26,27]. We believe that the recent results on the coherent states constructions [6,7,10,11] (see also [28]) can be used to understand the quantum properties as well as the classical limit of the Pöschl–Teller coherent states. On the other hand, in view of the results discussed through this Letter, a natural question arises. For an arbitrary quantum system, what formalism unifies the description of the various kinds of coherent states within a common frame? We believe that the answer to this will establish relations between different types of coherent states to understand the physical basis of their mathematical properties. Our work is a first step in this sense. More details and further developments on this subject will be submitted for publication in the near future [29]..