Classical motion and coherent states for Pöschl–Teller potentials

Physics Letters A 372 (2008) 1391–1405 www.elsevier.com/locate/pla

Classical motion and coherent states for Pöschl–Teller potentials S. Cruz y Cruz a,b,1 , S. ¸ Kuru a,c , J. Negro a,∗ a Departamento de Física Teórica, Atómica y Óptica, Universidad de Valladolid, 47071 Valladolid, Spain b Departamento de Física, Cinvestav, AP 14-740, México 07000 DF, Mexico c Department of Physics, Faculty of Science, Ankara University, 06100 Ankara, Turkey

Received 14 June 2007; received in revised form 24 September 2007; accepted 1 October 2007 Available online 9 October 2007 Communicated by P.R. Holland

Abstract The trigonometric and hyperbolic Pöschl–Teller potentials are dealt with from the point of view of classical and quantum mechanics. We show that there is a natural correspondence between the algebraic structure of these two approaches for both kind of potentials. Then, the coherent states are constructed and the appropriate classical variables are compared with the expected values of their corresponding quantum operators. © 2007 Elsevier B.V. All rights reserved. PACS: 03.65.Fd; 03.65.-w; 45.20.Jj Keywords: Pöschl–Teller potential; Normalized solutions; Ladder operators; Factorization; Coherent states

1. Introduction In this work, we consider the trigonometric (also known as Scarf ) and hyperbolic (sometimes called modified) Pöschl–Teller (PT) potentials which can be seen as a sort of deformations of the harmonic oscillator potential. Near the minimum the three potentials have the same behaviour, but far from the origin, the trigonometric PT is much narrower while the hyperbolic PT is much wider than the reference harmonic oscillator potential. We remark that, in the frame of quantum mechanics, while the harmonic oscillator potential eigenvalues are equally-spaced, the eigenvalues of PT potentials are not. We will discuss the consequences that these properties have in the behaviour of coherent states (CS) for these systems. We will start by studying the motion and algebraic properties of the classical PT systems. In particular, we show that there are two time-dependent integrals of motion which determine the phase trajectories [1]. Afterwards, we will consider the corresponding quantum systems and define lowering and raising operators by means of factorizations [2]. Such operators determine a spectrum generating Lie algebra and they have as classical analogues two complex conjugate functions closing the same algebra (together with the Hamiltonian) by means of Poisson brackets which characterize the classical systems. Next, the coherent states for both kind of PT potentials will be constructed with the help of lowering and raising operators according to the Barut–Girardello (BG) [3] and Klauder–Perelomov (KP) [4,5] approaches. While the method of Barut–Girardello is not applicable in the case of compact groups, that of Klauder–Perelomov is relevant to general Lie groups. As the discrete spectrum of the Schrödinder equation with trigonometric PT potential is infinite and the su(1, 1) underlying algebra is non-compact, we can use both descriptions. A discussion on several types of CS’s for this potential can also be found in the literature [6–10]. On the other hand, the spectrum corresponding to the hyperbolic PT potential is finite, so we will construct the CS using the Klauder–Perelomov approach. For both potentials the minimum-uncertainty CS formalism has also been used [11,12]. However, in this Letter our main * Corresponding author.

E-mail addresses: [email protected] (S. Cruz y Cruz), [email protected] (S. ¸ Kuru), [email protected] (J. Negro). 1 On leave of absence from: Ciencias Básicas, UPIITA-IPN, Av. IPN 2508, CP 07340 México DF, Mexico.

0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.10.010

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objective will be to compare the classical and quantum PT systems by means of the underlying algebraic structure. In this way, by taking into account the CS’s, we will compute the expected values of the appropriate quantum operators and draw some conclusions from the similarities and differences with the corresponding classical variables. The organization of this Letter is as follows. In Section 2 we will characterize the classical PT potentials from an algebraic point of view. Next, in Section 3 we will deal with the quantum trigonometric PT potential: its spectrum, normalized eigenfunctions, and lowering and raising operators closing the su(1, 1) algebra. Then, we build the BG and KP coherent states and compute the relevant expected values for this potential. In the following section the same program is carried out for the hyperbolic PT potential. We will end the Letter with some conclusions and remarks on the results here obtained and also we will discuss other results of previous works on CS’s dealing with the same potentials. 2. Classical Pöschl–Teller potentials In this section we will discuss some algebraic properties of the classical PT systems that will have their counterparts in the corresponding quantum systems. This will serve as a key reference in studying the coherent states. Let us consider the classical Hamiltonian p2 + V (x) (1) 2m where x, p are canonical coordinates, i.e., {x, p} = 1, being {·,·} the notation for the usual Poisson brackets, and V (x) is the potential. Henceforth, we will set 2m = 1 for simplicity. In some instances, the phase-space trajectories (x(t), p(t)) can be found from the algebraic properties of the classical system. This is the case for the trigonometric and hyperbolic PT potentials that we will characterize by means of the factorization method adapted to classical mechanics (for more details on the general set up see [1]). The trigonometric PT potential is given by H (x, p) =

π π ( − 1) , − 1. The corresponding Hamiltonian can be factorized in terms of two complex conjugate functions as V (x) =

H = A+ A− + ( − 1) with the factor functions

(3)

√ H sin x.

A± = ∓ip cos x +

These functions, together with the Hamiltonian, close a deformed algebra with Poisson brackets √ √ {H, A± } = ∓i2 H A± , {A+ , A− } = i2 H . √ By defining A0 ≡ − H , these brackets can be easily rewritten as follows {A0 , A± } = ±iA± ,

{A+ , A− } = −i2A0

(4)

(5)

(6)

which correspond to the su(1, 1) Poisson algebra. We remark that for the quantum system with trigonometric PT potential, the spectrum generating algebra is the su(1, 1) Lie algebra as we will see in the next section. The two complex conjugate functions (4) give rise to two time-dependent integrals of motion, √ Ht

Q± = A± e∓i2

.

(7)

Let us denote by E the value of the time-independent constant of motion H , then from these integrals of motion we easily get the expressions of the trajectories (x(t), p(t)) fixed by   √  √    E − ( − 1) sin x = (8) p cos x = − E − ( − 1) sin ϕ0 + 2 Et , cos ϕ0 + 2 Et , E where ϕ0 is a constant (determined by initial conditions) and the energy E must be greater than the minimum of the potential, E > ( − 1). The hyperbolic PT potential has the expression ( + 1)

, −∞ < x < ∞, (9) cosh2 x where we assume that  > 0. For this potential, as the bound motions have negative energy, the factor functions take the form √ A± = ∓ip cosh x + −H sinh x (10) V (x) = −

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and the Hamiltonian can be written in terms of these functions in the following way H = A+ A− − ( + 1).

(11)

Now, we have the Poisson brackets √ {H, A± } = ∓i2 −H A± ,

(12)

√ {A+ , A− } = i2 −H √ that can also be rewritten, by defining A0 ≡ −H , as follows {A0 , A± } = ±iA± ,

{A+ , A− } = i2A0

(13)

which correspond to the su(2) Poisson algebra. We notice that for the quantum PT potential the spectrum generating algebra is an su(2) Lie algebra as it will be shown in Section 4. In a similar way to the previous case, the trajectories for this potential are given by   √ √     E + ( + 1) p cosh x = − E + ( + 1) sin ϕ0 + 2 −Et , cos ϕ0 + 2 −Et , sinh x = − (14) E where the energy must be negative, but greater than the potential minimum, −( + 1) < E < 0. Therefore, the PT potentials have an underlying “dynamical algebra” of Poisson brackets that give rise to two time-dependent integrals of motion. From the functions A± of this algebra, we can get two new variables (8) and (14), depending on x and p, which have a harmonic behavior and determine the motion. We also remark that the coordinates (x(t), p(t)) for both potentials are periodic with the frequency depending on the energy. In the trigonometric case the frequency grows √ as the energy of the motion increase, while it diminishes with the energy for the hyperbolic case. These frequencies, ωc = 2 ±E, appear in the Poisson brackets (5) and (12), so that they have an algebraic origin. A similar formula will hold for the frequency of the stationary quantum states. 3. Coherent states for the trigonometric Pöschl–Teller potential 3.1. Normalized eigenfunctions The Hamiltonian for the trigonometric PT system is given by d2 ( − 1) + dx 2 cos2 x and the Schrödinger (eigenvalue) equation reads H = −

(15)

H ψ(x) = Eψ(x), = κ2

(16)

where E is the energy of a bound state and ψ(x) is the corresponding eigenfunction. Here, we have taken ¯ = 2m = 1, also to simplify the notation. By performing the following change of variables   1 − sin x , ψ x(y) = (1 − y)/2 y /2 φ(y) 2 the eigenvalue equation (16) can be transformed into a hypergeometric type equation,   d 2 φ(y) 1 dφ(y)  2 y(1 − y) + (−2 − 1)y +  + + κ − 2 φ(y) = 0. 2 2 dy dy y=

h2

(17)

(18)

In order to get finite solutions at y = 0 and y = 1, we have to choose  − κ = −n, where n = 0, 1, 2, . . . , giving rise to a polynomial of degree n [13,14]. This condition determines the energy eigenvalue in terms of the parameters  and n κ 2 = En = ( + n)2 . The physical eigenfunctions of Eq. (16) corresponding to the eigenvalues (19) are written as 

1 − sin x 1 ψn (x) = N2− cos x 2 F1 −n, 2 + n, + ; , 2 2

(19)

(20)

where N is a normalization constant that will be determined by considering the relation between hypergeometric and Gegenbauer polynomials (see [13,15,16]) and taking into account the integral formulas for Gegenbauer polynomials [16]. We obtain the normalized eigenfunctions of (16), ( + n)n! () (2) n cos xCn() (sin x). ψ (x) = (21) 1/2 π (2 + n) ( + 12 )

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They can also be expressed in terms of associated Legendre functions [15]  ( + n) (2 + n) √ ( 1 −) n cos xP 2 (sin x). ψ (x) = (+n− 12 ) n!

(22)

3.2. Lowering and raising operators for the trigonometric PT potential Let us rewrite the eigenvalue equation (16), substituting the energy (19), as follows hn ψn = −( − 1)ψn ,

(23)

where hn = − cos2 x∂x2 − ( + n)2 cos2 x

(24)

and ∂x = d/dx denotes the derivative with respect to x. The second order differential operator hn can be factorized in terms of the first order operators Bn+ = − cos x∂x + ( + n) sin x,

(25)

Bn−

(26)

= cos x∂x + ( + n + 1) sin x

satisfying the conditions + − Bn−1 + γn−1 , hn = Bn− Bn+ + γn = Bn−1

γn = −( + n)( + n + 1).

(27)

These factorizations imply the following intertwining relations Bn+ hn = hn+1 Bn+ ,

Bn− hn+1 = hn Bn− .

(28)

Then, by using (23), (27) and (28) we can show that Bn+ ψn = bn+ ψn+1 ,

Bn− ψn+1 = bn− ψn ,

(29)

bn±

Bn+

Bn−

where are proportionality coefficients. This means that raises and lowers the energy eigenvalue when acting on the eigenfunctions in the same potential. Remark that these are not the same as the factorization operators used in some references (see − for instance [7] or [9]). We can determine the square-integrable wave function annihilated by B−1 , ( + 1) − cos x B−1 (30) ψ0 = 0, ψ0 (x) = 1/2 π ( + 1/2) which is the normalized ground state of (16) with the energy E0 . The excited states (22) with energy En = ( + n)2 are then obtained by the consecutive application of the raising operators on this state, + + Bn−2 · · · B0+ ψ0 . ψn ∝ Bn−1

(31) Bn− Bn+

Taking into account (23) and (27), the action of the operator product on the eigenfunction  Bn− Bn+ ψn = bn− bn+ ψn = −( − 1) − γn ψn = (2 + n)(n + 1)ψn .

ψn

is immediately obtained, (32)

Since (Bn+ )† = Bn− , the coefficients bn± cannot be determined directly in an algebraic way. Here, we will find them by means of the recurrence relations of the associated Legendre functions [19,20], which are related with the eigenfunctions by (22). To get the action of Bn+ on ψn , we consider the following recurrence relation [15] 

 dPμ ν (υ)

μ

= (ν + 1)υPμ ν (υ) − (ν − μ + 1)Pν+1 (υ). dυ Having in mind μ = 1/2 − , ν =  + n − 1/2, υ = sin x and using (22) in (33), it is found that (n + 1)(2 + n)( + n) n+1 + n Bn ψ = ψ . ( + n + 1) 1 − υ2

In a similar way, we get n(2 + n − 1)( + n) n−1 − Bn−1 ψn = ψ . ( + n − 1)

(33)

(34)

(35)

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We can check that the coefficients bn± in (34)–(35) satisfy (32). In order to construct two adjoint operators, i.e., (B˜ n+ )† = B˜ n− , we will modify Bn± as follows   +n+1 + +n−1 − − + ˜ ˜ Bn−1 = Bn , Bn−1 . Bn = (36) +n +n − Now, the actions of B˜ n+ and B˜ n−1 on the eigenfunctions are  B˜ n+ ψn = (n + 1)(2 + n)ψn+1 , (37) − ψn = B˜ n−1



n(2 + n − 1)ψn−1

(38)

displaying explicitly the adjointness relationship. To make clear the underlying algebraic structure, let us define the linear operators B˜ ± and B˜ in the Hilbert space generated by the eigenfunctions as  B˜ + ψn ≡ B˜ n+ ψn = (n + 1)(2 + n)ψn+1 , (39)  − n−1 − n n B˜ ψ ≡ B˜ n−1 ψ = n(2 + n − 1)ψ , (40) ˜ n ≡ ( + n)ψ n . Bψ  

(41)

Then, from (27) and (39)–(40), we have the relation (B˜ n− B˜ n+ − commutator

+ ˜− B˜ n−1 Bn−1 )ψn

= 2( + n)ψn , which can be identified as the

[B˜ − , B˜ + ] = 2B˜

(42)

when it acts on any ψn . It is also easy to get from (39)–(41) the following two commutators ˜ B˜ ± ] = ±B˜ ± . [B,

(43)

Hence, we see that the commutation relations (42) and (43) close the su(1, 1) algebra, whose Casimir operator is 1 C˜ = B˜ 2 − (B˜ + B˜ − + B˜ − B˜ + ). 2 ˜ The action of C can be found by means of the definitions (39)–(41):

(44)

˜ n = ( − 1)ψ n . Cψ  

(45)

Substituting the differential expressions for B˜ + and B˜ − (obtained from (25)–(26) and (36)) in the Casimir (44), and having in mind that it is acting on ψn , we arrive at the differential expression C˜ = cos2 x∂x2 + ( + n)2 cos2 x = −hn .

(46)

From this relation we understand the meaning of the starting differential operator hn given by (24): it can be identified, up to a constant factor, as the Casimir operator of the su(1, 1) algebra. We remark that H and C˜ given respectively by (15) and (46) ˜ = 0, and they have the common eigenfunctions ψ n (x). commute, [H , C] 

3.3. Coherent states In this section we will use two different ways to construct coherent states. One of them is the Barut–Girardello approach, which only applicable to non-compact groups [3] and the other one, which is relevant to arbitrary Lie groups, is known as Klauder– Perelomov [5]. 3.3.1. Barut–Girardello coherent states According to BG, the CS’s are defined as eigenvectors of the lowering operator B˜ − of the Lie algebra of our system (42)–(43). Let us introduce the ket notation for eigenfunctions ψn ≡ |, n , since it is more appropriate for studying CS’s. Rewriting (40) in this notation  B˜ − |, n = n(2 + n − 1)|, n − 1 (47) as usual, we will express the eigenvectors |z of B˜ − as a linear combination of the vectors |, n , i.e., B˜ − |z = z|z ,

|z =

∞

n=0

cn |, n ,

(48)

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where z is a complex number. Using (47) in (48), and the normalization z|z = 1, the BG CS takes the form |z = 

∞

zn |, n . √ 2 0 F1 (2, |z| ) n=0 n!(2)n 1

(49)

The completeness relation  |z z| dμ(z) = 1

(50)

C

of these set of CS’s, is obtained in a standard way [3,7,21]. The measure dμ(z) is defined as dμ(z) =

1 0 F1 (2, |z|2 )  2  2 σ |z| d z, π (2)

(51)

√ where σ (y) is given by σ (y) = 2y −1/2 K−+1/2 (2 y ), and Kν (x) is the modified Bessel function of the third kind. If at time t = 0, we have a CS |z ≡ |z, 0 , then its temporal evolution is given by |z, t = e−iH t |z = 

n ∞

zn e−iE t |, n . √ 2 0 F1 (2, |z| ) n=0 n!(2)n

1

(52)

Therefore, in the coordinate representation, the wavefunction of a CS in terms of eigenfunctions is n ∞

zn e−iE t n

x|z, t = ψz,t (x) =  ψ (x). √ 2 0 F1 (2, |z| ) n=0 n!(2)n

1

(53)

Since En = ( + n)2 , the CS wavefunction change its shape with time and it does not cohere as the harmonic oscillator CS’s. The distribution function (or probability of detecting the nth excited state in the CS |z ) Pn = | , n|z |2 takes the form Pn =

|z|2n |z|2n+2−1 = , n!0 F1 (2, |z|2 )(2)n n!I2−1 (2|z|) (2 + n)

(54)

where I2 is the modified Bessel function with the following asymptotic behaviour [13] √ |z|2 , 0 < |z| 2 + 1, (2 + 1) √   e2|z| , |z| 2 + 1. I2 2|z| ≈ √ 4π|z|   I2 2|z| ≈

(55)

Therefore, if |z| → 0, the distribution function Pn (z) decreases as n grows starting from the maximum value at n = 0. On the other hand if |z| → ∞, Pn (z) increases with n. This behaviour will be useful to interpret some plots given below. The expected value of the energy in this CS is 

∞

2I2 (2|z|) Pn ( + n)2 = 2 + |z|2 1 + .

H z ≡ z|H |z = (56) |z|I2−1 (2|z|) n=0

Having in mind the asymptotic formulas (55), it can be seen from (56) that if |z| → 0, then H z ≈ 2 (which is the ground state energy), and if |z| → ∞, H z ≈ ( + |z|)2 . Let us define the number operator N˜ |, n = n|, n

(57)

which gives the excitation number of an eigenstate. Then, the related expected values are

N˜ z =

∞

n=0

Pn n =

|z|I2 (2|z|) , I2−1 (2|z|)

∞

 |z|[I2 (2|z|)+|z|I2+1 (2|z|)] Pn n2 = N˜ 2 z = I2−1 (2|z|)



(58)

n=0

and the dispersion (or uncertainty) of the number operator is given by      |z|I2 I2−1 + |z|2 (I2+1 I2−1 − I 2 ) 2 2 2 ˜ ˜ , σN˜ = N z − N z =  2 I2−1

(59)

S. Cruz y Cruz et al. / Physics Letters A 372 (2008) 1391–1405

where the argument of I2−1 , etc., has been suppressed. Its asymptotic behaviour is  √|z| , |z| → 0, σN˜ ≈ √2 |z|, |z| → ∞.

1397

(60)

From (60) it can be easily seen that in the limit |z| → 0, the uncertainty takes the same form as in the harmonic oscillator. This is not an unexpected situation because in this case, from (54), the CS is composed mainly by the ground state, so the wavefunction takes significative values in x ≈ 0, where the PT potential is very close to the harmonic oscillator. On the other hand if |z| → ∞, the distribution function Pn (z) takes significant values for n 1 (or E 2 ) where the properties of the trigonometric PT potential are quite different from those of the harmonic oscillator. Next, we want to compare the classical motion, as given in (8), with the expected values of the corresponding quantum operators sin x and −i cos x∂x . From the differential expressions of B˜ ± (see (25)–(26) and (36)), it is easy to get   − B˜ n−1 1 B˜ n+ +√ , sin x = (61) √ 2 ( + n + 1)( + n) ( + n − 1)( + n)     + n ˜−  + n ˜+ 1 cos x∂x = (62) − B B . 2  + n − 1 n−1 +n+1 n However, we can give a generalized expression for the expected values of Hermitian operators O having the following form (which includes (61) and (62)) O=

1  ˜ ˜ + ˜ − ˜ ∗ f (N )B + B f (N ) 2

(63)

˜ is a complex function of the number operator such that where f (N) f (N˜ )|, n = f (n)|, n .

(64)

Using the notation z, t| · |z, t = · z,t and the polar expression z = |z|eiθ , it is straightforward to show that the expected value of O is given by

O z,t =

∞

  n Pn |z| f (n + 1)ei(ω t−θ) .

(65)

n=0

In the case O = sin x we have f (N˜ ) = 

1 ( + N˜ )( + N˜ − 1)

(66)

therefore, the evolution of the expected value of sin x is

sin x z,t =

∞

|z| cos(ωn t − θ ) , Pn √ ( + n + 1)( + n) n=0

(67)

where ωn = En+1 − En = 2 + 2n + 1.

(68)

Since the operator −i cos x∂x = cos xp is not Hermitian, we will look for the expected value of its symmetrization, (cos xp) + (p cos x) . 2 For O = P, P=

f (N˜ ) =

i 2 + 2N˜ − 1  2 ( + N˜ )( + N˜ − 1)

(69)

(70)

and its time dependent expected value is

P z,t = −

∞

|z|ωn sin(ωn t − θ ) Pn √  . 2 ( + n + 1)( + n) n=0

(71)

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3.3.2. Klauder–Perelomov coherent states According to the Klauder–Perelomov approach [5] a CS is defined by |z = D(z)|, 0 ,

(72)

where |, 0 = ψ0 is the ground state of H , and the displacement operator D(z) in the su(1, 1) algebra has the form ˜ + −z∗ B˜ −

D(z) = ezB

= e−ξ

∗B ˜−

˜

˜+

e−ηB eξ B

(73)

with   η = − ln 1 − |ξ |2 ,

ξ = tanh |z|eiϕ ,

eiϕ =

z . |z|

(74)

Then, using (72) and (73) the CS can be written as   ˜ + |ξ = 1 − |ξ |2 eξ B |, 0 .

(75)

Here, the complex number ξ given by (74) satisfies |ξ | < 1. Expanding the exponential and making use of (39) in (75), we obtain the decomposition of the CS over the orthonormal basis  ∞   2  n (2)n ξ |, n . |ξ = 1 − |ξ | (76) n! n=0

This set of CS’s also satisfies a completeness relation with the measure [22] dμ(ξ ) =

1 2 − 1 dξ 2 . π (1 − |ξ |2 )2

(77)

Its temporal evolution in the coordinate representation is given by  ∞   2  n (2)n −iEn t n ξ ψ (x). e

x|ξ, t = ψξ,t (x) = 1 − |ξ | n!

(78)

n=0

As in the BG CS’s, the evolution in time does not cohere due to the quadratic expression (19) of En . From the definition (76) of CS, the distribution function Pn = | n, |ξ |2 is 2  (2)n Pn = 1 − |ξ |2 |ξ |2n n!

(79)

and the expected value of the energy reads

H ξ =

∞

Pn ( + n)2 =

n=0

(2|ξ |2 + (1 + |ξ |2 )2 ) . (1 − |ξ |2 )2

(80)

In this case, the dispersion is also found from (59). Recalling the definition (57), we get 2 ˜ ξ = 2|ξ | ,

N 1 − |ξ |2

 2|ξ |2 (1 + 2|ξ |2 ) N˜ 2 ξ = . (1 − |ξ |2 )2



Then, the asymptotic behaviour of the dispersion is given by  √ √ |ξ | 2, |ξ | → 0, 2|ξ | ≈  1 σN˜ = 2 1 − |ξ | 2 1−|ξ | , |ξ | → 1.

(81)

(82)

Here, the same comments as in the BG case apply about the behaviour of distribution function, expected value of the energy, or uncertainty. In this case the expected value of O is given by a similar expression to (65), with the notation (74) |ξ, t ≡ |z, t

O z,t =

∞

n=0

  n Pn (2 + n)| tanh z| f (n + 1)ei(ω t−ϕ) .

(83)

S. Cruz y Cruz et al. / Physics Letters A 372 (2008) 1391–1405

(a)

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(b)

Fig. 1. Plot of the variables sin x (a) and P (b), for the trigonometric PT potential. Continuous lines denote the expected values in KP CS’s and dashing lines denote the classical variables for z = 0.02,  = 5.5 with E ≈ 30.

(a)

(b)

Fig. 2. Plot of the variables sin x (a) and P (b), for the trigonometric PT potential. Continuous lines denote the expected values in KP CS’s and dashing lines denote the classical variables for z = 0.6,  = 5.5 with E ≈ 146.

We find directly the expected values of sin x and P, having in mind the value of the operators f (N˜ ) given in (66) and (70), respectively,

sin x z,t =

∞

tanh |z| cos(ωn t − ϕ) Pn (2 + n) √ ( + n + 1)( + n) n=0

(84)

and

P z,t = −

∞

tanh |z|ωn sin(ωn t − ϕ) Pn (2 + n) √ 2 ( + n + 1)( + n) n=0

(85)

where ωn is given by (68). We remark that the expected values of sin x and P in the BG approach given by (67) and (71) have a similar form as the corresponding formulas (84) and (85), for the KP formalism. The difference come from the parametrization (74) and the distribution functions (54) and (79). Both have the same character as can be seen by fixing the same expected value of the energy and drawing them. Here, these expected values have been plotted together with the corresponding classical variable for a Klauder–Perelomov CS in Figs. 1 and 2. Now, we will compare the classical motion and quantum expectation values in the CS of sin x and P with the same energy: √ • The CS expected values (84) and (85) are superpositions of classical motions (8) where the classical frequency ωc = 2 E has been replaced by ωn = 2 + 2n + 1. • For small |z|, we see from Fig. 1 that since these CS’s are constructed mainly from the ground state and a few excited eigenstates, the expected value oscillations have a harmonic behaviour. Due to the very small values of Pn for n > 0, the amplitude of the oscillations is much smaller and the frequency is slightly greater than the classical counterparts. From formulas (84) and (85) the main contribution to these expected values comes from P0 with frequency ω0 = 2 + 1, while the ‘classical’ frequency in the ground state energy (14) is ωc = 2. • For big |z|, the expected values in time are composed of two types of contributions. One of them has high frequency similar to classical oscillations. The other one modulates the amplitude giving rise to collapses and revivals with period π . The origin of this behaviour comes from the non-equispaced energy spectrum and the distribution function Pn . As we increase the value of |z| the coherent state is composed of many excited states whose evolution gives rise to colapses where the wave-packet is spread in the whole space interval and revivals where the wave-packet is recovered after a period when the amplitudes cohere again [6,17,18]. Then the expected values in this CS also inherits the same character: colapses when the wavefunction is spread and revivals when it is tight around a point. This phenomenon can be seen in Fig. 2.

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4. Coherent states for hyperbolic Pöschl–Teller potential 4.1. Normalized eigenfunctions Now consider the hyperbolic PT Hamiltonian, d2 ( + 1) − ,  > 0, −∞ < x < ∞. (86) 2 dx cosh2 x The eigenfunctions ψ(x) and eigenvalues E satisfy the Schrödinger equation (16). Since we are interested in bound states, in this case we will take E = −κ 2 assuming, without loss of generality, that κ > 0. The corresponding differential equation can be transformed by means of the following change of variables H = −

  1 − tanh x , ψ x(y) = y κ/2 (1 − y)κ/2 φ(y) 2 into the hypergeometric equation y(x) =

y(1 − y)

(87)

dφ(y)  d 2 φ(y)  + κ + 1 − (2κ + 2)y + ( + 1) − κ(κ + 1) φ(y) = 0. 2 dy dy

(88)

Observe that, the asymptotic limits of ψ(x) are obtained by taking y → 0, 1. It is clear from (87) that, in order to construct functions with the proper behavior as x → ±∞, we have to look for the finite solutions of (88) near y = 0 and y = 1. This is only possible for κ =  − n, n <  (n = 0, 1, 2, . . .). Then, the physical eigenfunctions ψ(x) for this system read 

1 − tanh x n n− n− , ψ (x) = N 2 (cosh x) (89) 2 F1 −n, 2 − n + 1;  − n + 1; 2 where N is the normalization constant to be determined. They correspond to the (finite) sequence of spectral values En = −( − n)2 ,

n = 0, 1, 2, . . . , n < .

(90)

These eigenfunctions can be also written in terms of Gegenbauer polynomials and associated Legendre functions. Hence, we can make use of the known results for integrals involving, e.g., Gegenbauer polynomials [23], in order to determine the normalization constant N . Finally, we obtain 1 ( − n)n! ( − n + 1/2) (2 − 2n + 1) n− (−n+ 2 ) (cosh x) ψn (x) = C (tanh x) n π 1/2 (2 − n + 1) ( − n + 1)  ( − n) (2 − n + 1) (n−) = (91) P(−−1) (tanh x). n! 4.2. Lowering and raising operators for the hyperbolic PT potential The eigenvalue equation (16) with H given by (86) and E = −( − n)2 , can also be written as hn ψn = ( + 1)ψn .

(92)

In this expression hn is the differential operator hn = − cosh2 x∂x2 + ( − n)2 cosh2 x

(93)

which can be factorized in terms of two non-adjoint n-dependent operators Bn± : − + Bn+1 + γn+1 hn = Bn+ Bn− + γn = Bn+1

(94)

with Bn− = cosh x∂x + ( − n) sinh x,

Bn+ = − cosh x∂x + ( − n + 1) sinh x,

γn = ( − n)( − n + 1).

(95)

In this case the intertwining relations Bn− hn = hn−1 Bn− ,

Bn+ hn−1 = hn Bn+

(96)

also follow as a consequence of (94), stating that Bn± act as raising and lowering operators on the eigenfunctions ψn (x). Hence we can write Bn− ψn = bn− ψn−1 ,

Bn+ ψn−1 = bn+ ψn .

(97)

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Since the operators Bn± are not adjoint to each other, expressions (92) and (94) permit only to obtain the product of the proportionality factors bn± , bn+ bn− = n(2 − n + 1).

(98)

Taking into account the first expression in (91) and using, the recurrence relation for the Gegenbauer polynomials (see [13]) d (α) (α+1) C (z) = 2αCn−1 (z) dz n

(99)

we can find bn− , and then bn+ can be obtained from (98)   n(2 − n + 1)( − n) n(2 − n + 1)( − n + 1) − + bn = , bn = . −n+1 −n Finally, the action of the raising and lowering operators on the eigenfunctions is   n(2 − n + 1)( − n) n−1 (n + 1)(2 − n)( − n) n+1 + − n n Bn+1 ψ = ψ , ψ Bn ψ = −n+1 −n−1

(100)

leading to ψn

=

( − n) (2 − n + 1) + + Bn Bn−1 · · · B1+ ψ0 , n! (2 + 1)

where ψ0 is the ground state determined by B0− ψ0 = 0, and whose explicit form is ( + 1/2) 0 cosh− x. ψ (x) = π 1/2 ()

(101)

(102)

Yet, this description is not completely satisfactory. Observe that the eigenfunctions with n   are not normalizable, so they cannot represent physical states. Then, there exist a ground state, annihilated by B0− , and a last bound state ψnmax that should be annihilated by the corresponding Bn+max , a fact which is not explicit in (100). Nevertheless, we can define new raising and lowering operators, adjoint of each other, just by multiplying Bn± by suitable coefficients depending on  and n. Observing that the last bound state in this case corresponds to n = ¯ − 1, where ¯ is an integer number such that  + 1 > ¯  , let us introduce ¯ − n) ( − n + 1)(  ( − n)(¯ − n) Bˆ n+ = Bn− , B +. Bˆ n− = (103) (2 − n + 1)( − n) (2 − n + 1)( − n + 1) n ˆ acting on the eigenfunction space as To establish the algebra generated by these operators, we define linear operators Bˆ ± , B,

   ¯ − 1 n−1 + n+1 − n − n + n n n ¯ ¯ ˆ ˆ ˆ ˆ ˆ B ψ ≡ Bn+1 ψ = (n + 1)( − n − 1)ψ , Bψ ≡ n − ψn . B ψ ≡ Bn ψ = n( − n)ψ , 2 (104) Now one can see that the above operators are more appropriate. For instance Bˆ − annihilates the ground state, while Bˆ + , in turn, ¯ annihilates the last bound state ψ−1 . Additionally, these operators satisfy the su(2) algebra ˆ [Bˆ + , Bˆ − ] = 2B,

ˆ Bˆ ± ] = ±Bˆ ± . [B,

(105)

Indeed, by defining the new quantum numbers j = (¯ − 1)/2,

m = n − j = −j, −j + 1, . . . , j − 1, j

(106)

and denoting the eigenfunctions ψn by |j, m , the expressions (104) transform into the usual action of the su(2) generators:  ˆ m = m|j, m . B|j, Bˆ ± |j, m = (j ∓ m)(j ± m + 1)|j, m ± 1 , (107) The su(2) Casimir operator Cˆ = Bˆ 2 + 12 (Bˆ + Bˆ − + Bˆ − Bˆ + ) satisfies ˆ m = j (j + 1)|j, m C|j,

(108) Bˆ ± ,

it coincides, up to some constants depending on , n, with our differential and substituting the differential expressions for operator hn . Therefore the operators Cˆ and hn commute and they have the common eigenfunctions ψn .

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4.3. Coherent states ¯ that Now we are in position of constructing the CS’s for this system. Recall that we have a finite set of bound states, namely , ¯ n  will be denoted hereafter in the ket notation {ψ ≡ |, n }n=0 . Then, we cannot build the Barut–Girardello CS’s as eigenvectors of Bˆ − , this is the reason why in this case we will restrict ourselves to the Klauder–Perelomov approach. 4.3.1. Klauder–Perelomov coherent states As mentioned before, these CS’s are constructed by applying the displacement operator D(z) to the ground state of the system (72). For the su(2) algebra D(z) has the form [5,24], ˆ + −z∗ Bˆ −

D(z) = ezB

ˆ+

ˆ

= eξ B eηB e−ξ

∗B ˆ−

(109)

,

where ξ = − tan |z|e−iϕ ,

  η = ln 1 + |ξ |2

(110)

with ϕ is defined by eiϕ = −|z|/z and |ξ | ranging over the interval [0, ∞). If we label the coherent state in terms of ξ , and by proceeding in a similar way to the trigonometric case, we find the CS ¯

1/2 −1

(1−)/2  ¯ ¯ − 1 ξ n |, n |ξ = 1 + |ξ |2 n

where

 m k

(111)

n=0

=

m! (m−k)!k!

dμ(ξ ) =

is the binomial coefficient. The completeness relation for these CS’s is fulfilled with the measure [22]

¯ 1 d 2 ξ. π (1 + |ξ |2 )2

(112)

Its time evolution in the coordinate representation ¯

1/2 −1

 (1−)/2 ¯ 2 ¯ − 1

x|ξ, t = ψξ,t (x) = 1 + |ξ |2 ξ n ei(−n) t ψn (x) n

(113)

n=0

does not cohere since the spectral values are not equally spaced. The probability distribution (the probability of detecting the state |, n in the coherent state |ξ ) is

  2  1−¯ ¯ − 1 Pn =  , n|ξ  = 1 + |ξ |2 (114) |ξ |2n n with the asymptotic behavior 

¯ − 1 |ξ |2n , |ξ | → 0, Pn ≈ n

Pn ≈

 ¯ − 1 ¯ |ξ |2(n−+1) , n

|ξ | → ∞.

(115)

This shows that, for small enough values of the argument |ξ |, the probability Pn decreases as n grows, meaning that only the lower energy states contribute significantly to the CS. On the other hand, for great values of |ξ |, Pn increases with n, and then the CS is basically composed by the states with greater energy. This fact will be important in comparing the expected values of some operators of interest with their classical counterparts. First, the energy expectation value is computed,

H ξ = −

¯ −1

n=0

Pn ( − n)2 =

 2    −1 ¯ − 1) + 22 |ξ |2 +  − (¯ − 1) 2 |ξ |4 .  + (1 − 2)(  (1 + |ξ |2 )2

It can be seen, from this expression that, in agreement with (115), H ξ ≈ −2 (the ground state energy) as |ξ | → 0, while for |ξ | → ∞, H ξ ≈ −[ − (¯ − 1)]2 (which is nothing but the energy of the last bound state). Now let us define the number operator Nˆ as in (57). The expected values and mean quadratic dispersion are  2 |ξ |2 (¯ − 1), Nˆ ξ = 1 + |ξ |2   |ξ |  ¯ |ξ | ¯ − 1, −1≈ 1  σNˆ = 2 ¯ 1 + |ξ | |ξ |  − 1,

Nˆ ξ =

(¯ − 1)[1 + (¯ − 1)ξ 2 ]ξ 2 , (1 + ξ 2 )2 |ξ | → 0, |ξ | → ∞.

(116) (117)

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Observe that the asymptotic limits of σNˆ are consistent with (115): for very small values of |ξ |, the CS basically coincides with the ground state, the system is in a state of a very low energy, where its behavior can be well approximated by the harmonic oscillator (remember that for the harmonic oscillator CS, the dispersion σN is proportional to |ξ |). In addition, in both limits, σNˆ → 0, indicating that the number n becomes well defined for the CS. This fact is in agreement with the expected value Nˆ ξ , which vanishes as |ξ | → 0 (because n = 0 for the ground state), and tends to ¯ − 1 for |ξ | → ∞ (the occupation number of the last bound state). Here its behaviour for large |ξ | is in sharp contrast to the trigonometric case. The evolution in time of the expected value of the operator (65) in this case is

O ξ,t =

¯ −2

  n Pn |ξ |(¯ − n − 1) f (n + 1)ei(ω t+ϕ0 )

(118)

n=0

where ϕ0 = ϕ + π , and ωn = 2 − 2n − 1. This allows us to determine, for instance, the expected values of the quantum operators corresponding to the classical variables in (14). We obtain O = sinh x by taking 2 − Nˆ + 1 f (Nˆ ) = (119) (¯ − Nˆ )( − Nˆ )( − Nˆ + 1) and O = P = 12 (cosh xp + p cosh x) with ˆ −1 2 − Nˆ + 1 2 − 2 N . f (Nˆ ) = i 2 (¯ − Nˆ )( − Nˆ )( − Nˆ + 1) Then, the corresponding expected values at time t read ¯ −2

  (¯ − 1 − n)(2 − n) Pn |ξ | cos ϕ + ωn t ,

sinh x ξ,t = ( − n)( − n − 1) n=0 ¯ −2

 (¯ − 1 − n)(2 − n) ωn 

P ξ,t = − Pn |ξ | sin ϕ + ωn t . ( − n)( − n − 1) 2

(120)

(121)

(122)

n=0

• Since the spectral values are not equally spaced, the frequencies ωn at which each bound state oscillate are different and the time dependent factors cannot be taken out of the summation. Therefore we have a ‘non-coherent’ superposition of the classical type of waves given by (14). • In the limits |ξ | → 0, ∞, for which the sums can be well approximated by the first, or the last term, respectively, the expected ¯ values oscillate harmonically with corresponding frequencies ω0 = (2 − 1) and ω−2 = 2 − 2¯ + 3. However, the classical motion is slightly  different. The oscillation frequencies in this framework are determined by the expected value of the quantum energy, ωc = 2 − H ξ . Then, for small enough |ξ |, ωc ≈ 2, while ωc ≈ 2 − 2¯ + 2 for large |ξ |. Note that, in both limits, the difference between the classical and quantum frequencies is the same (1 frequency unit). However, for large  the relative difference becomes significant as for very large |ξ |, but it is negligible for very small |ξ |. Figs. 3–5 illustrate well these phenomena, they compare the classical motion (14) (dotted curves) with the expected values (121) and (122) (solid curves), for different values of |ξ | and  = 8.5. √ • Notice that in the hyperbolic PT system when ξ → 0 the classical frequency ωc = 2 E is slightly bigger than that of the CS expected values, while in the trigonometric case it was slower (see Figs. 1 and 3). • Fig. 4 shows the plots for |ξ | = 0.5. It can be seen that there exist some marked quantum revivals due to the superposition of many oscillation modes. • As in the trigonometric case, the amplitude of oscillations also depends on |ξ |. In Figs. 3 and 5 we can observe that the amplitude of the expected value of the quantum operators is smaller than the corresponding classical functions, indicating that for this |ξ | regimes, the CS wave-packet is made of almost one eigenfunction.

5. Conclusions In this work we have shown that the classical PT potentials have underlying algebras, su(1, 1) for the trigonometric and su(2) for the hyperbolic case, realized by means of Poisson brackets of the functions A± , A0 depending on the canonical variables x and p. In the frame of quantum mechanics the corresponding spectrum generating algebras close the same Lie algebras, which are generated by the lowering and raising operators B ± , B by means of commutators. Thus, the functions A± , A0 can be seen as the classical analogues of the quantum operators B ± , B.

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(a)

(b)

Fig. 3. Plot of the variables sinh x (a) and P (b), for the hyperbolic PT potential for ξ = 0.05,  = 8.5 and E ≈ −73. Continuous lines denote the expected values in CS’s (ωcs = 17) and dashing lines denote the classical variables (ωc = 16).

(a)

(b)

Fig. 4. Plot of the variables sinh x (a) and P (b), for the hyperbolic PT potential for ξ = 0.5,  = 8.5 and E ≈ −49. Continuous lines denote the expected values in CS’s and dashing lines denote the classical variables.

(a)

(b)

Fig. 5. Plot of the variables sinh x (a) and P (b), for the hyperbolic PT potential for ξ = 15,  = 8.5 and E = −0.25. Continuous lines denote the expected values in CS ωcs = 2 and dashing lines denote the classical variables ωc = 1.

Next, we have built the CS’s, |z , for both PT potentials with the help of the underlying Lie algebra. Therefore, depending on the potential we have used the BG or KP approaches to construct them. The time evolution of these CS’s |z, t constitute CS’s of the same Lie algebra of the operators that have also evolved with time: B ± (t), B(t). For instance, in the BG approach we have B˜ − (t)|z, t = z|z, t ,

B˜ − (t) = e−iH t B˜ − eiH t .

(123)

For the KP CS’s of the trigonometric and hyperbolic PT potentials, we have |z, t = D(z, t)|, 0 ,

D(z, t) = e−iH t D(z)eiH t .

(124)

As a conclusion, we can see the states |z, t as a one-parametric class of coherent states labelled by t , corresponding to a oneparametric realization of the same Lie algebra (su(2) or su(1, 1)). The CS’s including a parameter corresponding to the time t, have been built in [6,7]. By means of the CS, we have computed expected values of appropriate Hermitian operators which take part in B ± . These expected values have been compared with the corresponding classical variables which take part in the factor functions A± . The correspondence between quantum operators and classical variables comes from the common algebraic structure shared by the classical and quantum systems and allows us the choice of the appropriate natural variables to be compared (remark that they have also been used in [11], but not with this algebraic meaning). For very small |z| the expected values of the natural variables in these CS’s evolve in a similar way as their classical counterparts. The amplitude of oscillations in these circumstances are much smaller than the classical ones. On the other hand for bigger values

S. Cruz y Cruz et al. / Physics Letters A 372 (2008) 1391–1405

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of |z| there appear some new features: there are colapses and revivals due to the non-equispaced energy spectrum, and in the hyperbolic case there appear again a harmonic behaviour for very high |z|, due to its finite spectrum. Notice that our work is compatible with the Gazeau–Klauder definition of CS, which are temporally stable [6,25]. A detailed study of the temporally stable CS’s for the (asymmetric) trigonometric PT potential and the infinite well has been done recently [6]. The authors consider the classical and quantum aspects by means of the evolution of the expected values x , p of the position and momentum operators. However, to establish the comparison between classical and quantum behaviour we preferred to use the algebraic structure of these systems by means of the functions A± and the operators B ± given above. In this sense our work can be considered as a complement of [6]. Remark that we have also studied the hyperbolic PT potential, a case rarely dealt with in the literature, paying special attention to the properties coming from the finite character of its spectrum. The infinite well also can be obtained in our study, with the correct boundary conditions, from the trigonometric PT system simply by taking  = 1. In summary, we have studied the behaviour of some expected values of the CS’s for PT potentials and have compared them with the motion of their classical analogues by using the underlying algebra of symmetries in these potentials. In this way we can easily appreciate the main differences that the shape of the trigonometric and hyperbolic PT potentials originate in their CS’s. This approach also facilitates the comparison with the well known harmonic oscillator CS’s. Finally, let us mention that we have not discussed the problem for the physical generation of these CS’s. However, there are many references dealing with this point, in particular some works have used trapped ions pumped with laser beams [26]. Acknowledgements This work is supported by the Spanish MEC (FIS2005-03989), AECI-MAEC (S.K. ¸ grant 0000169684), Junta de Castilla y León (Excellence project VA013C05) and Conacyt-Mexico (grant SEP-2004-C01-47200/A-1 and projects 50766 and 49253-F). S.K. ¸ acknowledges Department of Physics, Ankara University. S.C.y.C. acknowledges the support from COTEPABE-, COFAA- and SIP-IPN. S.K. ¸ and S.C.y.C. acknowledge also the warm hospitality at Department of Theoretical Physics, University of Valladolid, Spain. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

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