Barut–Girardello coherent states for the Morse potential

Physics Letters A 310 (2003) 1–8 www.elsevier.com/locate/pla

Barut–Girardello coherent states for the Morse potential H. Fakhri a,b , A. Chenaghlou a,c,∗ a Institute for Studies in Theoretical Physics and Mathematics (IPM), PO Box 19395-5531, Tehran, Iran b Department of Theoretical Physics and Astrophysics, Physics Faculty, Tabriz University, PO Box 51664, Tabriz, Iran c Physics Department, Faculty of Science, Sahand University of Technology, PO Box 51335-1996, Tabriz, Iran

Received 28 October 2002; received in revised form 24 December 2002; accepted 27 December 2002 Communicated by P.R. Holland

Abstract Using the shape invariance idea, it is shown that the quantum states of Morse potential represent an infinite-dimensional Lie algebra the so-called Morse algebra. Then, we derive a representation of the Lie algebra u(1, 1) by means of using the generators of the Morse algebra. Meanwhile, we obtain the Barut–Girardello coherent states which are constructed as a linear combination of the quantum states corresponding to the Morse potential. Finally, we realise the resolution of the identity condition for the coherent states.  2003 Elsevier Science B.V. All rights reserved. PACS: 03.65.-w; 03.65.Fd; 02.20.Sv Keywords: Coherent states; Shape invariance; Special functions

1. Introduction For the one-dimensional Morse potential, which was first introduced as a useful model for the diatomic molecules in 1929 [1], one may follow detailed discussions in Refs. [2,3]. The Morse potential has been widely used in many areas such as molecular systems, quantum chemistry and, in particular, chemical bonds [4–9]. In algebraic models of the molecular structure, the Morse potential is related to the Lie algebra su(2) and the bound states are labeled by the rep-

* Corresponding author.

E-mail addresses: [email protected] (H. Fakhri), [email protected] (A. Chenaghlou).

resentations of the algebra [10]. Later on, the Morse potential became one of the most successful models for the states of the diatomic molecules [11,12]. In the context of supersymmetric quantum mechanics, the energy spectrum and the eigen-function of the Morse potential have been calculated by variational method for the various diatomic molecules in Ref. [13]. In the approaches of the potential group, the Lie algebra su(1, 1) and other dynamical algebras much studies have been done for obtaining the energy spectrum and the eigenfunctions of the Morse potential [14–17]. Furthermore, applying the factorisation technique in the framework of the supersymmetric quantum mechanics, the generalised Morse potential and its solutions in terms of the Morse potential solutions have been derived [18–20].

0375-9601/03/$ – see front matter  2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00125-7

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H. Fakhri, A. Chenaghlou / Physics Letters A 310 (2003) 1–8

In Ref. [2] on the basis of the classical motion method of the particle, by defining the generalised position and momentum operators in terms of the raising and lowering operators, approximate coherent states have been calculated for the Morse potential. A representation for the algebraic Hamiltonian of the Morse potential as a quadratic form of the position and momentum operators has been obtained [21]. On the basis of the representation, approximate coherent states of Klauder–Perelomov (the displacement operator method on the ground state) and their time evolution have been deduced by using the structure of the Lie algebra su(1, 1) [22]. Meanwhile, using approximate coherent states constructed in Ref. [22], the Barut–Girardello coherent states [23] for the Morse potential have been obtained in the harmonic limit [24]. At the same time for the Morse potential with the help of the supersymmetry techniques, generalised Klauder–Perelomov coherent states have been calculated and the resolution of the identity has been realised as well [25]. Moreover, like the relation between the Heisenberg–Weyl group and the harmonic oscillator coherent states, it has been shown that the Klauder–Perelomov coherent states of the Morse potential are related to the representation of the real line affine group and some of its extensions too [26]. Finally, the Gazeau–Klauder coherent states and some of their properties for the Morse potential have been obtained in Ref. [27]. In Ref. [28], using the Laguerre polynomials differential equation, the Schrödinger equation corresponding to the Morse potential and the Barut–Girardello coherent states (in other words, the coherent states for the generalised Laguerre functions) of the Morse potential have been derived. However, there is a problem in the section corresponding to the Barut–Girardello coherent states of the mentioned paper, i.e., the ladder operators are functions of the quantum numbers which, in turn, leads to some difficulties. This Letter is organised as follows: in Section 2 by reviewing the Laguerre polynomials labeled by the main quantum number n and the associated Laguerre functions which in addition to n labeled by the secondary quantum number m, we obtain the Schrödinger–Morse equation from the shape invariance with respect to the main quantum number n. Following this approach, we obtain the well-known Morse potential, Morse partner potentials as well as the energy spectrum and their

eigenfunctions as functions of n and m. It is shown that the Morse quantum states represent an infinitedimensional Morse Lie algebra. In Section 3 by means of introducing a new auxiliary variable φ, the shape invariance leads to a representation of the Lie algebra u(1, 1) by the Morse quantum states. Here, we show that the Casimir operator of the Lie algebra u(1, 1), unlike the dynamical methods, does not obtain the two-dimensional new quantum solvable models [29]. Then in Section 4, we construct the Barut–Girardello coherent states for the lowering operator of the Lie algebra u(1, 1) which is, in fact, the corrected form of the states in Ref. [28].

2. Morse potential and the approach of its quantum states to the representation of infinite-dimensional Morse Lie algebra In the mathematical physics text books [30], the Laguerre differential equation is introduced by xL n

(α,β)

(x) + (1 + α − βx)L n

(α,β)

(x)

(x) = 0, + nβL(α,β) n

(1)

where n is a non-negative integer. For the above equation, the polynomial solutions of order n in the Language of Rodrigues representation are given by
n  n+α −βx  d (α,β) −α βx x . e Ln (x) = an,0 (α, β)x e dx (2) For α > −1 and β > 0, it is clear that the Laguerre (α,β) polynomials Ln (x) with respect to the inner product with the weight function x α e−βx are orthogonal in the interval 0
x < +∞. Thus, choosing the normalisation coefficient αn,0 (α, β) as  β α+1 , an,0 (α, β) = (3) (n + 1)(n + α + 1) one may obtain ∞

(α,β)

L(α,β) (x)Ln n

(x)x α e−βx dx = δnn .

(4)

0

If we take the derivative of the differential equation (1) m-times and by using the change of function with

H. Fakhri, A. Chenaghlou / Physics Letters A 310 (2003) 1–8

the multiple x m/2 , the associated Laguerre differential equation will be obtained as [31] xL n,m (x) + (1 + α − βx)L n,m (x) 
  
m m 1 (α,β) m β− α+ L (x) = 0. + n− 2 2 2 x n,m (5) (α,β)

(α,β)

The derivation process shows that 0
m
n. Comparing the differential equations (1) and (5) with each other, one can derive the Rodrigues representation for the associated Laguerre functions which is
 an,m (α, β) d n−m  n+α −βx  (α,β) Ln,m (x) = α+ m e x x 2 e−βx dx m (α+m,β) an,m (α, β) = x 2 Ln−m (x). (6) an−m,0 (α + m, β) According to Eq. (6), the associated Laguerre functions are not polynomials of integer powers from x unless m = 0 is chosen. In what follows, we will obtain the quantum states of the Morse potential in terms of the associated Laguerre functions. Choosing the coefficients an,m (α, β) as  β α+m+1 m , an,m (α, β) = (−1) (n − m + 1)(n + α + 1) (7) and by using the relations (4) and (6), one can deduce that the associated Laguerre functions form an orthonormal set with the same m but with different n, that is, ∞

α −βx L(α,β) dx = δnn . n,m (x)Ln ,m (x)x e (α,β)

(8)

0

Note that with or without the coefficient (−1)m in Eq. (7), Eq. (8) is satisfied, however, the existence of the coefficient (−1)m is for realising of the shape invariance equations with respect to m [32]. The asso(α,β) ciated Laguerre functions Ln,m (x) are reduced to the Laguerre polynomials when m = 0, whereas the parameters α > −1 and β > 0 are still free. Considering the representation (6) for the associated Laguerre functions, it is understood that the highest power of x and its coefficient are x n−m/2 and (−β)n−m an,m (α, β), respectively.

3

Using the change of variable x = eθ and the following change of function α β

(α,β) ψn,m (9) (θ ) = x 2 e− 2 x L(α,β) n,m (x) x=eθ , the associated Laguerre differential equation (5) can be converted to a one-dimensional Schrödinger equation on the axis −∞ < θ < +∞. Moreover, the Schrödinger equation obtained by this method may be written down as the shape invariance equations with respect to m, i.e., as multiplication by the raising and lowering operators: (α,β) (θ ) A+ (n, m)A− (n, m)ψn,m (α,β) (θ ), = (n − m)(n + α)ψn,m (α,β) A− (n, m)A+ (n, m)ψn−1,m (θ ) (α,β) = (n − m)(n + α)ψn−1,m (θ ),

(10a)

(10b)

where d + Wn,m (θ ). (11) dθ The superpotential Wn,m (θ ) derived by this technique is the well-known Morse superpotential: A± (n, m) = ±

1 −βeθ + 2n + α − m . (12) 2 At this stage it is necessary to mention some important facts. According to the master function theory [31], the differential equations (1) and (5) have been obtained for the master function A(x) = x and subsequently, the partner equations (10a) and (10b) are derived by using the change of variable x = eθ which is the solution of the differential equation dx x = dθ . It is evident that if we choose the master function A(x) = γ x instead of A(x) = x, we will find the solution x = eγ θ for the differential equation γdxx = dθ . In this case the Morse superpotential will be a function of eγ θ instead of eθ . The other significant point is the fact that the Morse superpotential is a function of e−γ θ (or e−θ when γ = 1), for example, in Refs. [1,25,33]. We must note that as it is seen from Ref. [31], this representation for the Morse superpotential is obtained by using the master function A(x) = −γ x and the shape invariance with respect to the parameter n as well. Because of the fact mentioned in the above, the weight function eθ in Eq. (14) has been appeared differently from the usual Wn,m (θ ) =

4

H. Fakhri, A. Chenaghlou / Physics Letters A 310 (2003) 1–8 (α,β)

normalisation. In fact, the wave functions ψn,m (θ ) given in Eq. (9) are determined in such a manner that the integrals (14) converge to the certain values of 0 or 1. The shape invariance equations (10a) and (10b) describe the Morse partner Hamiltonians with the following partner potentials, respectively: 1 Vn,m,± (θ ) = β 2 e2θ − 2β(2n + α − m ± 1)eθ 4

+ (2n + α − m)2 . (13) Clearly, the parameter β > 0 of the weight function plays the role of the Morse potential depth. Now, using Eqs. (8) and (9), it is easily shown that the set of quantum states corresponding to the Morse superpotential form an orthonormal set for a given m as: ∞ (α,β) (α,β) (14) ψn,m (θ )ψn ,m (θ )eθ dθ = δnn . −∞

For a given m, we define the Hilbert space Hm as the closure of the span generated by all the quantum states (α,β) of the Morse potential, Hm = span{ψn,m (θ )}n
m . This Hilbert space is equipped with the inner product (14) and its bases are orthogonal, and they have unit length too. An important result obtained from Eqs. (10) is the role of raising and lowering for the operators A+ (n, m) and A− (n, m), respectively:  (α,β) (α,β) A+ (n, m)ψn−1,m (θ ) = (n − m)(n + α) ψn,m (θ ), (α,β) (θ ) = A− (n, m)ψn,m

(15a)  (α,β) (n − m)(n + α) ψn−1,m (θ ).

(15b) It can be easily verified that choosing the normalisation coefficients an,m (α, β) as the relation (7) is suitable for realising of Eqs. (15a) and (15b). For this purpose it is sufficient to compare the highest power coefficients of eθ on the both sides of Eqs. (15a) and (15b). The Hilbert space Hm spanned by the quantum states of the Morse potential, in algebraic method, is generated by the action of the infinite number of the raising operators {A+ (n, m)}n
m+1 from the ground quantum (α,β) state ψm,m (θ ) as well:  (α + m + 1) (α,β) A+ (n, m) ψn,m (θ ) = (n − m + 1)(n + α + 1)

× A+ (n − 1, m) · · · A+ (m + 1, m) (α,β) (θ ). × ψm,m

(16)

Substituting n = m in Eq. (15b), the ground state (α,β) ψm,m (θ ) can be calculated by solving a first-order differential equation to yield:  1 β α+m+1 θ (α,β) e 2 ((α+m)θ−βe ) ψm,m (17) (θ ) = (α + m + 1) where the result is in agreement with the analytic solution given in (9) for the ground state. In contrast to the harmonic oscillator, the raising and lowering operators of the Morse quantum states are functions of the main quantum number n for a given m: A± (n, m) + n = A± (n , m) + n.

(18)

Therefore, we deal with infinite number of the raising and lowering operators {A± (n, m)}n
m in the Morse problem. The set of these operators along with the identity operator I = 1 form an infinite-dimensional Morse Lie algebra with the following commutation relations:

A+ (n, m), A− (n , m) = A+ (n, m) + A− (n , m) − (n + n + α − m)I,

A+ (n, m), A+ (n , m)



= A− (n, m), A− (n , m) = A± (n, m), I = 0. (19) Considering Eq. (18), it is concluded that the Hilbert space Hm carrying the representation of the infinitedimensional Morse Lie algebra (with the commutation relations (19)) as: (α,β)

A+ (n, m)ψn ,m (θ )  (α,β) = (n − m + 1)(n + α + 1) ψn +1,m (θ ) + (n − n − 1)ψn ,m (θ ), (α,β)

(20a)

(α,β) A− (n, m)ψn ,m (θ )

=



(α,β)

(n − m)(n + α) ψn −1,m (θ )

+ (n − n )ψn ,m (θ ), (α,β)

(α,β)

(α,β)

I ψn ,m (θ ) = ψn ,m (θ ).

(20b) (20c)

According to the above discussions, the lowering operators A− (n, m) are functions of the quantum

H. Fakhri, A. Chenaghlou / Physics Letters A 310 (2003) 1–8

number n which, in turn, leads to an extra term (second term) on the right-hand side of the representation equation (20b). So, the Barut–Girardello coherent states introduced in Ref. [28] have some difficulties. In fact, Section 3 in [28] and, in particular, Eq. (18) in [28] have signs of incorrectness. It seems that for correcting of the Barut–Girardello coherent states corresponding to the Morse potential introduced in Ref. [28], at first we must show that how one may derive the representation of the Lie algebra u(1, 1) from the solvability of the Morse potential. Then, we shall discuss the Barut–Girardello coherent states.

3. Representation of Lie algebra u(1, 1) by the Morse potential quantum states

5

span{|n, m }n
m . So, ∞

|n, m n, m| = 1.

(25)

n=m

Defining L3 = −i

∂ , ∂φ

I = 1,

(26)

it is evident that we have L3 |n, m = n|n, m ,

I |n, m = |n, m .

(27)

On the other hand, the four operators L+ , L− , L3 and I satisfy the following commutation relations of the Lie algebra u(1, 1): [L+ , L− ] = −2L3 − (α − m + 1)I,

For the following states

[L3 , L± ] = ±L± ,

einφ

(α,β) (θ ), |n, m := √ ψn,m 2π

(21)

which are functions of θ and a free auxiliary variable 0
φ < 2π , using Eqs. (15) one can deduce:  L+ |n − 1, m = (n − m)(n + α) |n, m , (22a)  L− |n, m = (n − m)(n + α) |n − 1, m , (22b) where the raising and lowering operators L+ and L− are defined by 


∂ 1 ∂ iφ θ −i + α − m − βe + 1 , L+ = e ∂θ ∂φ 2


∂ 1 ∂ −iφ θ −i + α − m − βe L− = e − . (23) ∂θ ∂φ 2 The new states |n, m form an orthonormal set with respect to the following inner product n, m|n , m 1 = 2π

2π

∞ dφ

0



(α,β) eθ dθ ei(n −n)φ ψn,m (θ )

−∞ (α,β)

× ψn ,m (θ ) = δnn .

(24)

The relation (24) immediately leads to the comm which pleteness relation on the Hilbert space H m = is spanned by the new bases |n, m i.e., H

[L, I ] = 0.

(28)

m represents Thus for a given m, the Hilbert space H the Lie algebra u(1, 1) given in Eqs. (28) as the relations (22) and (27). Note that the basis elements |n, m and consequently the representation of the Lie algebra u(1, 1) are obtained in terms of the Laguerre functions. It is also clear that, according to Eqs. (22), the u(1, 1) generators L+ and L− change n into n + 1 and n − 1, respectively, so that both of the potential and the energy eigenvalue are changed. As it has been introduced in Refs. [19,34–36], this is a characteristic property of the satellite algebras. The satellite algebras are symmetry algebras corresponding to the different potentials of a given family. The Casimir operator of the Lie algebra u(1, 1) is given by 1 H = −L+ L− + L23 + (α − m)L3 − (α − m + 1)I 2 2  ∂2 1 ∂ = 2 − iβeθ − βeθ − α + m ∂φ 4 ∂θ  1 θ + βe − α + m − 1 , (29) 2 so that it has the following eigenvalue equation on the m : Hilbert space H 1 H |n, m = (2mα − α + m − 1)|n, m . 2

(30)

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H. Fakhri, A. Chenaghlou / Physics Letters A 310 (2003) 1–8

The Casimir operator (29) via the relation − 12 H = − 12 DjA D Aj + V , where DjA represents the gaugecovariant derivative with the connection Ai , cannot describe a two-dimensional dynamics on a manifold with the coordinates θ and φ since it does not have a term with the second-order derivative with respect to ∂ acts only as φ [29]. Therefore, the term including ∂φ an intermediate formula which after reducing it and eliminating the dependence of the both sides of the eigenvalue equation (30) on einφ gives the Schrödinger equation on the θ -axis with the familiar form of the Morse potential as [1]:   2 (α,β) 1 d2 (θ ) − 2 + βeθ − 2n − α + m − 1 ψn,m 4 dθ 1 (α,β) (θ ). (31) = (2n − 2m + 1)(2n + 2α + 1)ψn,m 4 Note that the potentials Vn,m,± (θ ) are the Morse partner potentials which are obtained by the factorised Morse equations, i.e., Eqs. (10) whereas, Eq. (31) yields the well-known form of the Morse potential. Anyway, the Morse partner potentials Vn,m,± (θ ) as well as the Morse potential of the form given in (α,β) Eq. (31) are described by the quantum states ψn,m (θ ) which span the Hilbert space Hm .

product (24):

4. Barut–Girardello coherent states

For the above purpose, we represent the complex variable z in the polar coordinates as z = reiϕ (0
ϕ < 2π , 0
r < ∞) and we suggest that

Now, using the representation of the Lie algebra u(1, 1) obtained from the Morse potential we construct the Barut–Girardello coherent states in the m as the eigen-states of the bosonic Hilbert space H lowering operator L− : L− |z m = z|z m ,

(32)

where z is an arbitrary complex variable. Applying Eq. (22b), we can obtain the coherent states |z m as a linear combination of the bases of the Hilbert space m : H  +∞ (α + m + 1) z−m |z m = gm (|z|) n=m (n − m + 1)(n + α + 1) × zn |n, m .

(33)

If we want the length of the coherent states |z m to be normalised to unit with respect to the inner

m z|z m

= 1,

(34)

−1 (|z|) we must choose the normalisation coefficient gm as the following real function: α+m

1 |z| 2 = , gm (|z|) (α + m + 1)Iα+m (2|z|)

(35)

where Iα+m (2|z|) is the modified Bessel function of the first kind with infinite radius of convergence for |z|: Iα+m (2|z|) =

+∞ n=0

|z|2n+α+m . (n + 1)(n + α + m + 1)

(36)

Considering Eq. (30), it is evident that |z m is the eigenstate of the Casimir operator H : 1 H |z m = (2mα − α + m − 1)|z m . (37) 2 In order to complete the discussion, we should introduce the appropriate measure dσ (z) so that the resolution of the identity is realised for the coherent m : states |z m in the Hilbert space H  dσ (z)|z m m z| = 1. (38)

2 Iα+m (2r)K α+m (2r)r dr dϕ, (39) 2 π where the functions K α+m (2r) are the modified Bessel 2 function of the second kind:

 π  α+m  I− α+m (2r) − I α+m (2r) . K α+m (2r) = 2 2 2 2 sin π 2 (40) Now, using the completeness relation (25) and the following integral relation dσ (z) =

∞ t

2

2n+α−m 2

 √  K α+m 2 t dt 2

0

= (n − m + 1)(n + α + 1), it is noticed that the relation (38) is satisfied.

(41)

H. Fakhri, A. Chenaghlou / Physics Letters A 310 (2003) 1–8

Considering the following relation A− (n, m) = n + eiφ L− − L3

(42)

and, also, the fact that φ is an intermediate and auxiliary variable, the coherent states |z m can be regarded as a linear combination of the quantum states (α,β) ψn,m (θ ) corresponding to the Morse potential so that the terms einφ have contribution in the expansion coefficient. Thus, it is found that A− (n, m)|z m  
∂ ∂ Ln gm (|z|) iφ − |z m . = n−m+z e − ∂z ∂z (43) Meanwhile, it should be noted that the overlapping of the Barut–Girardello coherent states |z m and |z m is calculated as follows  √  z α+m 4 Iα+m 2 z ¯ z ( zz¯ ¯ z )  , (44) m z|z m =  Iα+m (2|z|)Iα+m (2|z |) where the relation (44) is converted to the relation (34) for z = z . As a consequence of the discussions of the previous section, the coherent states |z m are a linear combination of the eigenstates belonging to the different Morse potentials corresponding to the different energy eigenvalues.

5. Conclusion In addition to the dependence on the two parameters α and β of the weight function, the Laguerre functions are introduced with their dependence on the main and secondary quantum numbers n  0 and 0
m
n. With the help of the appropriate changes of variable and function on the associated Laguerre differential equation, using the shape invariance with respect to n as well, we derive the Morse superpotential, the Morse partner potentials, the Morse quantum states and also the spectrum of the energy. The set of infinite raising and lowering operators of the Morse quantum states together with the identity operator constitute an infinite-dimensional Lie algebra the so-called Morse algebra which is represented by the Morse quantum states. The mentioned shape invariance leads to the derivation of the Lie algebra u(1, 1) and its representation by the Morse quantum states. Reducing the eigen-

7

value equation of the Casimir operator of the Lie algebra u(1, 1) with respect to the auxiliary variable φ, the Schrödinger equation is obtained with the wellknown Morse potential. For the Hilbert space carrying the representation of the Lie algebra u(1, 1), we construct the Barut–Girardello coherent states as a linear combination of the Morse quantum states. Meanwhile, the resolution of the identity is realised on the whole of the complex plane.

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