Exact solutions and ladder operators for a new anharmonic oscillator

Physics Letters A 340 (2005) 94–103 www.elsevier.com/locate/pla

Exact solutions and ladder operators for a new anharmonic oscillator Shi-Hai Dong a,∗ , Guo-Hua Sun b , M. Lozada-Cassou a a Programa de Ingeniería Molecular, Instituto Mexicano del Petróleo, Lázaro Cárdenas 152, 07730 México DF, Mexico b Instituto de Investigaciones en Matematicas Aplicadas y en Sistemas, UNAM,

A.P. 20-726, Del. Alvaro Obregón, 01000 México DF, Mexico Received 23 February 2005; received in revised form 11 April 2005; accepted 11 April 2005 Available online 19 April 2005 Communicated by R. Wu

Abstract In this Letter, we propose a new anharmonic oscillator and present the exact solutions of the Schrödinger equation with this oscillator. The ladder operators are established directly from the normalized radial wave functions and used to evaluate the closed expressions of matrix elements for some related functions. Some comments are made on the general calculation formula and recurrence relation for off-diagonal matrix elements. Finally, we show that this anharmonic oscillator possesses a hidden symmetry between E(r) and E(ir) by substituting r → ir.  2005 Elsevier B.V. All rights reserved. PACS: 03.65.-w; 03.65.Ge Keywords: Exact solutions; Anharmonic oscillator; Ladder operators

1. Introduction The study of exactly solvable problems has attracted much attention since the early development of quantum mechanics. For instance, the exact solutions of the Schrödinger equation for a hydrogen atom and for a harmonic oscillator in 3D [1,2] represent two typical examples in quantum mechanics. Generally speaking, one is able to construct the ladder operators with the factorization method [3,4] provided that the exact solutions of given quan-

* Corresponding author.

E-mail address: [email protected] (S.-H. Dong). 0375-9601/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.04.024

S.-H. Dong et al. / Physics Letters A 340 (2005) 94–103

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tum system are known. This elegant method1 was formulated in many different ways and played an important role in various domains [6,7]. Based on this method, recently we have been successful in establishing the ladder operators for some important molecular potentials like the Morse potential, the Pöschl–Teller potential, the pseudoharmonic potential and others [8–10]. From those ladder operators we can obtain the closed expressions of matrix elements for some related functions. Even though it is possible to obtain those matrix elements through using other approaches such as the direct integration for some special functions, the advantage of factorization method lies in its elegance and intuitivity. The purpose of this Letter is two-fold. First, due to recent interest of the exact solutions of the Schrödinger equation with non-central potentials such as the non-central electromagnetic, Aharonov–Bohm and magnetic monopole potentials carried out by Alhaidari [11] and our recently proposed new non-spherical potential [12] composed of the Coulomb potential and the angular-dependent potential (r sin θ )−2 cos2 θ , we first propose a new anharmonic oscillator, where the Coulomb part of non-spherical potential [12] is replaced by a harmonic oscillator plus an inverse squared potential and then establish the ladder operators. Second, we derive the general calculation formula and recurrence relation for off-diagonal matrix elements since some theoretical calculations like the vibrational transitions require the knowledge of matrix elements of powers of the radial coordinate. Such a new proposed model potential is possibly useful in studying the ring-shaped molecule. As we know, the harmonic oscillator as an ideal model potential describing the interaction between the atoms in molecule is proved to be anharmonic in practice. We may interprete this model in such a way that the second term describes the dipole-like interaction and the third represents the angular-dependent interaction between the atoms apart from the principal harmonic oscillator interaction term. This Letter is organized as follows. In Section 2 we shall present the exact solutions of this system. Section 3 is devoted to constructing the ladder operators for the radial wave functions. In Section 4 we derive the general calculation formula and recurrence relation for off-diagonal matrix elements. Some concluding remarks are given in Section 5.

2. Eigenvalues and eigenfunctions We begin by considering the Schrödinger equation      
1 ∂ 1 ∂ 1 ∂2 h¯ 2 1 ∂ 2 ∂ + V (r) − E Φ(r, θ, ϕ) = 0, r + sin θ + 2 − 2µ r 2 ∂r ∂r r 2 sin θ ∂θ ∂θ sin θ ∂ϕ 2

(1)

where we take V (r) as a new proposed anharmonic oscillator h¯ 2 α h¯ 2 β cos2 θ 1 + V (r, θ ) = µω2 r 2 + . 2 2µ r 2 2µ r 2 sin2 θ

(2)

Here µ, ω, α and β denote the mass of the particle, the frequency and two dimensionless parameters, respectively. Taking Φ(r, θ, ϕ) = r −1 R(r)Θ(θ )e±imϕ (m = 0, 1, 2, . . .) and substituting it into Eq. (1) lead to the following differential equations (h¯ = µ = ω = 1)   α+λ d 2 R(r) 2 R(r) = 0, + 2E − r − (3) dr 2 r2     1 d dΘ(θ ) β cos2 θ + m2 (4) Θ(θ ) = 0, sin θ + λ− sin θ dθ dθ sin2 θ 1 The roots of this method may be traced back to the great mathematician Cauchy. We may find some detailed lists of references illustrating

the history from the book written by Schlesinger [5]. Also, we should mention that it is impossible to refer to all contributions in this field.

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with a constant λ to be determined below. Introduce  m = β + m2 , λ + β =  (  + 1), x = cos θ.

(5)

Inserting this into Eq. (4) yields     d 2 Θ(θ (x)) dΘ(θ (x)) (m )2   Θ θ (x) = 0, +

1 − x2 − 2x (

+ 1) − 2 2 dx dx 1−x

(6)

whose normalized solutions are given by [12]

Θ  m (θ ) =

  −m

2

(2  + 1) (  − m )! (−1)k (2  − 2k + 1)   m (sin θ ) (cos θ ) −m −2k ,   

   2 ( + m + 1) 2 k!( − m − 2k)!( − k + 1)

k=0

(7)

with

 = m + k,

k = 0, 1, 2, . . . .

We now turn to study Eq. (3). Substitution of Eqs. (5) and (8) into Eq. (3) leads to   d 2 R(r) L(L + 1) 2 R(r) = 0, + 2E − r − dr 2 r2 with

     1 L≡ 1 + 4 α − β + β + m2 + k β + m2 + k + 1 − 1 . 2

Defining a new variable ρ = r 2 , we may rearrange Eq. (9) as   √ 1 L(L + 1) d 2 √ 1 d √ E R ρ − + R ρ + − R ρ = 0. 2 2 2ρ dρ 4 2ρ dρ 4ρ ρ √ Taking R( ρ ) = ρ s e− 2 F (ρ) (s = (L + 1)/2) and substituting this into Eq. (11) yield     d E 1 1 d2 F (ρ) + −s − F (ρ) = 0, ρ 2 F (ρ) + 2s + − ρ 2 dρ 2 4 dρ

(8)

(9)

(10)

(11)

(12)

whose solutions are the confluent hypergeometric functions F (s − E/2 + 1/4, 2s + 1/2; ρ). Finally, one can obtain the eigenfunctions as √ ρ R ρ = Nρ s e− 2 F (s − E/2 + 1/4, 2s + 1/2; ρ). (13) In view of the finiteness of the solutions, we see from Eq. (12) that the general quantum condition is given by s − E/2 + 1/4 = −n, from which we have 1 3 (14) + 2n + 2s = 2n + L + , n = 0, 1, 2, . . . . 2 2 This means that the energy levels are equidistant. Recall that when n = E/2 − s − 1/4 is a non-negative integer, using the following two known relations [13] En =

Lαn (y) =

(α + n + 1) F (−n, α + 1; y), n!(α + 1)

(15)

S.-H. Dong et al. / Physics Letters A 340 (2005) 94–103

∞

y α e−y Lαn (y)Lαm (y) dy =

(n + α + 1) δnm , n!

97

(16)

0

we can finally obtain the orthonormality relation for the radial wave functions as ∞ Rn(L)

√ (L) √ dρ ρ Rm ρ √ = δnm , 2 ρ

0

where the radial wave function is given by √ ρ L+ 1 L+1 Rn(L) ρ = Nn(L) ρ 2 e− 2 Ln 2 (ρ),

Nn(L) =

2n! . (L + n + 3/2)

(17)

3. Ladder operators We are now in the position to construct the ladder operators. As shown in our previous work [8–10], the ladder operators can be constructed directly from Eq. (17), i.e., we intend to find the differential operators Lˆ ± with the following properties: d + B± (ρ), Lˆ ± = A± (ρ) dρ

√ √ Lˆ ± Rn(L) ρ = l± (n)Rn(L) ρ , ±1

where we stress that these operators only depend on the physical variable ρ. For this purpose, we start by acting the operator d/dρ on the wave functions (17)   √ ρ d s 1 d (L) √ 2s−1/2 ρ = (ρ), Rn − Rn(L) ρ + Nn(L) ρ s e− 2 Ln dρ ρ 2 dρ

(18)

(19)

where we have used 2s = L + 1. It should be pointed out that we shall use this expression to construct the ladder operators Lˆ ± by using the recurrence relations among the associated Laguerre functions. Thus, we may establish (L) √ (L) √ the relations between Rn ( ρ ) and Rn±1 ( ρ ). To this end, let us first recall the following two useful relations [13]
α nLn (x) − (n + α)Lαn−1 (x), d α x Ln (x) = (20) (n + 1)Lαn+1 (x) − (n + α + 1 − x)Lαn (x). dx Substitutions of them into (19) enable us to obtain the following relations:   (L) √ n+s 1 n + 2s − 1/2 Nn d (L) √ − + Rn(L) ρ = − Rn−1 ρ , (L) dρ ρ 2 ρ Nn−1   √ n + 1 Nn(L) (L) √ n + s + 1/2 1 d + − Rn(L) ρ = R ρ , dρ ρ 2 ρ N (L) n+1 n+1 (L)

from which, together with Nn

(21)

(22)

given in Eq. (17), we can define the following operators:

ρ d + s + nˆ − , Lˆ − = −ρ dρ 2

1 ρ d Lˆ + = ρ + s + nˆ + − , dρ 2 2

where we have introduced the number operator nˆ with the property [10] √ √ nR ˆ n(L) ρ = nRn(L) ρ .

(23)

(24)

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S.-H. Dong et al. / Physics Letters A 340 (2005) 94–103

The ladder operators Lˆ ± have the following properties by acting them on wave functions: √ √ (L) √ (L) √ Lˆ − Rn(L) ρ = l− (n)Rn−1 ρ , Lˆ + Rn(L) ρ = l+ (n)Rn+1 ρ ,

(25)

where l− (n) =



n(n + 2s − 1/2),

l+ (n) =



(n + 1)(n + 2s + 1/2).

(26)

On the other hand, we find that the radial wave functions can be directly obtained by acting the creation operator (L) √ Lˆ + on the ground state R0 ( ρ ), i.e., Rn(L)

√ (L) √ ρ = Nn Lˆ n+ R0 ρ ,

with

Nn =

(27)

(L) √ R0 ρ =

(L + 3/2) , n!(n + L + 3/2)

ρ L+1 1 ρ 2 e− 2 . (L + 3/2)

(28)

As a byproduct, we are able to obtain the following related functions in terms of the ladder operators Lˆ ± in order to demonstrate the simplification in practical calculation,   1 d 1 1 ˆ L+ − Lˆ − − , ρ = 2nˆ + 2s + − Lˆ − − Lˆ + , (29) ρ = 2 dρ 2 2 from which, together with Eqs. (25) and (26), we have ∞

  1 (L) δn ,n Rn (r)r 2 Rn(L) (r) dr = 2n + 2s + 2

0

− ∞

(L)

Rn (r)



n(n + 2s − 1/2) δn ,(n−1) −

 (n + 1)(n + 2s + 1/2) δn ,(n+1) ,

r d (L) 1 Rn (r) dr = (n + 1)(n + 2s + 1/2) δn ,(n+1) 2 dr 2

0

−

1 1 n(n + 2s − 1/2) δn ,(n−1) − δn ,n . 2 4

(30)

4. General calculation formula and recurrence relation for off-diagonal matrix elements As we know, the vibrational transitions involving two different electronic states require the calculation of matrix elements, which is usually done by direct numerical integration or by an approximate method. Nevertheless, the analytical expression of matrix elements shall be elegant and stands in its own right. For this purpose, we first derive a general calculation formula for off-diagonal matrix elements and then derive the recurrence relation among them. We denote |n, L ≡ Rn(L) (r) for simplicity. Thus, we can express the calculation formula of off-diagonal matrix elements for r κ as (L)

(L) ∞

Nn Nn n|r |n  = 2 κ



0



ρ (L+L +κ+1)/2 e−ρ Ln

L+1/2

L +1/2

(ρ)Ln

(ρ) dρ,

(31)

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99

where we have used 2s = L + 1. Using the following relation [14] ∞





x λ e−x Lun (x)Lun (x) dx = (−1)n+n (1 + λ)

 λ − u   λ − u   λ + i  i

0

n − i

n−i

i

,

(32)

with the condition e(λ) > −1, we have 

(−1)n+n Nn Nn   (L + L + κ + 3)/2 n|r |n  = 2 κ



(L)

(L)

 L +κ−L   L+κ−L   1+L+L +κ+2i  2

n−i

i

κ > −(L + L + 3),

2

n − i

2

i

, (33)

from which we obtain a simple calculation formula for diagonal matrix elements n|r κ |n =

 κ 2  1+2L+κ+2i  (L) (Nn )2  2 2 ,  (2L + κ + 3)/2 2 n − i i i

(34)

which implies that n|n = 1 for κ = 0. Now, let us derive the recurrence relation among off-diagonal matrix elements following the so-called Kramers’ approach.2 For two different states |n and |n , we may rewrite Eq. (9) as   (L) L(L + 1) (L) d 2 Rn (r) 2 Rn (r) = 0, + (4n + 2L + 3) − r − dr 2 r2   (L) d 2 Rn (r) L(L + 1) (L)  2 Rn (r) = 0. + (4n + 2L + 3) − r − dr 2 r2

(35) (36)

According to the asymptotic behaviors of the radial wave functions (17) at the origin and at infinity, we have 

(L)

Rn (r) → r L+1 ,

when r → 0,

2 (L) Rn (r) → e−r /2 ,

when r → ∞.

(37)

If taking κ > −(L + L + 1), we have the following definite integral results:  (L) dRn (r) ∞  = 0, dr 0 ∞ (L) dRn (r) (L)  rκ Rn (r) = 0, dr 0 (L)

r κ Rn (r)

∞  (L) r κ−1 Rn (r)Rn(L) (r) = 0, 0

r

(L) κ+1 dRn (r)

dr

 dRn(L) (r) ∞ dr



= 0.

(38)

0

(L)

First, multiplying each term of Eq. (35) by r κ Rn (r) and then taking integration, the three terms in the brackets are matrix elements of r κ , r κ−1 and r κ−2 , respectively. Integrating the first term twice by parts and taking Eq. (38) into account, we may obtain 2 This is a method that uses the Schrödinger equation multiplied by an appropriate function, integral and differential operators and boundary

conditions [15].

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S.-H. Dong et al. / Physics Letters A 340 (2005) 94–103

∞ (L) dR  (r) κr κ−1 Rn(L) (r) n dr dr 0

∞ (L) (L) dRn (r) dRn (r) dr + n|r κ+2 |n  − (4n + 2L + 3)n|r κ |n  = rκ dr dr 0

 + L(L + 1) − κ(κ − 1) n|r κ−2 |n .

(39) (L)

Similarly, multiplying each term of Eq. (36) by r κ Rn (r) and then taking integration. Integrating the first term by parts and considering Eq. (38), we have ∞ −

(L)

κr κ−1 Rn(L) (r)

dRn (r) dr dr

0

∞ =

(L)

(L)

rκ

dRn (r) dRn (r) dr + n|r κ+2 |n  − (4n + 2L + 3)n|r κ |n  + L (L + 1)n|r κ−2 |n . dr dr

(40)

0

Comparing Eq. (39) with Eq. (40) leads to ∞ −

(L)

rκ

dRn(L) (r) dRn (r) dr dr dr

0

= n|r κ+2 |n  − (2n + 2n + L + L + 3)n|r κ |n  1 + L(L + 1) + L (L + 1) − κ(κ − 1) n|r κ−2 |n , 2 ∞

(41)

(L)

κr κ−1 Rn(L) (r)

dRn (r) dr dr

0

= −(2n − 2n + L − L )n|r κ |n  +

1 L(L + 1) − L (L + 1) − κ(κ − 1) n|r κ−2 |n . 2

(42)

Second, multiplying each term of Eq. (35) by r κ+1 dRn(L)  (r)/dr. Integrating the first term by parts and considering Eq. (38) again, we have ∞ −

(L)

r κ+1

dRn(L) (r) dRn (r) dr dr dr

0

∞ =

r

κ+3

(L) dRn (r) (L) Rn (r)

dr

∞ (L) dRn (r) κ+1 (L) dr − (4n + 2L + 3) r dr Rn (r) dr

0

0

∞ + L(L + 1) r κ−1 Rn(L) (r) 0

(L)

dRn (r) dr + dr

∞ (1 + κ)r κ 0

(L)

(L)

dRn (r) dRn (r) dr. dr dr

(43)

S.-H. Dong et al. / Physics Letters A 340 (2005) 94–103

101

(L)

In a similar way, multiplying each term of Eq. (36) by r κ+1 dRn (r)/dr. Integrating the three terms in the brackets by parts and using the conditions Eq. (38) allow one to obtain ∞

(L)

(L)

r κ+1

dRn (r) d 2 Rn (r) dr dr dr 2

0

∞ =−

r

κ+3

(L) dR  (r) Rn(L) (r) n

dr

dr − (κ + 3)n|r

κ+2

∞ (L) dR  (r) |n  + (4n + 2L + 3) r κ+1 Rn(L) (r) n dr dr 





0

0 









+ (4n + 2L + 3)(κ + 1)n|r |n  − L (L + 1)(κ − 1)n|r ∞ (L) dR  (r)   dr. − L (L + 1) r κ−1 Rn(L) (r) n dr κ

κ−2



|n  (44)

0

Incorporating Eq. (44) into (43) yields ∞ (L) (L) κ dRn (r) dRn (r) − (1 + κ)r dr dr dr 0 



= −L (L + 1)(κ − 1)n|r

κ−2

 |n  + L(L + 1) − L (L + 1) 

∞

(L)

r κ−1 Rn(L) (r)

dRn (r) dr dr

0

∞ (L) dR  (r) dr + (1 + κ)(4n + 2L + 3)n|r |n  − (4n − 4n + 2L − 2L ) r κ+1 Rn(L) (r) n dr 



κ







0

− (κ + 3)n|r

κ+2



|n .

(45)

Replacing the second term integral on the right-hand side of Eq. (45) by Eq. (42), while replacing the fourth term integral by Eq. (42), in which κ is replaced by κ + 2 and then simplifying, one is able to obtain ∞ −

(L)

rκ 0

=

(L)

dRn (r) dRn (r) dr dr dr




 2[2(n − n ) + L − L ]2 κ + 3 − n|r κ+2 |n  (κ + 1)(κ + 2) κ +1 
 2[L(L + 1) − L (L + 1)](2(n − n ) + L − L ) n|r κ |n  + 2(n + n ) + L + L + 3 − κ(κ + 2)
 [L(L + 1) − L (L + 1)]2 (κ − 1)[L(L + 1) + L (L + 1)] + − n|r κ−2 |n . 2κ(1 + κ) 2(1 + κ)

(46)

In comparison Eq. (41) with Eq. (46), we obtain a very useful recurrence relation among off-diagonal matrix elements
 κ[L(L + 1) + L (L + 1)] [L(L + 1) − L (L + 1)]2 κ(κ − 1) − − n|r κ−2 |n  2(κ + 1) 4κ(1 + κ) 4 
 [L(L + 1) − L (L + 1)](2(n − n ) + L − L )   n|r κ |n  + − 2(n + n ) + L + L + 3 + κ(κ + 2)

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S.-H. Dong et al. / Physics Letters A 340 (2005) 94–103

 (2(n − n ) + L − L )2 κ + 2 − n|r κ+2 |n  = 0, − (κ + 1)(κ + 2) κ +1


(47)

from which we may obtain a simple recurrence relation among diagonal matrix elements κ[(2L + 1)2 − κ 2 ] (48) n|r κ−2 |n. 4 For the special case α = β = 0, however, we have L = l. As a result, the general calculation formula (33) for off-diagonal matrix elements and recurrence relation (47) reduce to those of the harmonic oscillator [12] in which the parameters L and L are replaced by l  and l, respectively. (κ + 2)n|r κ+2 |n = (κ + 1)(4n + 2L + 3)n|r κ |n −

5. Concluding remarks In this Letter we have proposed a new anharmonic oscillator and presented the exact solutions of this system. We have constructed the ladder operators directly from the wave functions (17) with the factorization method. The matrix elements of some related functions have also been obtained analytically from the ladder operators Lˆ ± . We found that this method represents a simple and elegant approach to obtain them. We have made some comments on the general calculation formula for off-diagonal matrix elements of r κ and the recurrence relation among them. Before ending this work, we make an interesting remark here. As discussed in Ref. [16], it is found that the eigenfunctions (17) (ρ = r 2 ) vanish not only as r → ∞, but also as r → i∞. The substitution r → ir demonstrates intimate connections between the eigenvalues E(r) and E(ir), that is to say, we find that the substitution r → ir reverses the sign of E in Eq. (9), leaving the parameter L invariant. Accordingly, we have E(r) = −E(ir).

Acknowledgements This work is partially supported by IMP. The authors would like to thank the kind referee for the positive and invaluable suggestions, from which the manuscript has been improved greatly.

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