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An algebraic approach to the ring-shaped non-spherical oscillator
Physics Letters A 328 (2004) 299–305 www.elsevier.com/locate/pla
An algebraic approach to the ring-shaped non-spherical 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, D.F., Mexico b Instituto de Investigaciones en Matematicas Aplicadas y en Sistemas UNAM, A.P. 20-726, Del. Alvaro Obregón, 01000 México, D.F., Mexico
Received 1 June 2004; accepted 15 June 2004 Available online 23 June 2004 Communicated by R. Wu
Abstract The eigenfunctions and eigenvalues of the Schrödinger equation with a ring-shaped non-spherical oscillator are obtained. A realization of the ladder operators for the radial wave functions is studied. It is found that these operators satisfy the commutation relations of an SU(1, 1) group. The closed analytical expressions for the matrix elements of different functions ρ d with ρ = r 2 are evaluated. and ρ dρ 2004 Published by Elsevier B.V. PACS: 03.65.Fd; 03.65.Ge; 02.20.Qs Keywords: Ring-shaped non-spherical oscillator; Ladder operators; SU(1, 1) group; Matrix elements
1. Introduction It is well known that the algebraic method has been the subject of the interest in the wide variety of fields of physics. Systems displaying a dynamical symmetry can be treated by algebraic techniques [1–4]. With the factorization method [5–7], we have established the ladder operators of a quantum system with some important potentials such as the Morse potential, the Pöschl–Teller one, the pseudoharmonic one, the infinitely square-well one and other quantum systems [8–12]. From those ladder operators we can finally constitute a suitable algebra and simultaneously obtain the matrix elements for some related functions. It should be addressed that our approach is different from the traditional one, where an auxiliary variable was introduced [13], namely, we can construct the ladder operators only from the physical variable without introducing any auxiliary variable. Recently, the quantum system for the ring-shaped non-spherical oscillator has been studied [14]. To our knowledge, however, the hidden symmetry of this quantum system is a gap to be filled in, which is the main purpose of the present work. This Letter is organized as follows. In Section 2 we derive the eigenvalues and eigenfunctions for this system. Section 3 is devoted to the construction of the ladder operators directly from the radial wave function with the * Corresponding author.
E-mail addresses: [email protected] (S.-H. Dong), [email protected] (G.-H. Sun), [email protected] (M. Lozada-Cassou). 0375-9601/$ – see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.physleta.2004.06.037
300
S.-H. Dong et al. / Physics Letters A 328 (2004) 299–305
d factorization method. The matrix elements of the different functions ρ and ρ dρ with ρ = r 2 are analytically obtained from these operators. The concluding remarks are given in Section 4.
2. The eigenvalues and eigenfunctions In this section we are now studying the exact solutions, which are necessary to be used to construct the ladder operators and constitute a suitable algebra. Consider the Schrödinger equation with a potential V (r) that depends only on the distance r from the origin 1 ∂2 h¯ 2 1 ∂ 2 ∂ r + − (1) Ψ (r, ϕ) + V (r)Ψ (r, θ, ϕ) = EΨ (r, θ, ϕ), 2m r 2 ∂r ∂r r 2 ∂ϕ 2 where m is the mass of the particle. In the present work, the potential V (r) is taken as the harmonic potential in the presence of the dipole-like interaction, h¯ 2 α h¯ 2 β 1 + , V (x) = mω2 r 2 + 2 2 2 2m r 2m r sin2 θ
(2)
where the ω is the frequency and α and β are the dimensionless parameters. For simplicity, the natural units h¯ = m = ω = 1 are employed throughout this Letter if not explicitly stated otherwise. Owing to the symmetry of the potential and following the procedure used by Cheng and Chen in Ref. [14], we take the wave functions with the form R(r) Θ(θ )e±imϕ , m = 0, 1, 2, . . .. r Substitution of this into Eq. (1) allows us to obtain the angular wave function and the radial one as α+λ d 2 R(r) 2 + 2E − r − 2 R(r) = 0, dr 2 r 1 d dΘ(θ ) β + m2 Θ(θ ) = 0, sin θ + λ− sin θ dθ dθ sin2 θ Ψ (r, θ, ϕ) =
with the constant λ to be determined below. Introduce λ = ( + 1), ξ = β + m2 ,
x = cos(θ ).
Substitution of Eq. (5) into Eq. (4b) leads to d 2 Θ(x) dΘ(x) ξ2 + 1 − x2 − 2x ( + 1) − Θ(x) = 0, dx 2 dx 1 − x2
(3)
(4a) (4b)
(5)
(6)
whose normalized solution can be obtained as [14] Θ ξ (θ ) =
−ξ
2 (2 + 1) ( − ξ )! (−1)κ (2 − 2κ + 1) ξ (sin θ ) (cos θ ) −ξ −2κ , 2
( + ξ + 1) 2 κ!( − ξ − 2κ)! ( − κ + 1)
(7)
κ=0
with = ξ + k,
k = 0, 1, 2, . . ..
(8)
S.-H. Dong et al. / Physics Letters A 328 (2004) 299–305
Substitution of this into Eq. (4a) allows us to obtain d 2 R(r) L(L + 1) 2 + 2E − r − R(r) = 0, dr 2 r2 with 1 L≡ 2
2 2 1+4 α+ β +m +k β +m +k+1 −1 .
Defining the new variable ρ = r 2 , Eq. (9) can be rearranged as d2 1 L(L + 1) 1 d E R(ρ) − + R(ρ) + − R(ρ) = 0. dρ 2 2ρ dρ 4 4ρ 2 2ρ
301
(9)
(10)
(11)
From the behaviors of the wave functions at the origin and at infinity, we can take the following ansatz for the wave functions ρ
R(ρ) = ρ s e− 2 F (ρ),
(12)
with L+1 (13) , 2 where another solution s = (−L/2) is not an acceptable one in physics. Substitution of Eq. (12) into Eq. (11) allows us to obtain d2 d E 1 1 ρ 2 F (ρ) + 2s + − ρ (14) F (ρ) = 0, F (ρ) + −s − 2 dρ 2 4 dρ s=
whose solution is nothing but the confluent hypergeometric solutions 1 F1 (s − E/2 + 1/4, 2s + 1/2; ρ). One can finally obtain the eigenfunctions as ρ R(ρ) = Nρ s e− 2 1 F1 s − E2 + 14 , 2s + 12 ; ρ , (15) with a normalized factor N to be determined below. From consideration of the finiteness of the solutions, it is shown from Eq. (13) that the general quantum condition is E 1 s − + = −n, n = 0, 1, 2, . . . , (16) 2 4 from which we have 1 1 En = 2 n + s + (17) = + 2n + 2s, 4 2 which implies that the energy level is equidistant. Recall that when n = E/2 − s − 1/4 is a non-negative integer, the confluent hypergeometric functions can be expressed by the associated Laguerre polynomials (see 8.972 of [14], p. 1062):
(α + n + 1) 1 F1 (−n, α + 1; x), n! (α + 1) from which, together with the following important formula (see 7.414 of [14], p. 849) Lαn (x) =
∞ 0
x α e−x Lαn (x)Lαm (x) dx =
(n + α + 1) δnm , n!
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S.-H. Dong et al. / Physics Letters A 328 (2004) 299–305
we can finally obtain the normalized wave functions ρ
Ψ (ρ) = Nn ρ s e− 2 Ln with
2s−1/2
(ρ),
(18)
Nn =
2n! . (2s + n − 1/2)!
(19)
3. The construction of the ladder operators In this section we address the problem of finding the creation and annihilation operators for the radial wave functions (12) with the factorization method. As shown in our previous works [8–12], the ladder operators can be constructed directly from the wave functions without introducing any auxiliary variable, namely, we intend to find differential operators Lˆ± with the following property: Lˆ ± Rn (ρ) = l± Rn±1 (ρ).
(20)
Specifically, we look for operators of the form d + B± (ρ), Lˆ ± = A± (ρ) dρ where we stress that these operators only depend on the physical variable ρ. d To this end we start by establishing the action of the differential operator dρ on the wave functions (12) ρ d d 1 s 2s+1/2 Rn (ρ) = − Ln (ρ). Rn (ρ) + Nn ρ s e− 2 dρ ρ 2 dρ
(21)
(22)
One possible relation for the first derivative of the associated Laguerre functions is given in (see 8.971 of Ref. [14], p. 1062) d α L (x) = nLαn (x) − (n + α)Lαn−1 (x). dx n The substitution of this expression into (22) enables us to obtain the following relation: n+s 1 n + 2s − 1/2 Nn d − + Rn (ρ) = − Rn−1 (ρ). dρ ρ 2 ρ Nn−1 x
(23)
(24)
Making use of Eq. (19) and introducing the number operator nˆ with the property nR ˆ n (ρ) = nRn (ρ),
(25)
we can define the following operator ρ d + s + nˆ − , Lˆ − = −ρ dρ 2 with the following effect over the wave functions Lˆ − Rn (ρ) = l− Rn−1 (ρ), where l− =
n(n + 2s − 1/2).
As we can see, this operator annihilates the ground state R0 (ρ), as expected from a step-down operator.
(26)
(27)
(28)
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303
We now proceed to find the corresponding creation operator. Before proceeding to do so, we should make use of another relation between the associated Laguerre functions (see 8.971 of Ref. [15], p. 1062) d α L (x) = (n + 1)Lαn+1 (x) − (n + α + 1 − x)Lαn (x). dx n Substitution of this expression into Eq. (22) admits us to obtain n + s + 1/2 1 d n + 1 Nn + − Rn (ρ) = Rn+1 (ρ). dρ ρ 2 ρ Nn+1 x
(29)
(30)
Using Eq. (19) again, we can define the following operator: 1 ρ d + s + nˆ + − , Lˆ + = ρ dρ 2 2
(31)
satisfying the equation Lˆ + Rn (ρ) = l+ Rn+1 (ρ), with l+ =
(32)
(n + 1)(n + 2s + 1/2).
(33)
Therefore, it is shown that the wave functions can be directly obtained from the creation operator Lˆ + acting on the ground state Ψ0 (ρ), namely, Rn (ρ) = Nn Lˆ n+ R0 (ρ), with
Nn =
(2s − 1/2)! , n!(n + 2s − 1/2)!
(34) R0 (ρ) =
ρ 2 ρ s e− 2 . (2s − 1/2)!
(35)
We now study the algebra associated to the operators Kˆ + and Lˆ − . Based on the results (27), (28) and (32), (33) we can calculate the commutator [Lˆ − , Lˆ + ]:
Lˆ− , Lˆ + Rn (ρ) = 2l0 Rn (ρ), (36) where we have introduced the eigenvalue 3 l0 = n + s + . 4 We can thus define the operator 1 ˆ . L0 = nˆ + s + 4 The operators Lˆ± and Lˆ 0 thus satisfy the commutation relations
Lˆ 0 , Lˆ − = −Lˆ − , Lˆ 0 , Lˆ + = Lˆ+ Lˆ− , Lˆ + = 2Lˆ 0 ,
(37)
(38)
(39)
which correspond to an su(1, 1) algebra for the radial wave functions. The Casimir operator can be also expressed as
ˆ n (ρ) = Lˆ 0 Lˆ0 − 1 − Lˆ+ Lˆ − Rn (ρ) = s 2 − s − 3 Rn (ρ). CR (40) 2 16
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The Hamiltonian Hˆ can be expressed as 1 |n. Hˆ |n = 2Lˆ 0 |n = 2n + 2s + 2
(41)
It is shown that there are four series of irreducible unitary representations for the SU(1, 1) algebra except for the identity representation [16]. They are the representation D ± (j ) with a spectrum bounded below and above, respectively; the supplementary series DS (Q, q0 ) and the principle series DP (Q, q0 ). Since the eigenvalues have the ground state, the representation of the dynamical group SU(1, 1) belongs to D + (j ): I0 |j, ν = ν|j, ν,
1/2 I− |j, ν = (ν + j )(ν − j − 1) |j, ν − 1,
1/2 I+ |j, ν − 1 = (ν + j )(ν − j − 1) |j, ν,
ν = −j + n,
n = 0, 1, 2, . . .,
j < 0.
(42)
In comparison with Eqs. (27), (28), (35), (36) and (39), (40) we have j = −(s + 1/4), ν = n + s + 1/4, and Rn (ρ) = |j, ν. On the other hand, the following expressions in terms of the creation and annihilation operators Lˆ ± and Lˆ 0 can be obtained as ρ = 2Lˆ 0 − Lˆ − − Lˆ+ , 1 ˆ 1 d ˆ = ρ L+ − L− − . dρ 2 2
(43a) (43b)
The matrix elements of these two functions can be analytically obtained in terms of Eqs. (27), (37) and (40) as 1 δm,n − n(n + 2s − 1/2) δm,(n−1) Rm (r)r 2 Rn (r) dr = 2n + 2s + 2 0 − (n + 1)(n + 2s + 1/2) δm,(n+1) , ∞ d r d 1 Rm (ρ) ρ Rn (ρ) ≡ Rm (r) Rn (r) dr = (n + 1)(n + 2s + 1/2) δm,(n+1) dρ 2 dr 2
Rm (ρ)ρ Rn (ρ) ≡
∞
(44a)
0
1 1 n(n + 2s − 1/2) δm,(n−1) − δm,n , (44b) 2 4 where the integral range r ∈ [0, ∞). It is shown that this is a very simple method to calculate the matrix elements from these ladder operators. −
4. Concluding remarks In this Letter we have studied the eigenvalues and eigenfunctions for the ring-shaped non-spherical oscillator and then established the creation and annihilation operators directly from the eigenfunctions (18) with the factorization method. We have derived a realization of dynamic group only in terms of the physical variable ρ, without introducing any auxiliary variable. It is shown that these operators satisfy an SU(1, 1) dynamic group. The representation of the bound states of this quantum system is described by the representation D + (j ) with a spectrum d bound below. The matrix elements of the different functions ρ and ρ dρ are also analytically obtained from the ˆ ˆ ladder operators L± and L0 . This method can be generalized to other wave functions and represents a simple and elegant approach to obtain these matrix elements.
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Acknowledgements One of the authors (S.-H. Dong) would like to thank Prof. Chang-Yuan Chen for sending their work [14]. This work is partially supported by CONACyT, Mexico, under projects L007E and C086A.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13] [14] [15] [16]
A. Arima, F. Iachello, Ann. Phys. (New York) 99 (1974) 253. A. Perelomov, Generalized Coherent States and Their Applications, Springer, New York, 1985. R.G. Wybourne, Classical Groups for Physicists, Wiley, New York, 1974. I.L. Cooper, J. Phys. A: Math. Gen. 26 (1993) 1601. E. Schrödinger, Proc. R. Irish Acad. A 46 (9) (1940) 183. L. Infeld, T.E. Hull, Rev. Mod. Phys. 23 (1951) 21. G. Haidinger, N. Bessis, G. Bessis, J. Math. Phys. 15 (1974) 716. S.H. Dong, R. Lemus, A. Frank, Int. J. Quantum Chem. 86 (2002) 433; S.H. Dong, R. Lemus, Int. J. Quantum Chem. 86 (2002) 265. S.H. Dong, Can. J. Phys. 80 (2002) 129; S.H. Dong, Z. Phys. Chem. 216 (2002) 103. S.H. Dong, Z.Q. Ma, Am. J. Phys. 70 (2002) 520; S.H. Dong, Z.Q. Ma, Int. J. Mod. Phys. E 11 (2002) 155. S. Dong, S.H. Dong, Czech. J. Phys. 52 (2002) 753; S. Dong, S.H. Dong, Int. J. Mod. Phys. E 11 (2002) 265. S.H. Dong, Appl. Math. Lett. 16 (2) (2003) 199; S.H. Dong, Comput. Math. Appl. 47 (6–7) (2004) 1035; S.H. Dong, G.H. Sun, Y. Tang, Int. J. Mod. Phys. E 12 (2003) 809. M. Berrondo, A. Palma, J. Phys. A 13 (1980) 773. T.L. Cheng, C.Y. Chen, Acta Sin. Quant. Opt. 7 (2001) 67. I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series, and Products, fifth ed., Academic Press, New York, 1994. B.G. Adams, J. Cizek, J. Paldus, in: P.-O. Löwdin (Ed.), Advances in Quantum Chemistry, vol. 19, Academic Press, New York, 1987.
An algebraic approach to the ring-shaped non-spherical 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, D.F., Mexico b Instituto de Investigaciones en Matematicas Aplicadas y en Sistemas UNAM, A.P. 20-726, Del. Alvaro Obregón, 01000 México, D.F., Mexico
Received 1 June 2004; accepted 15 June 2004 Available online 23 June 2004 Communicated by R. Wu
Abstract The eigenfunctions and eigenvalues of the Schrödinger equation with a ring-shaped non-spherical oscillator are obtained. A realization of the ladder operators for the radial wave functions is studied. It is found that these operators satisfy the commutation relations of an SU(1, 1) group. The closed analytical expressions for the matrix elements of different functions ρ d with ρ = r 2 are evaluated. and ρ dρ 2004 Published by Elsevier B.V. PACS: 03.65.Fd; 03.65.Ge; 02.20.Qs Keywords: Ring-shaped non-spherical oscillator; Ladder operators; SU(1, 1) group; Matrix elements
1. Introduction It is well known that the algebraic method has been the subject of the interest in the wide variety of fields of physics. Systems displaying a dynamical symmetry can be treated by algebraic techniques [1–4]. With the factorization method [5–7], we have established the ladder operators of a quantum system with some important potentials such as the Morse potential, the Pöschl–Teller one, the pseudoharmonic one, the infinitely square-well one and other quantum systems [8–12]. From those ladder operators we can finally constitute a suitable algebra and simultaneously obtain the matrix elements for some related functions. It should be addressed that our approach is different from the traditional one, where an auxiliary variable was introduced [13], namely, we can construct the ladder operators only from the physical variable without introducing any auxiliary variable. Recently, the quantum system for the ring-shaped non-spherical oscillator has been studied [14]. To our knowledge, however, the hidden symmetry of this quantum system is a gap to be filled in, which is the main purpose of the present work. This Letter is organized as follows. In Section 2 we derive the eigenvalues and eigenfunctions for this system. Section 3 is devoted to the construction of the ladder operators directly from the radial wave function with the * Corresponding author.
E-mail addresses: [email protected] (S.-H. Dong), [email protected] (G.-H. Sun), [email protected] (M. Lozada-Cassou). 0375-9601/$ – see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.physleta.2004.06.037
300
S.-H. Dong et al. / Physics Letters A 328 (2004) 299–305
d factorization method. The matrix elements of the different functions ρ and ρ dρ with ρ = r 2 are analytically obtained from these operators. The concluding remarks are given in Section 4.
2. The eigenvalues and eigenfunctions In this section we are now studying the exact solutions, which are necessary to be used to construct the ladder operators and constitute a suitable algebra. Consider the Schrödinger equation with a potential V (r) that depends only on the distance r from the origin 1 ∂2 h¯ 2 1 ∂ 2 ∂ r + − (1) Ψ (r, ϕ) + V (r)Ψ (r, θ, ϕ) = EΨ (r, θ, ϕ), 2m r 2 ∂r ∂r r 2 ∂ϕ 2 where m is the mass of the particle. In the present work, the potential V (r) is taken as the harmonic potential in the presence of the dipole-like interaction, h¯ 2 α h¯ 2 β 1 + , V (x) = mω2 r 2 + 2 2 2 2m r 2m r sin2 θ
(2)
where the ω is the frequency and α and β are the dimensionless parameters. For simplicity, the natural units h¯ = m = ω = 1 are employed throughout this Letter if not explicitly stated otherwise. Owing to the symmetry of the potential and following the procedure used by Cheng and Chen in Ref. [14], we take the wave functions with the form R(r) Θ(θ )e±imϕ , m = 0, 1, 2, . . .. r Substitution of this into Eq. (1) allows us to obtain the angular wave function and the radial one as α+λ d 2 R(r) 2 + 2E − r − 2 R(r) = 0, dr 2 r 1 d dΘ(θ ) β + m2 Θ(θ ) = 0, sin θ + λ− sin θ dθ dθ sin2 θ Ψ (r, θ, ϕ) =
with the constant λ to be determined below. Introduce λ = ( + 1), ξ = β + m2 ,
x = cos(θ ).
Substitution of Eq. (5) into Eq. (4b) leads to d 2 Θ(x) dΘ(x) ξ2 + 1 − x2 − 2x ( + 1) − Θ(x) = 0, dx 2 dx 1 − x2
(3)
(4a) (4b)
(5)
(6)
whose normalized solution can be obtained as [14] Θ ξ (θ ) =
−ξ
2 (2 + 1) ( − ξ )! (−1)κ (2 − 2κ + 1) ξ (sin θ ) (cos θ ) −ξ −2κ , 2
( + ξ + 1) 2 κ!( − ξ − 2κ)! ( − κ + 1)
(7)
κ=0
with = ξ + k,
k = 0, 1, 2, . . ..
(8)
S.-H. Dong et al. / Physics Letters A 328 (2004) 299–305
Substitution of this into Eq. (4a) allows us to obtain d 2 R(r) L(L + 1) 2 + 2E − r − R(r) = 0, dr 2 r2 with 1 L≡ 2
2 2 1+4 α+ β +m +k β +m +k+1 −1 .
Defining the new variable ρ = r 2 , Eq. (9) can be rearranged as d2 1 L(L + 1) 1 d E R(ρ) − + R(ρ) + − R(ρ) = 0. dρ 2 2ρ dρ 4 4ρ 2 2ρ
301
(9)
(10)
(11)
From the behaviors of the wave functions at the origin and at infinity, we can take the following ansatz for the wave functions ρ
R(ρ) = ρ s e− 2 F (ρ),
(12)
with L+1 (13) , 2 where another solution s = (−L/2) is not an acceptable one in physics. Substitution of Eq. (12) into Eq. (11) allows us to obtain d2 d E 1 1 ρ 2 F (ρ) + 2s + − ρ (14) F (ρ) = 0, F (ρ) + −s − 2 dρ 2 4 dρ s=
whose solution is nothing but the confluent hypergeometric solutions 1 F1 (s − E/2 + 1/4, 2s + 1/2; ρ). One can finally obtain the eigenfunctions as ρ R(ρ) = Nρ s e− 2 1 F1 s − E2 + 14 , 2s + 12 ; ρ , (15) with a normalized factor N to be determined below. From consideration of the finiteness of the solutions, it is shown from Eq. (13) that the general quantum condition is E 1 s − + = −n, n = 0, 1, 2, . . . , (16) 2 4 from which we have 1 1 En = 2 n + s + (17) = + 2n + 2s, 4 2 which implies that the energy level is equidistant. Recall that when n = E/2 − s − 1/4 is a non-negative integer, the confluent hypergeometric functions can be expressed by the associated Laguerre polynomials (see 8.972 of [14], p. 1062):
(α + n + 1) 1 F1 (−n, α + 1; x), n! (α + 1) from which, together with the following important formula (see 7.414 of [14], p. 849) Lαn (x) =
∞ 0
x α e−x Lαn (x)Lαm (x) dx =
(n + α + 1) δnm , n!
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we can finally obtain the normalized wave functions ρ
Ψ (ρ) = Nn ρ s e− 2 Ln with
2s−1/2
(ρ),
(18)
Nn =
2n! . (2s + n − 1/2)!
(19)
3. The construction of the ladder operators In this section we address the problem of finding the creation and annihilation operators for the radial wave functions (12) with the factorization method. As shown in our previous works [8–12], the ladder operators can be constructed directly from the wave functions without introducing any auxiliary variable, namely, we intend to find differential operators Lˆ± with the following property: Lˆ ± Rn (ρ) = l± Rn±1 (ρ).
(20)
Specifically, we look for operators of the form d + B± (ρ), Lˆ ± = A± (ρ) dρ where we stress that these operators only depend on the physical variable ρ. d To this end we start by establishing the action of the differential operator dρ on the wave functions (12) ρ d d 1 s 2s+1/2 Rn (ρ) = − Ln (ρ). Rn (ρ) + Nn ρ s e− 2 dρ ρ 2 dρ
(21)
(22)
One possible relation for the first derivative of the associated Laguerre functions is given in (see 8.971 of Ref. [14], p. 1062) d α L (x) = nLαn (x) − (n + α)Lαn−1 (x). dx n The substitution of this expression into (22) enables us to obtain the following relation: n+s 1 n + 2s − 1/2 Nn d − + Rn (ρ) = − Rn−1 (ρ). dρ ρ 2 ρ Nn−1 x
(23)
(24)
Making use of Eq. (19) and introducing the number operator nˆ with the property nR ˆ n (ρ) = nRn (ρ),
(25)
we can define the following operator ρ d + s + nˆ − , Lˆ − = −ρ dρ 2 with the following effect over the wave functions Lˆ − Rn (ρ) = l− Rn−1 (ρ), where l− =
n(n + 2s − 1/2).
As we can see, this operator annihilates the ground state R0 (ρ), as expected from a step-down operator.
(26)
(27)
(28)
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We now proceed to find the corresponding creation operator. Before proceeding to do so, we should make use of another relation between the associated Laguerre functions (see 8.971 of Ref. [15], p. 1062) d α L (x) = (n + 1)Lαn+1 (x) − (n + α + 1 − x)Lαn (x). dx n Substitution of this expression into Eq. (22) admits us to obtain n + s + 1/2 1 d n + 1 Nn + − Rn (ρ) = Rn+1 (ρ). dρ ρ 2 ρ Nn+1 x
(29)
(30)
Using Eq. (19) again, we can define the following operator: 1 ρ d + s + nˆ + − , Lˆ + = ρ dρ 2 2
(31)
satisfying the equation Lˆ + Rn (ρ) = l+ Rn+1 (ρ), with l+ =
(32)
(n + 1)(n + 2s + 1/2).
(33)
Therefore, it is shown that the wave functions can be directly obtained from the creation operator Lˆ + acting on the ground state Ψ0 (ρ), namely, Rn (ρ) = Nn Lˆ n+ R0 (ρ), with
Nn =
(2s − 1/2)! , n!(n + 2s − 1/2)!
(34) R0 (ρ) =
ρ 2 ρ s e− 2 . (2s − 1/2)!
(35)
We now study the algebra associated to the operators Kˆ + and Lˆ − . Based on the results (27), (28) and (32), (33) we can calculate the commutator [Lˆ − , Lˆ + ]:
Lˆ− , Lˆ + Rn (ρ) = 2l0 Rn (ρ), (36) where we have introduced the eigenvalue 3 l0 = n + s + . 4 We can thus define the operator 1 ˆ . L0 = nˆ + s + 4 The operators Lˆ± and Lˆ 0 thus satisfy the commutation relations
Lˆ 0 , Lˆ − = −Lˆ − , Lˆ 0 , Lˆ + = Lˆ+ Lˆ− , Lˆ + = 2Lˆ 0 ,
(37)
(38)
(39)
which correspond to an su(1, 1) algebra for the radial wave functions. The Casimir operator can be also expressed as
ˆ n (ρ) = Lˆ 0 Lˆ0 − 1 − Lˆ+ Lˆ − Rn (ρ) = s 2 − s − 3 Rn (ρ). CR (40) 2 16
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The Hamiltonian Hˆ can be expressed as 1 |n. Hˆ |n = 2Lˆ 0 |n = 2n + 2s + 2
(41)
It is shown that there are four series of irreducible unitary representations for the SU(1, 1) algebra except for the identity representation [16]. They are the representation D ± (j ) with a spectrum bounded below and above, respectively; the supplementary series DS (Q, q0 ) and the principle series DP (Q, q0 ). Since the eigenvalues have the ground state, the representation of the dynamical group SU(1, 1) belongs to D + (j ): I0 |j, ν = ν|j, ν,
1/2 I− |j, ν = (ν + j )(ν − j − 1) |j, ν − 1,
1/2 I+ |j, ν − 1 = (ν + j )(ν − j − 1) |j, ν,
ν = −j + n,
n = 0, 1, 2, . . .,
j < 0.
(42)
In comparison with Eqs. (27), (28), (35), (36) and (39), (40) we have j = −(s + 1/4), ν = n + s + 1/4, and Rn (ρ) = |j, ν. On the other hand, the following expressions in terms of the creation and annihilation operators Lˆ ± and Lˆ 0 can be obtained as ρ = 2Lˆ 0 − Lˆ − − Lˆ+ , 1 ˆ 1 d ˆ = ρ L+ − L− − . dρ 2 2
(43a) (43b)
The matrix elements of these two functions can be analytically obtained in terms of Eqs. (27), (37) and (40) as 1 δm,n − n(n + 2s − 1/2) δm,(n−1) Rm (r)r 2 Rn (r) dr = 2n + 2s + 2 0 − (n + 1)(n + 2s + 1/2) δm,(n+1) , ∞ d r d 1 Rm (ρ) ρ Rn (ρ) ≡ Rm (r) Rn (r) dr = (n + 1)(n + 2s + 1/2) δm,(n+1) dρ 2 dr 2
Rm (ρ)ρ Rn (ρ) ≡
∞
(44a)
0
1 1 n(n + 2s − 1/2) δm,(n−1) − δm,n , (44b) 2 4 where the integral range r ∈ [0, ∞). It is shown that this is a very simple method to calculate the matrix elements from these ladder operators. −
4. Concluding remarks In this Letter we have studied the eigenvalues and eigenfunctions for the ring-shaped non-spherical oscillator and then established the creation and annihilation operators directly from the eigenfunctions (18) with the factorization method. We have derived a realization of dynamic group only in terms of the physical variable ρ, without introducing any auxiliary variable. It is shown that these operators satisfy an SU(1, 1) dynamic group. The representation of the bound states of this quantum system is described by the representation D + (j ) with a spectrum d bound below. The matrix elements of the different functions ρ and ρ dρ are also analytically obtained from the ˆ ˆ ladder operators L± and L0 . This method can be generalized to other wave functions and represents a simple and elegant approach to obtain these matrix elements.
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Acknowledgements One of the authors (S.-H. Dong) would like to thank Prof. Chang-Yuan Chen for sending their work [14]. This work is partially supported by CONACyT, Mexico, under projects L007E and C086A.
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