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The series solutions of the non-relativistic equation with the Morse potential
Physics Letters A 314 (2003) 261–266 www.elsevier.com/locate/pla
The series solutions of the non-relativistic equation with the Morse potential Shi-Hai Dong a,∗ , Guo-Hua Sun b 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 27 May 2003; received in revised form 3 June 2003; accepted 4 June 2003 Communicated by R. Wu
Abstract In this Letter the analytical solutions of the two-dimensional Schrödinger equation with the Morse potential are obtained by the series expansion method. We then generalize this method to the D-dimensional Schrödinger equation case. The studied Morse potential itself is expanded in the series about the origin. 2003 Elsevier B.V. All rights reserved. PACS: 03.65.Pm; 03.65.-w Keywords: Schrödinger equation; Morse potential; Two dimensions; Series method
1. Introduction The study of exactly solvable problems has attracted attention since the early development of quantum mechanics. A fundamental method in this direction was called the series expansion method, which has been widely used to study some quantum problems, especially to investigate the exact solutions of the hydrogen atom in the classical textbooks [1–3]. Because of its importance in the field of molecular physics [4–6], the Morse potential has been the subject of many studies since its introduction by Morse in 1929 [7]. This potential is solvable, hence the interest is to deal with it using different approaches such as the factorization method [8–11], the supersymmetry approach [12,13], the Green’s function approach [14], etc. It should be addressed that the almost all works mentioned above have been studied in one dimension. Recently, the approximate method has been used to obtain the solutions of Schrödinger equation with the Morse potential in three dimensions [15,16], where an identity has been introduced to incorporate
* Corresponding author.
E-mail addresses: [email protected] (S.-H. Dong), [email protected] (G.-H. Sun). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0375-9601(03)00895-8
262
S.-H. Dong, G.-H. Sun / Physics Letters A 314 (2003) 261–266
the effect of the centrifugal barrier. That is to say, in their calculation, the following identity was used 1 1 1 = [1 + x]−2 = 2 C0 + C1 exp(−αx) + C2 exp(−2αx) . r 2 r02 r0
(1)
Thus they obtained the relations between the coefficients Ci (i = 0, 1, 2) and the parameter α and presented the solutions. On the other hand, Rouse [17] has carried out the analytic solutions of the three-dimensional Schrödinger equation with the Morse potential by the series approach, but the derivation is flawed, namely, the term involving the 2b2φ appearing in Eq. (10) of Ref. [17] should be removed. The successive calculations related with this important equation are inevitably incorrect. The main purpose of this Letter is to carry out the analytical solutions of the two-dimensional Schrödinger equation with the Morse potential applicable to lower-dimensional field theory and condensed matter physics and then generalize the Schrödinger equation with the Morse potential to the Ddimensional case with the interest in higher-dimensional field theory. This Letter is organized as follows. In Section 2, we obtain the analytical solutions of the Schrödinger equation with the Morse potential in two dimensions by the series expansion method. Section 3 is devoted to generalizing the Schrödinger equation with the Morse potential to the D-dimensional case. Finally, we present the conclusions in Section 4.
2. The Morse potential in two dimensions Choosing the separated atoms limit as the zero of energy, the Morse potential has the following form [7] V (r) = V0 e−2βr − 2e−βr ,
(2)
where V0 > 0 corresponds to its depth of the potential well, β is related with the range of the potential, and r gives the relative distance from the equilibrium position of the atoms. We first study the two-dimensional Schrödinger equation with the Morse potential. For simplicity, the natural units h¯ = m = 1 are employed throughout this Letter if not explicitly stated otherwise. Consider the Schrödinger equation with a potential V (r) that depends only on the distance r from the origin 1 ∂2 1 1 ∂ ∂ r + 2 2 Ψ (r, ϕ) + V (r)Ψ (r, ϕ) = EΨ (r, ϕ). − (3) 2 r ∂r ∂r r ∂ϕ Owing to the symmetry of the potential, we take the wave functions with the form Ψ (r, ϕ) = r −1/2 Rm (r)e±imϕ ,
m = 0, 1, 2, . . . ,
where the radial wave function Rm (r) satisfies the following equation m2 − 1/4 d 2 Rm (r) Rm (r) = 0. + 2 E − V (r) − dr 2 r2
(4)
(5)
In this Letter, we only study the bound states of this quantum system, namely, say E < 0. Following Ref. [17], we take the ansatz for the wave function with the form Rm (r) = r σ e−br F (r), where we have introduced the dimensionless parameter √ b = −2E.
(6)
(7)
S.-H. Dong, G.-H. Sun / Physics Letters A 314 (2003) 261–266
263
Substitution of Eq. (6) into (5) leads to rσ
dF d 2F + 2σ r σ −1 − 2br σ 2 dr dr + σ (σ − 1)r σ −2 − 2bσ r σ −1 + b 2 + 2E − 2V r σ − m2 − 1/4 r σ −2 F = 0,
(8)
which can be re-arranged as 2 d F dF dF − 2bσ F r + − 2V F r 2 = 0. σ (σ − 1) − m2 − 1/4 F + 2σ − 2b dr dr 2 dr
(9)
From the behavior of the wave functions at the origin, it is shown from Eqs. (8) and (9) that 1 σ =m+ , (10) 2 where another solution σ = 1/2 − m is not acceptable in physics. Before further proceeding to do so, it is necessary to obtain the series expansion of the Morse potential about the origin. Expansion of the exponential terms with the r allows us to obtain V (r) = 2V0
∞
cl r l ,
(11)
l=0
where we have used an important formula ∞ ∞ 2 βl l r =2 e−2βr − 2e−βr = e−βr − 1 − 1 = 2 (−1)l 2l−1 − 1 cl r l . l! l=0
(12)
l=0
We now take the standard series for F (r) with the form F (r) =
∞
βκ r κ ,
β0 = 0.
(13)
κ=0
Substituting this, together with Eq. (12), into Eq. (9) and setting the coefficients of the powers of r n (n = l + κ + 2) to zero, one can obtain ∞ ∞ ∞ ∞ κ(2m + κ) βκ r κ − 2b(κ + m + 1/2)βκ r κ+1 − 4V0 cl βκ r l+κ+2 = 0, κ=0
κ=0
(14)
κ=0 l=0
from which we have β1 = bβ0,
β2 =
b(2m + 3)β1 + 4V0 β0 c0 , 4 + 4m
β3 =
b(2m + 5)β2 + 4V0 (β1 c0 + β0 c1 ) . 9 + 6m
(15)
Similarly continuing to use Eq. (14), we can finally obtain the expansion coefficients βn as βn =
b(2m + 2n − 1)βn−1 + 4V0 Dn,κ , n(2m + n)
n 2,
(16)
with Dn,κ =
n−2=N l=0,l2
cl βN−l ,
(17)
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S.-H. Dong, G.-H. Sun / Physics Letters A 314 (2003) 261–266
which was defined by Rouse [17]. As shown in β3 , the sum of the terms β1 c0 + β0 c1 is from the double sum appearing in Eq. (14). Hence, when n = l + κ + 2, the number of the product terms are from the combinations of l and κ such that l + κ = n − 2. We thus obtain the analytical solutions of the Schrödinger equation with the Morse potential in two dimensions as Rm (r) = r 1/2+m e−br
∞
βn r n ,
β0 = 0,
(18)
n=0
where βn is given in Eq. (16). It should be addressed that the cl can be easily obtained from Eq. (12). For example, we can obtain c0 = −1/2, c1 = 0, c2 = 12 β 2 , c3 = − 12 β 3 , etc. On the other hand, as addressed in [17], for the given Morse potential, we can always find the suitable eigenvalue to make the wave function Rm (r) convergent in the case of r → ∞.
3. The Morse potential in D dimensions In this section we generalize the Schrödinger equation with the Morse potential to the arbitrary dimension D. For simplicity, how to derive the radial equation of the Schrödinger equation in D dimensions is ignored since the detailed information can be found in Refs. [18–22]. The D-dimensional Schrödinger equation with a spherically symmetric potential V (r) can be written as 1 2 Ψ (r) = E − V (r) Ψ (r). − ∇D (19) 2 Following Refs. [18–22], the wave function Ψ (r) with a given angular momentum l can be now decomposed as a product of the radial function Rl (r) and the generalized spherical harmonics YllD−2 ...l1 (ˆx) Ψ (r) = r −(D−1)/2Rl (r)YllD−2 ...l1 (ˆx). Substitution Eq. (20) into Eq. (19) allows us to obtain radial Schrödinger equation 2 d l(l + D − 2) + (D − 1)(D − 3)/4 − Rl (r) = −2 E − V (r) Rl (r), 2 2 dr r which can be re-arranged as 2 d λ2 − 1/4 − Rλ (r) = −2 E − V (r) Rλ (r), dr 2 r2
(20)
(21)
(22)
where λ ≡ (l − 1 + D/2), which implies that λ depends on the angular momentum l and the spatial dimension D. Similarly, we can take the ansatz for the wave function with the form √ Rλ (r) = r η e−br G(r), b = −2E, E < 0.
(23)
(24)
Substitution of Eq. (24) into (22) leads to rη
dG d 2G + 2ηr η−1 − 2br η dr 2 dr η−2 η−1 + η(η − 1)r − 2bηr + b 2 + 2E − 2V r η − λ2 − 1/4 r η−2 G = 0,
(25)
S.-H. Dong, G.-H. Sun / Physics Letters A 314 (2003) 261–266
265
which is re-arranged as 2 2 d G dG dG − 2bηG r + − 2V G r 2 = 0. η(η − 1) − λ − 1/4 G + 2η − 2b dr dr 2 dr
(26)
From the behavior of the wave functions at the origin, it is shown from Eqs. (25) and (26) that D−1 1 =l+ , (27) 2 2 where another solution η = 1/2 − λ is ignored in physics. Likewise, it is necessary to obtain expansion of the Morse potential about the origin as given in Eq. (12). We take the standard series for G(r) with the form η=λ+
G(r) =
∞
γk r k ,
γ0 = 0.
(28)
k=0
Substituting this, together with Eq. (12), into Eq. (26) and setting the coefficients of the powers of r n (n = l + k + 2) to zero, one can obtain ∞ ∞ ∞ ∞ k k+1 k(2λ + k) γk r − 2b(k + λ + 1/2)γk r − 4V0 cl γk r l+k+2 = 0, k=0
k=0
(29)
k=0 l=0
from which we have γ1 = bγ0 ,
γ2 =
b(2m + 3)γ1 + 4V0 γ0 c0 , 4 + 4m
γ3 =
b(2m + 5)γ2 + 4V0 (γ1 c0 + γ0 c1 ) . 9 + 6m
(30)
Similarly continuing to use Eq. (29), we can finally obtain the expansion coefficients γn as γn =
b(2m + 2n − 1)γn−1 + 4V0 Sn,k , n(2m + n)
(31)
with Sn,k =
n−2=N
cl γN−l .
(32)
l=0,l2
Accordingly, we can finally obtain the analytical solutions of the Schrödinger equation with the Morse potential in two dimensions as Rλ (r) = r 1/2+λ e−br
∞
γn r n ,
γ0 = 0.
(33)
n=0
Likewise, for a given Morse potential, we can always find the suitable eigenvalue to make the wave functions Rλ (r) convergent when r → ∞.
4. Conclusions In this Letter we have carried out the analytical solutions of the Schrödinger equation with the Morse potential in two dimensions and arbitrary dimension D with the recent interest in lower-dimensional and higher-dimensional field theory by the series expansion method.
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S.-H. Dong, G.-H. Sun / Physics Letters A 314 (2003) 261–266
Acknowledgements One of the authors (D.S.-H.) thanks Prof. M. Lozada-Cassou for the hospitality shown at IMP. This work is partly supported by CONACyT, Mexico, under projects L007E and C086A. This work was started at Instituto de Ciencias Nucleares of UNAM.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
L.I. Schiff, Quantum Mechanics, Third Edition, McGraw-Hill, New York, 1955. P.A. Dirac, The Principles of Quantum Mechanics, Fourth Edition, Oxford Univ. Press, 1958. L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory, Third Edition, Pergamon, 1977. P. Jensen, Mol. Phys. 98 (2000) 1253. M.S. Child, L. Halonen, Adv. Chem. Phys. 62 (1984) 1. L. Pauling, E.B. Wilson Jr., Introduction to Quantum Mechanics with Applications to Chemistry, Dover, New York, 1985. P.M. Mörse, Phys. Rev. 34 (1929) 57. J.W. Dabrowska, A. Khare, U.P. Sukhatme, J. Phys. A: Math. Gen. 21 (1988) L195. A.B. Balentekin, Phys. Rev. A 57 (1998) 4188. S.H. Dong, R. Lemus, A. Frank, Int. J. Quantum Chem. 86 (2002) 433, and references therein. F. Fernández, A. Castro, Algebraic Methods in Quantum Chemistry and Physics, in: Mathematical Chemistry Series, CRC Press, 1996. G. Lévai, J. Phys. A: Math. Gen. 27 (1994) 3809. M.G. Benedict, B. Molnár, Phys. Rev. A 60 (1999) R1737. L. Chetouani, L. Guechi, T.F. Hammann, Helv. Phys. Acta 65 (1992) 1069; L. Chetouani, L. Guechi, T.F. Hammann, Czech. J. Phys. 43 (1993) 13. S. Flügge, P. Walger, A. Weiguni, J. Mol. Spectrosc. 23 (1967) 243. B. Talukdar, M. Chatterji, P. Banerjee, Pramana 13 (1979) 15. C.A. Rouse, Phys. Rev. A 36 (1987) 1. A. Erdelyi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953, p. 232. J.D. Louck, J. Mol. Spectrosc. 4 (1960) 298; A. Chatterjee, Phys. Rep. 186 (1990) 249. S.H. Dong, Z.Q. Ma, Phys. Rev. A 65 (2002) 042717; S.H. Dong, Found. Phys. Lett. 15 (2002) 385. S.H. Dong, J. Phys. A: Math. Gen. 36 (2003) 4977. S.H. Dong, X.Y. Gu, Z.Q. Ma, Int. J. Mod. Phys. E, in press.
The series solutions of the non-relativistic equation with the Morse potential Shi-Hai Dong a,∗ , Guo-Hua Sun b 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 27 May 2003; received in revised form 3 June 2003; accepted 4 June 2003 Communicated by R. Wu
Abstract In this Letter the analytical solutions of the two-dimensional Schrödinger equation with the Morse potential are obtained by the series expansion method. We then generalize this method to the D-dimensional Schrödinger equation case. The studied Morse potential itself is expanded in the series about the origin. 2003 Elsevier B.V. All rights reserved. PACS: 03.65.Pm; 03.65.-w Keywords: Schrödinger equation; Morse potential; Two dimensions; Series method
1. Introduction The study of exactly solvable problems has attracted attention since the early development of quantum mechanics. A fundamental method in this direction was called the series expansion method, which has been widely used to study some quantum problems, especially to investigate the exact solutions of the hydrogen atom in the classical textbooks [1–3]. Because of its importance in the field of molecular physics [4–6], the Morse potential has been the subject of many studies since its introduction by Morse in 1929 [7]. This potential is solvable, hence the interest is to deal with it using different approaches such as the factorization method [8–11], the supersymmetry approach [12,13], the Green’s function approach [14], etc. It should be addressed that the almost all works mentioned above have been studied in one dimension. Recently, the approximate method has been used to obtain the solutions of Schrödinger equation with the Morse potential in three dimensions [15,16], where an identity has been introduced to incorporate
* Corresponding author.
E-mail addresses: [email protected] (S.-H. Dong), [email protected] (G.-H. Sun). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0375-9601(03)00895-8
262
S.-H. Dong, G.-H. Sun / Physics Letters A 314 (2003) 261–266
the effect of the centrifugal barrier. That is to say, in their calculation, the following identity was used 1 1 1 = [1 + x]−2 = 2 C0 + C1 exp(−αx) + C2 exp(−2αx) . r 2 r02 r0
(1)
Thus they obtained the relations between the coefficients Ci (i = 0, 1, 2) and the parameter α and presented the solutions. On the other hand, Rouse [17] has carried out the analytic solutions of the three-dimensional Schrödinger equation with the Morse potential by the series approach, but the derivation is flawed, namely, the term involving the 2b2φ appearing in Eq. (10) of Ref. [17] should be removed. The successive calculations related with this important equation are inevitably incorrect. The main purpose of this Letter is to carry out the analytical solutions of the two-dimensional Schrödinger equation with the Morse potential applicable to lower-dimensional field theory and condensed matter physics and then generalize the Schrödinger equation with the Morse potential to the Ddimensional case with the interest in higher-dimensional field theory. This Letter is organized as follows. In Section 2, we obtain the analytical solutions of the Schrödinger equation with the Morse potential in two dimensions by the series expansion method. Section 3 is devoted to generalizing the Schrödinger equation with the Morse potential to the D-dimensional case. Finally, we present the conclusions in Section 4.
2. The Morse potential in two dimensions Choosing the separated atoms limit as the zero of energy, the Morse potential has the following form [7] V (r) = V0 e−2βr − 2e−βr ,
(2)
where V0 > 0 corresponds to its depth of the potential well, β is related with the range of the potential, and r gives the relative distance from the equilibrium position of the atoms. We first study the two-dimensional Schrödinger equation with the Morse potential. For simplicity, the natural units h¯ = m = 1 are employed throughout this Letter if not explicitly stated otherwise. Consider the Schrödinger equation with a potential V (r) that depends only on the distance r from the origin 1 ∂2 1 1 ∂ ∂ r + 2 2 Ψ (r, ϕ) + V (r)Ψ (r, ϕ) = EΨ (r, ϕ). − (3) 2 r ∂r ∂r r ∂ϕ Owing to the symmetry of the potential, we take the wave functions with the form Ψ (r, ϕ) = r −1/2 Rm (r)e±imϕ ,
m = 0, 1, 2, . . . ,
where the radial wave function Rm (r) satisfies the following equation m2 − 1/4 d 2 Rm (r) Rm (r) = 0. + 2 E − V (r) − dr 2 r2
(4)
(5)
In this Letter, we only study the bound states of this quantum system, namely, say E < 0. Following Ref. [17], we take the ansatz for the wave function with the form Rm (r) = r σ e−br F (r), where we have introduced the dimensionless parameter √ b = −2E.
(6)
(7)
S.-H. Dong, G.-H. Sun / Physics Letters A 314 (2003) 261–266
263
Substitution of Eq. (6) into (5) leads to rσ
dF d 2F + 2σ r σ −1 − 2br σ 2 dr dr + σ (σ − 1)r σ −2 − 2bσ r σ −1 + b 2 + 2E − 2V r σ − m2 − 1/4 r σ −2 F = 0,
(8)
which can be re-arranged as 2 d F dF dF − 2bσ F r + − 2V F r 2 = 0. σ (σ − 1) − m2 − 1/4 F + 2σ − 2b dr dr 2 dr
(9)
From the behavior of the wave functions at the origin, it is shown from Eqs. (8) and (9) that 1 σ =m+ , (10) 2 where another solution σ = 1/2 − m is not acceptable in physics. Before further proceeding to do so, it is necessary to obtain the series expansion of the Morse potential about the origin. Expansion of the exponential terms with the r allows us to obtain V (r) = 2V0
∞
cl r l ,
(11)
l=0
where we have used an important formula ∞ ∞ 2 βl l r =2 e−2βr − 2e−βr = e−βr − 1 − 1 = 2 (−1)l 2l−1 − 1 cl r l . l! l=0
(12)
l=0
We now take the standard series for F (r) with the form F (r) =
∞
βκ r κ ,
β0 = 0.
(13)
κ=0
Substituting this, together with Eq. (12), into Eq. (9) and setting the coefficients of the powers of r n (n = l + κ + 2) to zero, one can obtain ∞ ∞ ∞ ∞ κ(2m + κ) βκ r κ − 2b(κ + m + 1/2)βκ r κ+1 − 4V0 cl βκ r l+κ+2 = 0, κ=0
κ=0
(14)
κ=0 l=0
from which we have β1 = bβ0,
β2 =
b(2m + 3)β1 + 4V0 β0 c0 , 4 + 4m
β3 =
b(2m + 5)β2 + 4V0 (β1 c0 + β0 c1 ) . 9 + 6m
(15)
Similarly continuing to use Eq. (14), we can finally obtain the expansion coefficients βn as βn =
b(2m + 2n − 1)βn−1 + 4V0 Dn,κ , n(2m + n)
n 2,
(16)
with Dn,κ =
n−2=N l=0,l2
cl βN−l ,
(17)
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S.-H. Dong, G.-H. Sun / Physics Letters A 314 (2003) 261–266
which was defined by Rouse [17]. As shown in β3 , the sum of the terms β1 c0 + β0 c1 is from the double sum appearing in Eq. (14). Hence, when n = l + κ + 2, the number of the product terms are from the combinations of l and κ such that l + κ = n − 2. We thus obtain the analytical solutions of the Schrödinger equation with the Morse potential in two dimensions as Rm (r) = r 1/2+m e−br
∞
βn r n ,
β0 = 0,
(18)
n=0
where βn is given in Eq. (16). It should be addressed that the cl can be easily obtained from Eq. (12). For example, we can obtain c0 = −1/2, c1 = 0, c2 = 12 β 2 , c3 = − 12 β 3 , etc. On the other hand, as addressed in [17], for the given Morse potential, we can always find the suitable eigenvalue to make the wave function Rm (r) convergent in the case of r → ∞.
3. The Morse potential in D dimensions In this section we generalize the Schrödinger equation with the Morse potential to the arbitrary dimension D. For simplicity, how to derive the radial equation of the Schrödinger equation in D dimensions is ignored since the detailed information can be found in Refs. [18–22]. The D-dimensional Schrödinger equation with a spherically symmetric potential V (r) can be written as 1 2 Ψ (r) = E − V (r) Ψ (r). − ∇D (19) 2 Following Refs. [18–22], the wave function Ψ (r) with a given angular momentum l can be now decomposed as a product of the radial function Rl (r) and the generalized spherical harmonics YllD−2 ...l1 (ˆx) Ψ (r) = r −(D−1)/2Rl (r)YllD−2 ...l1 (ˆx). Substitution Eq. (20) into Eq. (19) allows us to obtain radial Schrödinger equation 2 d l(l + D − 2) + (D − 1)(D − 3)/4 − Rl (r) = −2 E − V (r) Rl (r), 2 2 dr r which can be re-arranged as 2 d λ2 − 1/4 − Rλ (r) = −2 E − V (r) Rλ (r), dr 2 r2
(20)
(21)
(22)
where λ ≡ (l − 1 + D/2), which implies that λ depends on the angular momentum l and the spatial dimension D. Similarly, we can take the ansatz for the wave function with the form √ Rλ (r) = r η e−br G(r), b = −2E, E < 0.
(23)
(24)
Substitution of Eq. (24) into (22) leads to rη
dG d 2G + 2ηr η−1 − 2br η dr 2 dr η−2 η−1 + η(η − 1)r − 2bηr + b 2 + 2E − 2V r η − λ2 − 1/4 r η−2 G = 0,
(25)
S.-H. Dong, G.-H. Sun / Physics Letters A 314 (2003) 261–266
265
which is re-arranged as 2 2 d G dG dG − 2bηG r + − 2V G r 2 = 0. η(η − 1) − λ − 1/4 G + 2η − 2b dr dr 2 dr
(26)
From the behavior of the wave functions at the origin, it is shown from Eqs. (25) and (26) that D−1 1 =l+ , (27) 2 2 where another solution η = 1/2 − λ is ignored in physics. Likewise, it is necessary to obtain expansion of the Morse potential about the origin as given in Eq. (12). We take the standard series for G(r) with the form η=λ+
G(r) =
∞
γk r k ,
γ0 = 0.
(28)
k=0
Substituting this, together with Eq. (12), into Eq. (26) and setting the coefficients of the powers of r n (n = l + k + 2) to zero, one can obtain ∞ ∞ ∞ ∞ k k+1 k(2λ + k) γk r − 2b(k + λ + 1/2)γk r − 4V0 cl γk r l+k+2 = 0, k=0
k=0
(29)
k=0 l=0
from which we have γ1 = bγ0 ,
γ2 =
b(2m + 3)γ1 + 4V0 γ0 c0 , 4 + 4m
γ3 =
b(2m + 5)γ2 + 4V0 (γ1 c0 + γ0 c1 ) . 9 + 6m
(30)
Similarly continuing to use Eq. (29), we can finally obtain the expansion coefficients γn as γn =
b(2m + 2n − 1)γn−1 + 4V0 Sn,k , n(2m + n)
(31)
with Sn,k =
n−2=N
cl γN−l .
(32)
l=0,l2
Accordingly, we can finally obtain the analytical solutions of the Schrödinger equation with the Morse potential in two dimensions as Rλ (r) = r 1/2+λ e−br
∞
γn r n ,
γ0 = 0.
(33)
n=0
Likewise, for a given Morse potential, we can always find the suitable eigenvalue to make the wave functions Rλ (r) convergent when r → ∞.
4. Conclusions In this Letter we have carried out the analytical solutions of the Schrödinger equation with the Morse potential in two dimensions and arbitrary dimension D with the recent interest in lower-dimensional and higher-dimensional field theory by the series expansion method.
266
S.-H. Dong, G.-H. Sun / Physics Letters A 314 (2003) 261–266
Acknowledgements One of the authors (D.S.-H.) thanks Prof. M. Lozada-Cassou for the hospitality shown at IMP. This work is partly supported by CONACyT, Mexico, under projects L007E and C086A. This work was started at Instituto de Ciencias Nucleares of UNAM.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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