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The exact solutions of the Schrödinger equation with the Morse potential via Laplace transforms
Physics Letters A 326 (2004) 55–57 www.elsevier.com/locate/pla
The exact solutions of the Schrödinger equation with the Morse potential via Laplace transforms Gang Chen Department of Physics, Shaoxing College of Arts and Sciences, Shaoxing 312000, PR China Received 23 March 2004; received in revised form 13 April 2004; accepted 14 April 2004 Available online 28 April 2004 Communicated by R. Wu
Abstract In this Letter, we reduce the second-order differential equation about the one-dimensional Schrödinger equation with the Morse potential reduced to the first-order differential equation in terms of Laplace transforms and then obtain the exact bound state solutions. 2004 Elsevier B.V. All rights reserved. PACS: 03.65.Ge Keywords: Bound state; Morse potential; Schrödinger equation; Laplace transforms
1. Introduction Since the appearance of the Schrödinger equation in quantum mechanics, there have been continual researches for solving the Schrödinger equation with any potential by using different methods because of the importance of such solutions in many branches of physics, especially molecules, atoms, nuclei, etc. All kinds of methods such as the factorization [1], the path integral [2], the group theoretical (algebraic method) [3], the 1/N expansion [4], the analytic continuation [5], the eigenvalue moment method [6], the power series expansion [7] and the supersymmetric
E-mail address: [email protected] (G. Chen). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.04.029
quantum mechanics [8] could be enumerated amongst other methods of the solutions of the wave equations. Laplace transforms, which are integral transforms, are comprehensively useful in physics and engineering [9]. Such techniques were used by Schrödinger in his first paper on the quantum mechanical hydrogen atom [10] and then by Swainson et al. on the recurrence relations of radial wave functions for hydrogen atom [11]. In this Letter, we will employ Laplace transforms to solve the one-dimensional Schrödinger equation with the Morse potential [12], which has played an important role in many different fields of the physics such as molecular physics, solid state physics and chemical physics, etc. This potential has been studied by many different approaches such as the standard confluent hypergeometric functions [13], the algebraic method [14], the supersymmetric method
56
G. Chen / Physics Letters A 326 (2004) 55–57
[15], the coherent states [16], the controllability [17], the series solutions of Morse potential with the mass distribution [18], etc. Fortunately, the second-order differential equation can be reduced to the first-order differential equation and therefore we may directly make use of integral to get the exact bound state solutions.
2. Bound state solutions The one-dimensional Schrödinger equation for any potential is given by [19] h¯ 2 d 2 − (1) + V (x) ψ(x) = Eψ(x). 2m dx 2 In the case of the Morse potential, Eq. (1) turns into 2 d 2mDe −2ax 4mDe −ax 2mE − e + e + ψ(x) dx 2 h¯ 2 h¯ 2 h¯ 2 = 0. (2)
where N is a constant. Noting that (1−
1 k−(2β+1) ) p+ 12
is a multi-valued function and the wave-functions are required to be single-valued, we must take k − (2β + 1) = 2n,
n = 0, 1, 2, 3, . . ..
(11)
Applying a simple series expansion to Eq. (10) yields F (p) = N
n (−1)j n!(p + 12 )−(2β+j +1) , (n − j )!j !
(12)
j =0
Put y = ke
∞ Using Laplace transforms L[g(t)] = 0 g(t)e−pt dt = G(p) [9] yields k 1 d F (p) + (2β + 1)p − F (p) = 0. p2 − 4 dp 2 (9) Eq. (9) is a first-order differential equation and so we may directly make use of integral to get k−(2β+1) 1 −(2β+1) 1 , F (p) = N P + 1− 2 p + 12 (10)
−ax
β2 = −
√ 2 2mDe , k= h¯ a
2mE
, h¯ 2 Eq. (2) becomes 2 y2 k d 2 d 2 − + y − β ψ(y) = 0. y +y dy 4 2 dy 2
(3) (4)
(5)
(6)
with A is a constant, and then inserting into Eq. (5) leads to equation y2 d2 d − y 2 2 + (2A + 1)y dy dy 4 k + y + A2 − β 2 f (y) = 0. (7) 2 Putting A = −β (the case A = β is not be acceptable in quantum mechanics because ψ(y) is finite when y → ∞) yields y k d2 d − + f (y) = 0. y 2 − (2β − 1) (8) dy dy 4 2
n
(−1)j n!(2β + 1) yj , (n − j )!j !(2β + j + 1) j =0 (13) where N is a constant. Comparing Eq. (13) with the series expansion of confluent hypergeometric functions 2β −y
f (y) = Ny e
Letting ψ(y) = y A f (y),
where N is a constant. Using a simple extension of inverse Laplace transforms [9] we can immediately deduce that
F (−n, γ , y) =
n j =0
(−1)j n!(γ ) yj (n − j )!j !(γ + j )
(14)
yields f (y) = Ny 2β e−y/2F (−n, 2β + 1, y).
(15)
Inserting Eq. (14) into Eq. (6) yields ψ(y) = Ny β e−y/2F (−n, 2β + 1, y).
(16)
On account of the relation between confluent hypergeometric functions and generalized Laguerre polynomials F (−m, u + 1, z) =
m!(u + 1) u L (z), (u + m + 1) m
(17)
G. Chen / Physics Letters A 326 (2004) 55–57
Eq. (16) is rewritten as ψ(y) = Nn y 2 −(n+ 2 ) e−y/2 Lnk−2n−1 (y), k
1
(18)
where Nn is a constant. Inserting ∞ 1(18)∗ into normalized condition of wave functions 0 ay ψ (y)ψ(y) dy = 1 yields an!(k − 2n − 1) 1/2 Nn = (19) . (k − n) In terms of Eqs. (3), (4) and (11), the bound state energy spectrum is given by √ h¯ 2 2mDe 2 1 En = (20) . n+ − 2m 2 a h¯ Eqs. (15) and (16) agree with those of Flügge [19], who obtained the results in terms of the factorization method.
3. Conclusions In general, we have easily obtained the exact bound state solutions of the one-dimensional Schrödinger equation with the Morse potential by the way of Laplace transforms. In terms of such technique the second-order differential equation can be come back to the first-order differential equation and therefore we may directly derive required results from making use of integral. The presented procedure in this work is simple and efficient. With the above considerations the
57
authors hope to stimulate further examples of applications for Laplace transforms in important problems of physics.
References [1] L. Infeld, T.E. Hull, Rev. Mod. Phys. 23 (1951) 21. [2] C. Grosche, J. Phys. A: Math. Gen. 28 (1995) 5889; C. Grosche, J. Phys. A: Math. Gen. 29 (1996) 365. [3] J. Wu, Y. Alhassid, F. Gürsey, Ann. Phys. (N.Y.) 196 (1989) 163; J. Wu, Y. Alhassid, J. Math. Phys. 31 (1990) 557. [4] C.H. Lai, J. Math. Phys. 28 (1987) 1801; S. Atag, Phys. Rev. A 37 (1988) 2280. [5] R.J.W. Hodgson, J. Phys. A: Math. Gen. 21 (1988) 1563. [6] R.L. Hall, N. Saad, J. Phys. A: Math. Gen. 29 (1996) 2127. [7] S.H. Dong, G.H. Sun, Phys. Lett. A 314 (2003) 261. [8] F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251 (1995) 267. [9] G. Doetsch, Guide to the Applications of Laplace Transforms, Princeton Univ. Press, Princeton, 1961. [10] E. Schrödinger, Ann. Phys. 79 (1926) 361. [11] R.A. Swainson, G.W.F. Drake, J. Phys. A: Math. Gen. 24 (1991) 79. [12] P.M. Morse, Phys. Rev. 34 (1929) 57. [13] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, NonRelativistic Theory, third ed., Pergamon, Elmsford, NY, 1977. [14] S.H. Dong, R. Lemus, A. Frank, Int. J. Quantum Chem. 86 (2002) 433. [15] M.G. Benedict, B. Molnar, Phys. Rev. A 60 (1999) R1737. [16] D. Popov, Phys. Lett. A 316 (2003) 369. [17] S.H. Dong, Y. Tang, G.H. Sun, Phys. Lett. A 314 (2003) 145. [18] J. Yu, S.H. Dong, G.H. Sun, Phys. Lett. A 322 (2004) 290. [19] S. Flügge, Practical Quantum Mechanics, Springer, Berlin, 1974.
The exact solutions of the Schrödinger equation with the Morse potential via Laplace transforms Gang Chen Department of Physics, Shaoxing College of Arts and Sciences, Shaoxing 312000, PR China Received 23 March 2004; received in revised form 13 April 2004; accepted 14 April 2004 Available online 28 April 2004 Communicated by R. Wu
Abstract In this Letter, we reduce the second-order differential equation about the one-dimensional Schrödinger equation with the Morse potential reduced to the first-order differential equation in terms of Laplace transforms and then obtain the exact bound state solutions. 2004 Elsevier B.V. All rights reserved. PACS: 03.65.Ge Keywords: Bound state; Morse potential; Schrödinger equation; Laplace transforms
1. Introduction Since the appearance of the Schrödinger equation in quantum mechanics, there have been continual researches for solving the Schrödinger equation with any potential by using different methods because of the importance of such solutions in many branches of physics, especially molecules, atoms, nuclei, etc. All kinds of methods such as the factorization [1], the path integral [2], the group theoretical (algebraic method) [3], the 1/N expansion [4], the analytic continuation [5], the eigenvalue moment method [6], the power series expansion [7] and the supersymmetric
E-mail address: [email protected] (G. Chen). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.04.029
quantum mechanics [8] could be enumerated amongst other methods of the solutions of the wave equations. Laplace transforms, which are integral transforms, are comprehensively useful in physics and engineering [9]. Such techniques were used by Schrödinger in his first paper on the quantum mechanical hydrogen atom [10] and then by Swainson et al. on the recurrence relations of radial wave functions for hydrogen atom [11]. In this Letter, we will employ Laplace transforms to solve the one-dimensional Schrödinger equation with the Morse potential [12], which has played an important role in many different fields of the physics such as molecular physics, solid state physics and chemical physics, etc. This potential has been studied by many different approaches such as the standard confluent hypergeometric functions [13], the algebraic method [14], the supersymmetric method
56
G. Chen / Physics Letters A 326 (2004) 55–57
[15], the coherent states [16], the controllability [17], the series solutions of Morse potential with the mass distribution [18], etc. Fortunately, the second-order differential equation can be reduced to the first-order differential equation and therefore we may directly make use of integral to get the exact bound state solutions.
2. Bound state solutions The one-dimensional Schrödinger equation for any potential is given by [19] h¯ 2 d 2 − (1) + V (x) ψ(x) = Eψ(x). 2m dx 2 In the case of the Morse potential, Eq. (1) turns into 2 d 2mDe −2ax 4mDe −ax 2mE − e + e + ψ(x) dx 2 h¯ 2 h¯ 2 h¯ 2 = 0. (2)
where N is a constant. Noting that (1−
1 k−(2β+1) ) p+ 12
is a multi-valued function and the wave-functions are required to be single-valued, we must take k − (2β + 1) = 2n,
n = 0, 1, 2, 3, . . ..
(11)
Applying a simple series expansion to Eq. (10) yields F (p) = N
n (−1)j n!(p + 12 )−(2β+j +1) , (n − j )!j !
(12)
j =0
Put y = ke
∞ Using Laplace transforms L[g(t)] = 0 g(t)e−pt dt = G(p) [9] yields k 1 d F (p) + (2β + 1)p − F (p) = 0. p2 − 4 dp 2 (9) Eq. (9) is a first-order differential equation and so we may directly make use of integral to get k−(2β+1) 1 −(2β+1) 1 , F (p) = N P + 1− 2 p + 12 (10)
−ax
β2 = −
√ 2 2mDe , k= h¯ a
2mE
, h¯ 2 Eq. (2) becomes 2 y2 k d 2 d 2 − + y − β ψ(y) = 0. y +y dy 4 2 dy 2
(3) (4)
(5)
(6)
with A is a constant, and then inserting into Eq. (5) leads to equation y2 d2 d − y 2 2 + (2A + 1)y dy dy 4 k + y + A2 − β 2 f (y) = 0. (7) 2 Putting A = −β (the case A = β is not be acceptable in quantum mechanics because ψ(y) is finite when y → ∞) yields y k d2 d − + f (y) = 0. y 2 − (2β − 1) (8) dy dy 4 2
n
(−1)j n!(2β + 1) yj , (n − j )!j !(2β + j + 1) j =0 (13) where N is a constant. Comparing Eq. (13) with the series expansion of confluent hypergeometric functions 2β −y
f (y) = Ny e
Letting ψ(y) = y A f (y),
where N is a constant. Using a simple extension of inverse Laplace transforms [9] we can immediately deduce that
F (−n, γ , y) =
n j =0
(−1)j n!(γ ) yj (n − j )!j !(γ + j )
(14)
yields f (y) = Ny 2β e−y/2F (−n, 2β + 1, y).
(15)
Inserting Eq. (14) into Eq. (6) yields ψ(y) = Ny β e−y/2F (−n, 2β + 1, y).
(16)
On account of the relation between confluent hypergeometric functions and generalized Laguerre polynomials F (−m, u + 1, z) =
m!(u + 1) u L (z), (u + m + 1) m
(17)
G. Chen / Physics Letters A 326 (2004) 55–57
Eq. (16) is rewritten as ψ(y) = Nn y 2 −(n+ 2 ) e−y/2 Lnk−2n−1 (y), k
1
(18)
where Nn is a constant. Inserting ∞ 1(18)∗ into normalized condition of wave functions 0 ay ψ (y)ψ(y) dy = 1 yields an!(k − 2n − 1) 1/2 Nn = (19) . (k − n) In terms of Eqs. (3), (4) and (11), the bound state energy spectrum is given by √ h¯ 2 2mDe 2 1 En = (20) . n+ − 2m 2 a h¯ Eqs. (15) and (16) agree with those of Flügge [19], who obtained the results in terms of the factorization method.
3. Conclusions In general, we have easily obtained the exact bound state solutions of the one-dimensional Schrödinger equation with the Morse potential by the way of Laplace transforms. In terms of such technique the second-order differential equation can be come back to the first-order differential equation and therefore we may directly derive required results from making use of integral. The presented procedure in this work is simple and efficient. With the above considerations the
57
authors hope to stimulate further examples of applications for Laplace transforms in important problems of physics.
References [1] L. Infeld, T.E. Hull, Rev. Mod. Phys. 23 (1951) 21. [2] C. Grosche, J. Phys. A: Math. Gen. 28 (1995) 5889; C. Grosche, J. Phys. A: Math. Gen. 29 (1996) 365. [3] J. Wu, Y. Alhassid, F. Gürsey, Ann. Phys. (N.Y.) 196 (1989) 163; J. Wu, Y. Alhassid, J. Math. Phys. 31 (1990) 557. [4] C.H. Lai, J. Math. Phys. 28 (1987) 1801; S. Atag, Phys. Rev. A 37 (1988) 2280. [5] R.J.W. Hodgson, J. Phys. A: Math. Gen. 21 (1988) 1563. [6] R.L. Hall, N. Saad, J. Phys. A: Math. Gen. 29 (1996) 2127. [7] S.H. Dong, G.H. Sun, Phys. Lett. A 314 (2003) 261. [8] F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251 (1995) 267. [9] G. Doetsch, Guide to the Applications of Laplace Transforms, Princeton Univ. Press, Princeton, 1961. [10] E. Schrödinger, Ann. Phys. 79 (1926) 361. [11] R.A. Swainson, G.W.F. Drake, J. Phys. A: Math. Gen. 24 (1991) 79. [12] P.M. Morse, Phys. Rev. 34 (1929) 57. [13] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, NonRelativistic Theory, third ed., Pergamon, Elmsford, NY, 1977. [14] S.H. Dong, R. Lemus, A. Frank, Int. J. Quantum Chem. 86 (2002) 433. [15] M.G. Benedict, B. Molnar, Phys. Rev. A 60 (1999) R1737. [16] D. Popov, Phys. Lett. A 316 (2003) 369. [17] S.H. Dong, Y. Tang, G.H. Sun, Phys. Lett. A 314 (2003) 145. [18] J. Yu, S.H. Dong, G.H. Sun, Phys. Lett. A 322 (2004) 290. [19] S. Flügge, Practical Quantum Mechanics, Springer, Berlin, 1974.