J Math Chem (2016) 54:1287–1295 DOI 10.1007/s10910-016-0621-z ORIGINAL PAPER A large class of bound-state solutions of the Schrödinger equation via Laplace transform of the confluent hypergeometric equation P. H. F. Nogueira1 · A. S. de Castro1 · D. R. M. Pimentel2 Received: 11 February 2016 / Accepted: 26 February 2016 / Published online: 7 March 2016 © Springer International Publishing Switzerland 2016 Abstract It is shown that analytically soluble bound states of theSchrödinger equation for a large class of systems relevant to atomic and molecular physics can be obtained by means of the Laplace transform of the confluent hypergeometric equation. It is also shown that all closed-form eigenfunctions are expressed in terms of generalized Laguerre polynomials. The generalized Morse potential is used as an illustration. Keywords Morse potential · Singular potential · Laplace transform · Confluent hypergeometric equation 1 Introduction Some exactly soluble systems with importance in atomic and molecular physics have been approached in the literature on quantum mechanics with a myriad of methods. Among such systems is the Morse potential a(e−αx − 2e−2αx ) [1–19], the pseudoharmonic potential a (x/b − b/x)2 [4,5,20–31], and the Kratzer–Fues potential a ( b2/x2 − 2b/x ) and itsmodified version a ( b2/x2 − b/x ) [4,5,12,30–34]. B A. S. de Castro castro@pq.cnpq.br P. H. F. Nogueira pedrofusconogueira@gmail.com D. R. M. Pimentel douglas.roberto.fis@gmail.com 1 Departamento de Física e Química, Universidade Estadual Paulista, Campus de Guaratinguetá, Guaratinguetá, SP 12516-410, Brazil 2 Departamento de Física Experimental, Instituto de Física, Universidade de São Paulo, São Paulo, SP 05508-090, Brazil 123 http://crossmark.crossref.org/dialog/?doi=10.1007/s10910-016-0621-z&domain=pdf 1288 J Math Chem (2016) 54:1287–1295 More general exactly soluble systems have also been appreciated: the general- ized Morse potential Ae−αx + Be−2αx [31,35–39], the singular harmonic oscillator Ax2 + Bx−2 [3–5,19,36,40–52], the singular Coulomb potential Ax−1 + Bx−2 [2–5,36,40,48,50,51,53–57], and some ring-shaped potentials (see, e.g. [19] and ref- erences therein). The integral transform methods have proven to be useful and powerful for solving ordinary differential equations because they can convert the original equation into a simpler differential equation or into an algebraic equation. The Laplace transform method applied to quantummechanics was used by Schrödinger into the discussion of radial eigenfunction of the hydrogen atom [58], and later Englefield approached the three-dimensional Schrödinger equation with diverse spherically symmetric poten- tials [59]. More than twenty years later the hydrogen atom was reexamined with the Laplace transform method [60]. Recently, some interest has been revived in searching bound-state solutions of the Schr ödinger equation via Laplace transform method. For some years now one-dimensional problems with the 1/x [61], Morse [17], generalized Morse [62], Dirac delta [63] and harmonic oscillator [64] potentials, three-dimensional problems with the singular harmonic oscillator, the singular Coulomb [51], some ring- shaped [65] potentials, and the D-dimensional harmonic oscillator [66], have been solved for the Laplace transform. With fulcrum on the relation mapping the behaviour of the eigenfunction near infinity and the Laplace transform near isolated singular points, Englefield [59] found the spectrum of three-dimensional problems by impos- ing that the radial eigenfunction vanishes at the origin. Englefield’s recipe, spiced up with the relation mapping the behaviour of the eigenfunction near the origin and its corresponding transform near infinity, was used in Ref. [64] in order to obtain the com- plete set of bound-state solutions for the one-dimensional harmonic oscillator without using the closed-form solution for the Laplace transform. Furthermore, the class of problems was enlarged to include eigenfunctions satisfying homogeneous Neumann conditions at the origin. In the present paper, the spiced Englefield’s recipe is followed with attention restricted to systems that after factorizing the behaviour at the neighbourhood of spe- cial points, the second-order differential equation for the eigenfunction can be reduced to the confluent hypergeometric equation. Then is shown that all well-behaved eigen- functions for that class of systems are expressed in terms of generalized Laguerre polynomials. Exactly solvable problems in this category include all the potentials mentioned in the first paragraph and the exactly soluble generalized Morse potential is used as an illustration. 2 Laplace transform and a few of its properties Let us beginwith a brief description of the Laplace transform and a few of its properties [67]. The Laplace transform of a function � is defined by F(s) = L {�} = ∫ ∞ 0 dξ e−sξ� (ξ) . (1) 123 J Math Chem (2016) 54:1287–1295 1289 If there is some positive constant σ such that � does not increase faster than eσξ for sufficiently large ξ then � is said to be of exponential order σ . In this case, the integral in Eq. (1) may exist for Re s > σ . Nevertheless, the Laplace transform may fail to exist because of a sufficiently strong singularity in the function � as ξ → 0. In particular, ξλ is of exponential order arbitrary and L { ξλ � (λ + 1) } = 1 sλ+1 , Re λ > −1, Re s > 0, (2) where � is the gamma function. Derivative properties involving the Laplace trans- form are convenient for solving differential equations. In this paper we shall use the following properties: L { d� dξ } = sF (s) − �|ξ=0 L { d2� dξ2 } = s2F (s) − s �|ξ=0 − d� dξ ∣∣∣ ∣ ξ=0 L {ξ�} = −dF (s) ds . (3) More than this, we shall use a pair of relations mapping limiting forms. If near an isolated singular point s0 the Laplace transform behaves as F (s) ∼ s→s0 1 (s − s0)ν , ν > 0, (4) then �(ξ) ∼ ξ→∞ 1 � (ν) ξν−1 es0ξ . (5) On the other hand, lim s→∞sF (s) = �(0) , (6) an result known as initial value theorem. 3 The generalized Morse potential The time-independent Schrödinger equation is an eigenvalue equation for the charac- teristic pair (E, ψ) with E ∈ R. For a particle of massm embedded in the generalized Morse potential it reads d2ψ (x) dx2 + 2m h̄2 ( E − V1e −αx − V2e −2αx ) ψ (x) = 0, (7) 123 1290 J Math Chem (2016) 54:1287–1295 where h̄ is Planck’s constant, α > 0, and ∫ +∞ −∞ dx |ψ |2 = 1 for bound states. The substitution ξ = 2 √ 2mV2 e−αx h̄α (8) and the definitions S = √−2mE h̄α , a = mV1 h̄α √ 2mV2 + S + 1 2 (9) convert Eq. (7) into d2ψ (ξ) dξ2 + 1 ξ dψ (ξ) dξ + ( −1 4 + S − a + 1/2 ξ − S2 ξ2 ) ψ (ξ) = 0, (10) whose solutions have asymptotic limits expressed asψ (ξ) →|ξ |→0 ξ±S andψ (ξ) →|ξ |→∞ e±ξ/2 . On account of the normalization condition, ∫ ∞ 0 d|ξ | |ψ (ξ) |2/|ξ | = α, one has that ψ behaves like ξ S as |ξ | → 0 and like e−ξ/2 as |ξ | → ∞ with ξ ∈ R (V2 > 0) and S > 0 (E < 0). The substitution ψ (ξ) = e−ξ/2ξ S�(ξ) (11) transforms Eq. (10) into ξ d 2�(ξ) dξ2 + (b − ξ) d�(ξ) dξ − a �(ξ) = 0, (12) with b = 2S + 1 > 1. Eq. (12) is the standard form of the confluent hypergeometric equation [68]. Notice that� is a nonzero constant at the origin and tends to infinity no more rapidly than exp (σ1ξ σ2), with σ2 < 1 and arbitrary σ1, for sufficiently large ξ . This occurs because σ1ξ σ2 − ξ/2 → −ξ/2 as ξ → ∞. The regular behaviour of � at the origin plus its behaviour for large ξ ensure the existence of its Laplace transform. In this case� is of exponential order arbitrary and consequently its Laplace transform exists for Re s > 0. 4 Laplace transform of the confluent hypergeometric equation Using the derivative properties of the Laplace transforms given by (3 ), the confluent hypergeometric equation is mapped onto s (s − 1) dF (s) ds + [(2 − b) s + a − 1] F (s) = (1 − b) � (0) , Re s > 0. (13) Note that this first-order differential equation has regular (nonessential) singularities at s = 0 and s = 1. Therefore, F (s) is either analytical, or possess a pole or branch point, 123 J Math Chem (2016) 54:1287–1295 1291 at the regular singular point (Fuchs theorem). The relation connecting the behaviour of F near an isolated singular point and the behaviour of � for large ξ dictates that � behaves like ξν−1 or ξν−1eξ , depending on where the isolated singularity of F is, whether near s = 0 or s = 1, respectively. Due to the asymptotic behaviour prescribed for � at the end of the previous section, one sees that F behaves like s−ν as s → 0, and (13) enforces ν = 1 − a. (14) On the other hand, using the initial value theorem one sees that F behaves like�(0) /s as s → ∞. Thus, we seek a particular solution of (13), regular at s = 1, in the form of a polynomial in inverse powers of s: F (s) = n∑ j=0 c j s j−ν = c0 sν + · · · + cn s , (15) with c0 �= 0 and ν = n + 1, (16) in such a way that s = 0 is a pole of order n + 1 and F (s) is the principal part of a finite Laurent series with residue cn = �(0) at s = 0. Comparing (14) with (16), one sees that a = −n. (17) Substituting (15) into (13) one obtains the following two-term recursive relation for the coefficients c j+1 = c j 1 + j − n − b j + 1 , j ≥ 0. (18) Inspection and induction yields c j = c0 (−1) j j ! � (n + b) �(n + b − j) , 0 ≤ j ≤ n. (19) This means that F (s) = c0 n∑ j=0 (−1) j j ! � (n + b) � (n + b − j) s j−n−1. (20) 123 1292 J Math Chem (2016) 54:1287–1295 Using (2), the termwise inverse transformation of (20) leads to the polynomial solution for �: �(ξ) = c0 n∑ j=0 (−1) j j ! � (n + b) � (n + b − j) (n − j)!ξ n− j . (21) Then, using Leibniz’s formula for the generalized Laguerre polynomials [68] L(b−1) n (ξ) = n∑ j=0 � (n + b) � ( j + b) (−ξ) j j ! (n − j)! , b > 0, (22) one obtains �n (ξ) = c0 (−1)n L(b−1) n (ξ) . (23) Actually, condition (17) transforms the confluent hypergeometric equation into gen- eralized Laguerre’s equation in such a way that the succeeding process involving the inversion of the Laplace transform is not surprising. For systemswhose eigenfunctions can be expressed in terms of a particular solution of the confluent hypergeometric equation, Eqs. (17) and (23) summarize all we need to determine the complete set of bound-state solutions. 5 Bound states in a generalized Morse potential We now turn our attention to the generalized Morse potential. Substitution of (17) into (9) leads to the quantization condition n + S + 1 2 = − mV1 h̄α √ 2mV2 . (24) Hence, V1 < 0 so that the generalized Morse potential is able to hold bound states only if it has a well structure (V1 < 0 and V2 > 0). Furthermore, because E < 0 one gets n < m|V1| h̄α √ 2mV2 − 1 2 . (25) This restriction on n limits the number of allowed states and requires m|V1|/( h̄α √ 2mV2 ) > 1/2 to make the existence of a bound state possible. Finally, we use the quantization condition (24) to write En = − V 2 1 4V2 [ 1 − h̄α √ 2mV2 m|V1| ( n + 1 2 )]2 . (26) 123 J Math Chem (2016) 54:1287–1295 1293 With the generalized Laguerre polynomial standardized as [68] L(μ) n (z) = n∑ j=0 � (n + μ + 1) � ( j + μ + 1) (−z) j j ! (n − j)! (27) and the integral [69] ∞∫ 0 dz e−z zγ−1L(μ) n (z) = � (γ ) � (1 + μ + n − γ ) n! � (1 + μ − γ ) , Reγ > 0, (28) one can show that ∞∫ 0 dz e−z zμ−1 [ L(μ) n (z) ]2 = � (μ + n + 1) μ n! , Re μ > 0. (29) Therefore, the normalization condition yields the normalized eigenfunction (firstly obtained in Ref. [7] for V2 = −2V1): ψn (ξ) = √ 2αS n! � (2S + n + 1) ξ Se−ξ/2L(2S) n (ξ) . (30) 6 Concluding remarks Bessel’s equation as well as differential equations with linear coefficients can be mapped onto simpler homogeneous first-order differential equations for the Laplace transformwhen the original functions are subject to special boundary conditions at the origin (see, e.g. [67]). Following the spiced Englefield’s recipe, we have shown that the bound-state solutions of the Schrödinger equation whose eigenfunctions are expressed in terms of particular solutions of the confluent hypergeometric equation, including the large class of systems with potentials relevant to atomic and molecular physics such as the generalized Morse, singular harmonic oscillator, and singular Coulomb potentials, beyond the ring-shaped potentials approached in Ref. [65], can be obtained by using the Laplace transform of the confluent hypergeometric equation. In those cases, the Laplace transform maps the confluent hypergeometric equation onto a non- homogeneous first-order differential equation. The source of nonhomogeneity is just the particular solution of the confluent hypergeometric equation at the origin but this fact does not represent a less favourable position because it is related asymptotically to the residue of the Laplace transform at the origin via the initial value theorem. It is worthwhile recall that the eigenfunction for the generalized Morse potential does not have values prescribed at the origin. 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(Academic, New York, 2007) 123 A large class of bound-state solutions of the Schrödinger equation via Laplace transform of the confluent hypergeometric equation Abstract 1 Introduction 2 Laplace transform and a few of its properties 3 The generalized Morse potential 4 Laplace transform of the confluent hypergeometric equation 5 Bound states in a generalized Morse potential 6 Concluding remarks Acknowledgments References