# Exact Analytical Solution of theN-Dimensional Radial Schrödinger Equation with Pseudoharmonic Potential via Laplace Transform Approach

Authors: Tapas Das; Altuğ Arda
Journal: Advances in High Energy Physics
 (2015)

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## Abstract

The second-orderN-dimensional Schrödinger equation with pseudoharmonic potential is reduced to a first-order differential equation by using the Laplace transform approach and exact bound state solutions are obtained using convolution theorem. Some special cases are verified and variations of energy eigenvalues En as a function of dimension N are furnished. To give an extra depth of this paper, the present approach is also briefly investigated for generalized Morse potential as an example.


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## Body

## 1. Introduction

Schrödinger equation has long been recognized as an essential tool for the study of atoms, nuclei, and molecules and their spectral behaviors. Much effort has been spent to find the exact bound state solution of this nonrelativistic equation for various potentials describing the nature of bonding or the nature of vibration of quantum systems. A large number of research workers all around the world continue to study the ever fascinating Schrödinger equation, which has wide application over vast areas of theoretical physics. The Schrödinger equation is traditionally solved by operator algebraic method [1], power series method [2, 3], or path integral method [4].There are various other alternative methods in the literature to solve Schrödinger equation such as Fourier transform method [5–7], Nikiforov-Uvarov method [8], asymptotic iteration method [9], and SUSYQM [10]. The Laplace transformation method is also an alternative method in the list and it has a long history. The LTA was first used by Schrödinger to derive the radial eigenfunctions of the hydrogen atom [11]. Later Englefield used LTA to solve the Coulomb, oscillator, exponential, and Yamaguchi potentials [12]. Using the same methodology, the Schrödinger equation has also been solved for various other potentials, such as pseudoharmonic [13], Dirac delta [14], and Morse-type [15, 16] and harmonic oscillator [17] specially on lower dimensions.Recently,N-dimensional Schrödinger equations have received focal attention in the literature. The hydrogen atom in five dimensions and isotropic oscillator in eight dimensions have been discussed by Davtyan and coworkers [18]. Chatterjee has reviewed several methods commonly adopted for the study of N-dimensional Schrödinger equations in the large N limit [19], where a relevant 1/N expansion can be used. Later Yáñez et al. have investigated the position and momentum information entropies of N-dimensional system [20]. The quantization of angular momentum in N-dimensions has been described by Al-Jaber [21]. Other recent studies of Schrödinger equation in higher dimension include isotropic harmonic oscillator plus inverse quadratic potential [22], N-dimensional radial Schrödinger equation with the Coulomb potential [23]. Some recent works on N-dimensional Schrödinger equation can be found in the references list [24–31].These higher dimension studies facilitate a general treatment of the problem in such a manner that one can obtain the required results in lower dimensions just dialing appropriateN. The pseudoharmonic potential is expressed in the form [32](1)Vr=Derre-rer2, where De=1/8Kere2 is the dissociation energy with the force constant Ke and re is the equilibrium constant. The pseudoharmonic potential is generally used to describe the rotovibrational states of diatomic molecules and nuclear rotations and vibrations. Moreover, the pseudoharmonic potential and some kinds of it for N-dimensional Schrödinger equation help to test the powerfulness of different analytical methods for solving differential equations. To give an example, the dynamical algebra of the Schrödinger equation in N-dimension has been studied by using pseudoharmonic potential [33]. Taşeli and Zafer have tested the accuracy of expanding of the eigenfunction in a Fourier-Bessel basis [34] and a Laguerre basis [35] with the help of different type of polynomial potentials in N-dimension.Motivated by these types of works, in this present paper we discuss the exact solutions of theN-dimensional radial Schrödinger equation with pseudoharmonic potential using the Laplace transform approach. To make this paper self-contained we briefly outline Laplace transform method and convolution theorem in the next section. Section 3 is for the bound state spectrum of the potential system. Section 4 is devoted to the results and discussion where we derive some well known results for special cases of the potential. The application of the present method is shown in Section 5 where we briefly show how generalized Morse potential could be solved. Finally the conclusion of the present work is placed in Section 6.


## 2. Laplace Transform Method and Convolution Theorem

The Laplace transformF(s) or L of a function χ(y) is defined by [36, 37](2)Fs=Lχy=∫0∞e-syχydy. If there is some constant σ∈R such that e-σyχ(y)≤M for sufficiently large y, the integral in (2) will exist for Re⁡s>σ. The Laplace transform may fail to exist because of a sufficiently strong singularity in the function χ(y) as y→0. In particular(3)LyαΓα+1=1sα+1,α>-1. The Laplace transform has the derivative properties(4)Lχny=snLχy-∑j=0n-1sn-1-jχj0,Lynχy=-1nFns, where the superscript (n) denotes the nth derivative with respect to y for χ(n)(y) and with respect to s for F(n)(s).The inverse transform is defined asL-1F(s)=χ(y). One of the most important properties of the Laplace transform is that given by the convolution theorem [38]. This theorem is a powerful tool to find the inverse Laplace transform. According to this theorem if we have two transformed functions g(s)=LG(y) and h(s)=LH(y), then the product of these two is the Laplace transform of the convolution (G∗H)(y), where(5)G∗Hy=∫0yGy-τHτdτ. So the convolution theorem yields(6)LG∗Hy=gshs. Hence(7)L-1gshs=∫0yGy-τHτdτ. If we substitute w=y-τ, then we find the important consequence G∗H=H∗G.


## 3. Bound State Spectrum

TheN-dimensional time-independent Schrödinger equation for a particle of mass M with orbital angular momentum quantum number l is given by [39](8)∇N2+2Mħ2E-Vrψnlmr,ΩN=0, where E and V(r) denote the energy eigenvalues and potential. ΩN within the argument of the nth state eigenfunctions ψnlm denotes angular variables θ1,θ2,θ3,…,θN-2,φ. The Laplacian operator in hyperspherical coordinates is written as(9)∇N2=1rN-1∂∂rrN-1∂∂r-ΛN-12r2, where(10)ΛN-12=-∑k=1N-21sin2⁡θk+1sin2⁡θk+2⋯sin2⁡θN-2sin2⁡φ1sink-1⁡θk∂∂θksink-1⁡θk∂∂θk+1sinN-2⁡φ∂∂φsinN-2⁡φ∂∂φ.ΛN-12 is known as the hyperangular momentum operator.We chose the bound state eigenfunctionsψnlm(r,ΩN) that are vanishing for r→0 and r→∞. Applying the separation variable method by means of the solution ψnlm(r,ΩN)=Rnl(r)Ylm(ΩN), (8) provides two separated equations:(11)ΛN-12YlmΩN=ll+N-2YlmΩN, where Ylm(ΩN) is known as the hyperspherical harmonics, and the hyperradial or in short the “radial” equation(12)d2dr2+N-1rddr-ll+N-2r2-2Mħ2Vr-ERnlr=0, where l(l+N-2)N>1 is the separation constant [40, 41] with l=0,1,2,….In spite of taking (1) we take the more general form of pseudoharmonic potential [42](13)Vr=a1r2+a2r2+a3, where a1,a2, and a3 are three parameters that can take any real value. If we set a1=De/re2, a2=Dere2, and a3=-2De, (13) converts into the special case which we have given in (1). Taking this into (12) and using the abbreviations(14)νν+1=ll+N-2+2Mħ2a2,μ2=2Mħ2a1,ϵ2=2Mħ2E-a3, we obtain(15)d2dr2+N-1rddr-νν+1r2-μ2r2+ϵ2Rnlr=0. We are looking for the bound state solutions for Rnl(r) with the following properties: Rnl(r→0)=r-k and Rnl(r→∞)=0. Let us assume a solution of type Rnl(r)=r-kf(r) with k>0. Here the term r-k ensures the fact that Rnl(r→∞)=0 and f(r) is expected to behave like f(r→0)=0.Now changing the variable asy=r2 and taking f(r)≡χ(y) from (15) we obtain(16)4yd2χdy2+2N-2kdχdy+kk+1-kN-1-νν+1y-μ2y+ϵ2χ=0. In order to get an exact solution of the above differential equation we remove the singular term by imposing the condition(17)kk+1-kN-1-νν+1=0. So we have(18)yd2χdy2-klN-N2dχdy-14μ2y-ϵ2χ=0, where klN is taken as the positive solution of k, which can be easily found by little algebra from (17). It is worth mentioning here that the condition given by (17) is necessary to get an analytical solution because otherwise only approximate or numerical solution is possible. Introducing the Laplace transform F(s)=L{χ(y)} with the boundary condition f(0)≡χ(0)=0 and using (4), (18) can read(19)s2-μ24dFds+ηlNs-ϵ24F=0, where ηlN=klN-N/2+2. This parameter can have integer or noninteger values and there will be integer or noninteger term(s) in energy eigenvalue according to values of ηlN which can be seen below.The solution of the last equation can be written easily as(20)Fs=Cs+μ2-ηlN/2-ϵ2/4μs-μ2-ηlN/2+ϵ2/4μ, where C is a constant. The term μ=2Ma1/ħ is a positive real number as we restrict ourselves to the choice a1>0. Now, since s is positive and μ>0, then the second factor of (20) could become negative if μ/2>s>0 and thus its power must be a positive integer to get singled valued eigenfunctions. This will also exclude the possibility of getting singularity in the transformation. So we have(21)ϵ24μ-12ηlN=n,n=0,1,2,…. Using (21), we have from (20)(22)Fs=Cs+μ2-as-μ2-b=Cgshs, where a=ηlN+n and b=-n. In order to find χ(y)=L-1F(s), we find [43](23)L-1s+μ2-a=Gy=ya-1e-μ/2yΓa,L-1s-μ2-b=Hy=yb-1eμ/2yΓb. Therefore using (7) and (23), we have(24)χy=L-1Fs=CG∗Hy=C∫0yGy-τHτdτ=Ce-μ/2yΓaΓb∫0yy-τa-1τb-1eμτdτ. The integration can be found in [44], which gives(25)∫0yy-τa-1τb-1eμτdτ=Ba,bya+b-1F11b,a+b,μy, where F11 is the confluent hypergeometric functions. Now using the beta function B(a,b)=Γ(a)Γ(b)/Γ(a+b), χ(y) can be written:(26)χy=CΓa+be-μ/2yya+b-1F11b,a+b,μy. So we have the radial eigenfunctions(27)RnlNr=r-klNfr=CnlNe-μ/2r2rηlN-N/2F11-n,ηlN,μr2, where CnlN is the normalization constant. It should be noted that because of the “quantization condition” given by (21), it is possible to write the radial eigenfunctions in a polynomial form of degree n, as 1F1(b,a+b,μy) converges for all finite μy with b=-n and (a+b) is not a negative integer or zero.Now the normalization constantCnlN can be evaluated from the condition [45](28)∫0∞RnlNr2rN-1dr=1. To evaluate the integration the formula 1F1(-q,p+1,z)=q!p!/(q+p)!Lqp(z) is useful here, where Lqp(u) are the Laguerre polynomials. It should be remembered that the formula is applicable only if q is a positive integer. Hence identifying q=n,  p=ηlN-1, and z=μr2 we have(29)F11-n,ηlN,μr2=n!ηlN-1!ηlN+n-1!LnηlN-1μr2. So using the following formula for the Laguerre polynomials,(30)∫0∞xAe-xLBAx2dx=ΓA+B+1B!, we write the normalization constant(31)CnlN=2μ1/2ηlNn!ΓηlN+nηlN+n-1!n!ηlN-1!. Finally, the energy eigenvalues are obtained from (21) along with (17) and (14):(32)EnlN=ħ22Mϵ2+a3=a3+8ħ2a1Mn+12+14N+2l-22+8Ma2ħ2, and we write the corresponding normalized eigenfunctions as (33)RnlNr=2μ1/2ηlNn!ΓηlN+nηlN+n-1!n!ηlN-1!e-μ/2r2rηlN-N/2F11-n,ηlN,μr2, or(34)RnlNr=2μ1/2ηlNn!ΓηlN+ne-μr2/2rηlN-N/2LnηlN-1μr2. Finally, the complete orthonormalized eigenfunctions of the N-dimensional Schrödinger equation with pseudoharmonic potential can be given by(35)ψr,θ1,θ2,…,θN-2,ϕ=∑n,l,mCnlNRnlNrYlmθ1,θ2,…,θN-2,ϕ.We give some numerical results about the variation of the energy on the dimensionalityN obtained from (32) in Figure 1. We summarize the plots for En(N) as a function of N for a set of physical parameters (a1,a2) by taking a3=0, especially.Figure 1
The dependence ofE(N) on N. Parameters: 2M=1,  ħ=1, and  a1=a2=0.5.
(a)
The variation of energy toN for n=0
(b)
The variation of energy toN for n=1
(c)
The variation of energy toN for n=2
(d)
The variation of energy toN for n=3


## 4. Results and Discussion

In this section we have shown that the results obtained in Section3 are very useful in deriving the special cases of several potentials for lower as well as for higher dimensional wave equation.


### 4.1. Isotropic Harmonic Oscillator

(1) Three Dimensions (
N
=
3
). For this case a1=1/2Mω2 and a2=a3=0 which gives from (32)(36)Enl3=ħω2n+l+32, where ω is the circular frequency of the particle. From (14) and (17) we obtain kl3=l+1. This makes ηl3=l+3/2 and we get radial eigenfunctions from (34). The result agrees with those obtained in [22].(2) Arbitrary
N
-Dimensions. Here N is an arbitrary constant and as before a1=1/2Mω2,  a2=a3=0. We have the energy eigenvalues from (32):(37)EnlN=ħω2n+l+N2. Solving (17) with the help of (14) we have klN=l+N-2 and one can easily obtain the normalization constant from (31):(38)CnlN=2Mω/ħl+N/2n!Γl+N/2+n1/2n+l+N-2/2!n!l+N-2/2!, where ηlN=l+N/2. The radial eigenfunctions are given in (34) with the above normalization constant. The results obtained here agree with those found in some earlier works [22, 46, 47].


### 4.2. Isotropic Harmonic Oscillator Plus Inverse Quadratic Potential

(1) Two Dimensions (
N
=
2
). Here we have a1=1/2Mω2,  a2≠0, and a3=0, where ω is the circular frequency of the particle. So from (31) we obtain(39)Cnl2=2Mω/ħkl2+1n!Γkl2+n+11/2kl2+n!n!kl2!, where ηl2=kl2+1. kl2 can be obtained from (14) and (17) as kl2=ν(ν+1)=l2+2M/ħ2a2. Hence (32) gives the energy eigenvalues of the system(40)Enl2=ħω2n+kl2+1. This result has already been obtained in [22, 48].(2) Three Dimensions (
N
=
3
). Here a1=1/2Mω2, a2≠0, and a3=0 which gives the energy eigenvalues as(41)Enl3=ħω24n+2+2l+12+8Ma2ħ2. Solving (17) we get kl3=ν+1 and hence ηl3=ν+3/2. The radial eigenfunction can hence be obtained from (34) which also corresponds to the result obtained in [22].


### 4.3.
3-Dimensional Schrödinger Equation with Pseudoharmonic Potential

Here (17) gives kl3=ν+1 and this makes ηl3=ν+3/2. So (31) provides(42)Cnl3=μ1/2ν+3/22Γν+n+3/2n!Γν+32-1, and hence the normalized eigenfunctions become(43)Rnl3r=μ1/2ν+3/22Γν+n+3/2n!Γν+32-1e-μ/2r2rν+1F11-n,ν+32,μr2, with the energy eigenvalues(44)Enl3=a3+ħ22Mϵ2=a3+8ħ2a1Mn+12+142l+12+8Ma2ħ2. This result corresponds exactly to the ones given in [13].


## 4.1. Isotropic Harmonic Oscillator

(1) Three Dimensions (
N
=
3
). For this case a1=1/2Mω2 and a2=a3=0 which gives from (32)(36)Enl3=ħω2n+l+32, where ω is the circular frequency of the particle. From (14) and (17) we obtain kl3=l+1. This makes ηl3=l+3/2 and we get radial eigenfunctions from (34). The result agrees with those obtained in [22].(2) Arbitrary
N
-Dimensions. Here N is an arbitrary constant and as before a1=1/2Mω2,  a2=a3=0. We have the energy eigenvalues from (32):(37)EnlN=ħω2n+l+N2. Solving (17) with the help of (14) we have klN=l+N-2 and one can easily obtain the normalization constant from (31):(38)CnlN=2Mω/ħl+N/2n!Γl+N/2+n1/2n+l+N-2/2!n!l+N-2/2!, where ηlN=l+N/2. The radial eigenfunctions are given in (34) with the above normalization constant. The results obtained here agree with those found in some earlier works [22, 46, 47].


## 4.2. Isotropic Harmonic Oscillator Plus Inverse Quadratic Potential

(1) Two Dimensions (
N
=
2
). Here we have a1=1/2Mω2,  a2≠0, and a3=0, where ω is the circular frequency of the particle. So from (31) we obtain(39)Cnl2=2Mω/ħkl2+1n!Γkl2+n+11/2kl2+n!n!kl2!, where ηl2=kl2+1. kl2 can be obtained from (14) and (17) as kl2=ν(ν+1)=l2+2M/ħ2a2. Hence (32) gives the energy eigenvalues of the system(40)Enl2=ħω2n+kl2+1. This result has already been obtained in [22, 48].(2) Three Dimensions (
N
=
3
). Here a1=1/2Mω2, a2≠0, and a3=0 which gives the energy eigenvalues as(41)Enl3=ħω24n+2+2l+12+8Ma2ħ2. Solving (17) we get kl3=ν+1 and hence ηl3=ν+3/2. The radial eigenfunction can hence be obtained from (34) which also corresponds to the result obtained in [22].


## 4.3.
3-Dimensional Schrödinger Equation with Pseudoharmonic Potential

Here (17) gives kl3=ν+1 and this makes ηl3=ν+3/2. So (31) provides(42)Cnl3=μ1/2ν+3/22Γν+n+3/2n!Γν+32-1, and hence the normalized eigenfunctions become(43)Rnl3r=μ1/2ν+3/22Γν+n+3/2n!Γν+32-1e-μ/2r2rν+1F11-n,ν+32,μr2, with the energy eigenvalues(44)Enl3=a3+ħ22Mϵ2=a3+8ħ2a1Mn+12+142l+12+8Ma2ħ2. This result corresponds exactly to the ones given in [13].


## 5. Short Review of Generalized Morse Potential: An Example

The generalized Morse potential [49] in terms of four parameters V1,V2,V3, and λ(>0) is(45)Vr=V1e-λr-re+V2e-2λr-re+V3, where λ describes the characteristic range of the potential, re is the equilibrium molecular separation, and V1,V2, and V3 are related to the potential depth. In this short section we will only show Laplace transformable differential equation like (18) can also be achieved if the exponential and singular terms 1/r and 1/r2 are properly handled. Looking back to the solution given by (34) we predetermine the solution for (8) for the potential given by (45) as(46)ψnlmr,ΩN=r-N-1/2RnlrYlmΩN. This substitution facilitates following easier differential equation (without the 1/r term) similar to (12) of Section 3:(47)d2dr2+2Mħ2E-Vr-ANlr2Rnlr=0, where ANl=(N+2l-1)(N+2l-3)/4.Let us introduce a new variablex=r-re/re. Hence inserting (45) into (47) we have(48)d2dx2+2Mre2ħ2E-V1e-αx-V2e-2αx-V3+ANl1+x2Rx=0, where α=λre.It is possible to expand the term1/(1+x)2 in power series as within the molecular point of view x≪1. So(49)11+x2≈1-2x+3x2-Ox3. This expansion can also be rewritten as(50)11+x2≈c0+c1e-αx+c2e-2αx. By comparing these two it is not hard to justify that c0=1-3/α+3/α2;  c1=4/α-6/α2;  c2=-1/α+3/α2.Inserting (50) into (48) and changing the variable as y=e-αx one can easily get(51)y2d2dy2+yddy+ϵα2+B1α2y+B2α2y2Ry=0. Now further assuming the solution of the above differential equation R(y)=y-ωg(y), as we did in the previous section with proper requirement for bound state scenario, finally we can construct the following differential equation just like (16) for the perfect platform for Laplace transformation:(52)yd2gdy2+1-2ωdgdy+ω2-ϵ/α2yg+B1α2+B3α2yg=0, where (53a)ϵ=2Mre2ħ2E-V3+c0ANl,(53b)B1=c1ANl-2Mre2ħ2V1,(53c)B2=c2ANl-2Mre2ħ2V2. Imposing the condition ω2-ϵ/α2=0 as previously and approaching the same way as we did in Section 3 one can obtain the energy eigenvalues and bound state wave functions in terms of confluent hypergeometric function. We have investigated that the results are exact match with [15] for V1=-2D, V2=D, and V3=0, where D describes the depth of the potential.


## 6. Conclusions

We have investigated some aspects ofN-dimensional hyperradial Schrödinger equation for pseudoharmonic potential by Laplace transformation approach. It is found that the energy eigenfunctions and the energy eigenvalues depend on the dimensionality of the problem. In this connection we have furnished few plots of the energy spectrum En(N) as a function of N for a given set of physical parameters a1,a2,l, and n=0,1,2,3 keeping a3=0. The general results obtained in this paper have been verified with earlier reported results, which were obtained for certain special values of potential parameters and dimensionality.The Laplace transform is a powerful, efficient, and accurate alternative method of deriving energy eigenvalues and eigenfunctions of some spherically symmetric potentials that are analytically solvable. It may be hard to predict which kind of potentials is solvable analytically by Laplace transform, but in general prediction of the eigenfunctions with the form likeRnl(r)=r-kf(r) always opens the gate of the possibility of closed form solutions for a particular potential model. This kind of substitution is calledUniversal Laplace transformation scheme. In this connection one might go through [50] to check out how Laplace transformation technique behaves over different potentials, specially for Schrödinger equation in lower dimensional domain. It is also true that the technique of Laplace transformation is useful if, inserting the potential into the Schrödinger equation and usingUniversal Laplace transformation scheme via some suitable parametric restrictions, one is able to get a differential equation with variable coefficient rj  (j≯1). This is not easy to achieve every time. However, for a given potential if there is no such achievement, iterative approach facilitates a better way to overcome the situation [51]. The results are sufficiently accurate for such special potentials at least for practical purpose.Before concluding we want to mention here that we have not succeeded in developing the scattering state solution for the pseudoharmonic potential. If we could develop those solutions using the LTA that would have been a remarkable achievement. The main barrier of this success lies on the realization of complex index in (20), because at scattering state situation, that is, E>limr→∞V(r), μ should be replaced with μ=iζ, where ζ=2Ma1/ħ2. Maybe this cumbersome situation would attract the researchers to study the subject further and we are looking forward to it.



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