Dimensional Radial Schrödinger Equation with Pseudoharmonic ...

Research ArticleExact Analytical Solution of the𝑁-Dimensional RadialSchrödinger Equation with Pseudoharmonic Potential viaLaplace Transform Approach

Tapas Das1 and AltuL Arda2

1Kodalia Prasanna Banga High School (H.S), South 24 Parganas, Sonarpur 700146, India2Department of Physics Education, Hacettepe University, 06800 Ankara, Turkey

Correspondence should be addressed to Tapas Das; [email protected]

Received 16 September 2015; Revised 1 December 2015; Accepted 7 December 2015

Academic Editor: Ming Liu

Copyright © 2015 T. Das and A. Arda. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Thepublication of this article was funded by SCOAP3.

The second-order 𝑁-dimensional Schrodinger equation with pseudoharmonic potential is reduced to a first-order differentialequation by using the Laplace transform approach and exact bound state solutions are obtained using convolution theorem. Somespecial cases are verified and variations of energy eigenvalues 𝐸

𝑛as a function of dimension𝑁 are furnished. To give an extra depth

of this paper, the present approach is also briefly investigated for generalized Morse potential as an example.

The author Tapas Das wishes to dedicate this work to his wife Sonali for her love and care

1. Introduction

Schrodinger equation has long been recognized as an essen-tial tool for the study of atoms, nuclei, and molecules andtheir spectral behaviors. Much effort has been spent to findthe exact bound state solution of this nonrelativistic equationfor various potentials describing the nature of bonding or thenature of vibration of quantum systems. A large number ofresearch workers all around the world continue to study theever fascinating Schrodinger equation, which has wide appli-cation over vast areas of theoretical physics. The Schrodingerequation 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 litera-ture to solve Schrodinger equation such as Fourier transformmethod [5–7], Nikiforov-Uvarov method [8], asymptoticiteration method [9], and SUSYQM [10]. The Laplace trans-formationmethod is also an alternativemethod in the list andit has a long history. The LTA was first used by Schrodingerto 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 samemethodology, the Schrodinger equation has also been solvedfor various other potentials, such as pseudoharmonic [13],Dirac delta [14], and Morse-type [15, 16] and harmonicoscillator [17] specially on lower dimensions.

Recently, 𝑁-dimensional Schrodinger equations havereceived focal attention in the literature. The hydrogen atomin five dimensions and isotropic oscillator in eight dimen-sions have been discussed by Davtyan and coworkers [18].Chatterjee has reviewed several methods commonly adoptedfor the study of 𝑁-dimensional Schrodinger equations inthe large 𝑁 limit [19], where a relevant 1/𝑁 expansion canbe used. Later Yanez et al. have investigated the positionand momentum information entropies of 𝑁-dimensionalsystem [20]. The quantization of angular momentum in𝑁-dimensions has been described by Al-Jaber [21]. Otherrecent studies of Schrodinger equation in higher dimensioninclude isotropic harmonic oscillator plus inverse quadraticpotential [22], 𝑁-dimensional radial Schrodinger equation

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015, Article ID 137038, 8 pageshttp://dx.doi.org/10.1155/2015/137038

2 Advances in High Energy Physics

with the Coulomb potential [23]. Some recent works on𝑁-dimensional Schrodinger equation can be found in thereferences list [24–31].

These higher dimension studies facilitate a general treat-ment of the problem in such amanner that one can obtain therequired results in lower dimensions just dialing appropriate𝑁. The pseudoharmonic potential is expressed in the form[32]

𝑉 (𝑟) = 𝐷𝑒(𝑟

𝑟𝑒

−𝑟𝑒

𝑟)

2

, (1)

where 𝐷𝑒= (1/8)𝐾

𝑒𝑟2

𝑒is the dissociation energy with the

force constant 𝐾𝑒and 𝑟

𝑒is the equilibrium constant. The

pseudoharmonic potential is generally used to describe therotovibrational states of diatomicmolecules and nuclear rota-tions and vibrations. Moreover, the pseudoharmonic poten-tial and some kinds of it for 𝑁-dimensional Schrodingerequation help to test the powerfulness of different analyt-ical methods for solving differential equations. To give anexample, the dynamical algebra of the Schrodinger equationin𝑁-dimension has been studied by using pseudoharmonicpotential [33]. Taseli and Zafer have tested the accuracy ofexpanding of the eigenfunction in a Fourier-Bessel basis [34]and a Laguerre basis [35] with the help of different type ofpolynomial potentials in𝑁-dimension.

Motivated by these types of works, in this present paperwe discuss the exact solutions of the 𝑁-dimensional radialSchrodinger equation with pseudoharmonic potential usingthe Laplace transform approach. To make this paper self-contained we briefly outline Laplace transform method andconvolution theorem in the next section. Section 3 is for thebound state spectrum of the potential system. Section 4 isdevoted to the results and discussion where we derive somewell known results for special cases of the potential. Theapplication of the presentmethod is shown in Section 5wherewe briefly show how generalized Morse potential could besolved. Finally the conclusion of the present work is placedin Section 6.

2. Laplace Transform Method andConvolution Theorem

TheLaplace transform𝐹(𝑠) orL of a function𝜒(𝑦) is definedby [36, 37]

𝐹 (𝑠) = L {𝜒 (𝑦)} = ∫

∞

0

𝑒−𝑠𝑦

𝜒 (𝑦) 𝑑𝑦. (2)

If there is some constant 𝜎 ∈ R such that |𝑒−𝜎𝑦𝜒(𝑦)| ≤

M for sufficiently large 𝑦, the integral in (2) will exist forRe 𝑠 > 𝜎. The Laplace transformmay fail to exist because of asufficiently strong singularity in the function 𝜒(𝑦) as 𝑦 → 0.In particular

L [𝑦𝛼

Γ (𝛼 + 1)] =

1

𝑠𝛼+1, 𝛼 > −1. (3)

The Laplace transform has the derivative properties

L {𝜒(𝑛)

(𝑦)} = 𝑠𝑛

L {𝜒 (𝑦)} −

𝑛−1

∑

𝑗=0

𝑠𝑛−1−𝑗

𝜒(𝑗)

(0) ,

L {𝑦𝑛

𝜒 (𝑦)} = (−1)𝑛

𝐹(𝑛)

(𝑠) ,

(4)

where the superscript (𝑛) denotes the 𝑛th derivative withrespect to 𝑦 for 𝜒(𝑛)(𝑦) and with respect to 𝑠 for 𝐹(𝑛)(𝑠).

The inverse transform is defined as L−1{𝐹(𝑠)} = 𝜒(𝑦).One of the most important properties of the Laplace trans-form is that given by the convolution theorem [38]. Thistheorem is a powerful tool to find the inverse Laplace trans-form. According to this theorem if we have two transformedfunctions 𝑔(𝑠) = L{𝐺(𝑦)} and ℎ(𝑠) = L{𝐻(𝑦)}, thenthe product of these two is the Laplace transform of theconvolution (𝐺 ∗ 𝐻)(𝑦), where

(𝐺 ∗ 𝐻) (𝑦) = ∫

𝑦

0

𝐺 (𝑦 − 𝜏)𝐻 (𝜏) 𝑑𝜏. (5)

So the convolution theorem yields

L (𝐺 ∗ 𝐻) (𝑦) = 𝑔 (𝑠) ℎ (𝑠) . (6)

Hence

L−1

{𝑔 (𝑠) ℎ (𝑠)} = ∫

𝑦

0

𝐺 (𝑦 − 𝜏)𝐻 (𝜏) 𝑑𝜏. (7)

If we substitute 𝑤 = 𝑦 − 𝜏, then we find the importantconsequence 𝐺 ∗ 𝐻 = 𝐻 ∗ 𝐺.

3. Bound State Spectrum

The𝑁-dimensional time-independent Schrodinger equationfor a particle of mass 𝑀 with orbital angular momentumquantum number ℓ is given by [39]

[∇2

𝑁+2𝑀

ℏ2(𝐸 − 𝑉 (𝑟))] 𝜓

𝑛ℓ𝑚(𝑟, Ω𝑁) = 0, (8)

where 𝐸 and 𝑉(𝑟) denote the energy eigenvalues and poten-tial. Ω

𝑁within the argument of the 𝑛th state eigenfunctions

𝜓𝑛ℓ𝑚

denotes angular variables 𝜃1, 𝜃2, 𝜃3, . . . , 𝜃

𝑁−2, 𝜑. The

Laplacian operator in hyperspherical coordinates is writtenas

∇2

𝑁=

1

𝑟𝑁−1

𝜕

𝜕𝑟(𝑟𝑁−1

𝜕

𝜕𝑟) −

Λ2

𝑁−1

𝑟2, (9)

where

Λ2

𝑁−1= −[

𝑁−2

∑

𝑘=1

1

sin2𝜃𝑘+1

sin2𝜃𝑘+2

⋅ ⋅ ⋅ sin2𝜃𝑁−2

sin2𝜑

⋅ (1

sin𝑘−1𝜃𝑘

𝜕

𝜕𝜃𝑘

sin𝑘−1𝜃𝑘

𝜕

𝜕𝜃𝑘

) +1

sin𝑁−2𝜑𝜕

𝜕𝜑

⋅ sin𝑁−2𝜑 𝜕

𝜕𝜑] .

(10)

Λ2

𝑁−1is known as the hyperangular momentum operator.

Advances in High Energy Physics 3

We chose the bound state eigenfunctions 𝜓𝑛ℓ𝑚

(𝑟, Ω𝑁)

that are vanishing for 𝑟 → 0 and 𝑟 → ∞. Applyingthe separation variable method by means of the solution𝜓𝑛ℓ𝑚

(𝑟, Ω𝑁) = 𝑅

𝑛ℓ(𝑟)𝑌𝑚

ℓ(Ω𝑁), (8) provides two separated

equations:

Λ2

𝑁−1𝑌𝑚

ℓ(Ω𝑁) = ℓ (ℓ + 𝑁 − 2) 𝑌

𝑚

ℓ(Ω𝑁) , (11)

where 𝑌𝑚

ℓ(Ω𝑁) is known as the hyperspherical harmonics,

and the hyperradial or in short the “radial” equation

[𝑑2

𝑑𝑟2+𝑁 − 1

𝑟

𝑑

𝑑𝑟−ℓ (ℓ + 𝑁 − 2)

𝑟2

−2𝑀

ℏ2[𝑉 (𝑟) − 𝐸]]𝑅

𝑛ℓ(𝑟) = 0,

(12)

where ℓ(ℓ+𝑁−2)|𝑁>1

is the separation constant [40, 41] withℓ = 0, 1, 2, . . ..

In spite of taking (1) we take the more general form ofpseudoharmonic potential [42]

𝑉 (𝑟) = 𝑎1𝑟2

+𝑎2

𝑟2+ 𝑎3, (13)

where 𝑎1, 𝑎2, and 𝑎

3are three parameters that can take any

real value. If we set 𝑎1= 𝐷𝑒/𝑟2

𝑒, 𝑎2= 𝐷𝑒𝑟2

𝑒, and 𝑎

3= −2𝐷

𝑒,

(13) converts into the special case which we have given in (1).Taking this into (12) and using the abbreviations

] (] + 1) = ℓ (ℓ + 𝑁 − 2) +2𝑀

ℏ2𝑎2,

𝜇2

=2𝑀

ℏ2𝑎1,

𝜖2

=2𝑀

ℏ2(𝐸 − 𝑎

3) ,

(14)

we obtain

[𝑑2

𝑑𝑟2+𝑁 − 1

𝑟

𝑑

𝑑𝑟−] (] + 1)

𝑟2− 𝜇2

𝑟2

+ 𝜖2

]𝑅𝑛ℓ(𝑟)

= 0.

(15)

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

−𝑘

𝑓(𝑟)

with 𝑘 > 0. Here the term 𝑟−𝑘 ensures the fact that 𝑅

𝑛ℓ(𝑟 →

∞) = 0 and 𝑓(𝑟) is expected to behave like 𝑓(𝑟 → 0) = 0.Now changing the variable as 𝑦 = 𝑟

2 and taking 𝑓(𝑟) ≡

𝜒(𝑦) from (15) we obtain

4𝑦𝑑2

𝜒

𝑑𝑦2+ 2 (𝑁 − 2𝑘)

𝑑𝜒

𝑑𝑦

+ [𝑘 (𝑘 + 1) − 𝑘 (𝑁 − 1) − ] (] + 1)

𝑦− 𝜇2

𝑦 + 𝜖2

] 𝜒

= 0.

(16)

In order to get an exact solution of the above differentialequation we remove the singular term by imposing thecondition

𝑘 (𝑘 + 1) − 𝑘 (𝑁 − 1) − ] (] + 1) = 0. (17)

So we have

𝑦𝑑2

𝜒

𝑑𝑦2− (𝑘ℓ𝑁

−𝑁

2)𝑑𝜒

𝑑𝑦−1

4(𝜇2

𝑦 − 𝜖2

) 𝜒 = 0, (18)

where 𝑘ℓ𝑁

is taken as the positive solution of 𝑘, which can beeasily found by little algebra from (17). It is worthmentioninghere that the condition given by (17) is necessary to getan analytical solution because otherwise only approximateor numerical solution is possible. Introducing the Laplacetransform 𝐹(𝑠) = L{𝜒(𝑦)} with the boundary condition𝑓(0) ≡ 𝜒(0) = 0 and using (4), (18) can read

(𝑠2

−𝜇2

4)𝑑𝐹

𝑑𝑠+ (𝜂ℓ𝑁𝑠 −

𝜖2

4)𝐹 = 0, (19)

where 𝜂ℓ𝑁

= 𝑘ℓ𝑁

− 𝑁/2 + 2. This parameter can have integeror noninteger values and there will be integer or nonintegerterm(s) in energy eigenvalue according to values of 𝜂

ℓ𝑁which

can be seen below.The solution of the last equation can be written easily as

𝐹 (𝑠) = 𝐶(𝑠 +𝜇

2)

−𝜂ℓ𝑁/2−𝜖2/4𝜇

(𝑠 −𝜇

2)

−𝜂ℓ𝑁/2+𝜖2/4𝜇

, (20)

where 𝐶 is a constant. The term 𝜇 = √2𝑀𝑎1/ℏ is a positive

real number aswe 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 powermustbe a positive integer to get singled valued eigenfunctions.Thiswill also exclude the possibility of getting singularity in thetransformation. So we have

𝜖2

4𝜇−1

2𝜂ℓ𝑁

= 𝑛, 𝑛 = 0, 1, 2, . . . . (21)

Using (21), we have from (20)

𝐹 (𝑠) = 𝐶(𝑠 +𝜇

2)

−𝑎

(𝑠 −𝜇

2)

−𝑏

= 𝐶𝑔 (𝑠) ℎ (𝑠) , (22)

where 𝑎 = 𝜂ℓ𝑁

+ 𝑛 and 𝑏 = −𝑛. In order to find 𝜒(𝑦) =

L−1{𝐹(𝑠)}, we find [43]

L−1

{(𝑠 +𝜇

2)

−𝑎

} = 𝐺 (𝑦) =𝑦𝑎−1

𝑒−(𝜇/2)𝑦

Γ (𝑎),

L−1

{(𝑠 −𝜇

2)

−𝑏

} = 𝐻 (𝑦) =𝑦𝑏−1

𝑒(𝜇/2)𝑦

Γ (𝑏).

(23)

Therefore using (7) and (23), we have

𝜒 (𝑦) = L−1

{𝐹 (𝑠)} = 𝐶 (𝐺 ∗ 𝐻) (𝑦)

= 𝐶∫

𝑦

0

𝐺 (𝑦 − 𝜏)𝐻 (𝜏) 𝑑𝜏

=𝐶𝑒−(𝜇/2)𝑦

Γ (𝑎) Γ (𝑏)∫

𝑦

0

(𝑦 − 𝜏)𝑎−1

𝜏𝑏−1

𝑒𝜇𝜏

𝑑𝜏.

(24)


The integration can be found in [44], which gives

∫

𝑦

0

(𝑦 − 𝜏)𝑎−1

𝜏𝑏−1

𝑒𝜇𝜏

𝑑𝜏

= 𝐵 (𝑎, 𝑏) 𝑦𝑎+𝑏−1

1𝐹1(𝑏, 𝑎 + 𝑏, 𝜇𝑦) ,

(25)

where1𝐹1is the confluent hypergeometric functions. Now

using the beta function 𝐵(𝑎, 𝑏) = Γ(𝑎)Γ(𝑏)/Γ(𝑎 + 𝑏), 𝜒(𝑦) canbe written:

𝜒 (𝑦) =𝐶

Γ (𝑎 + 𝑏)𝑒−(𝜇/2)𝑦

𝑦𝑎+𝑏−1

1𝐹1(𝑏, 𝑎 + 𝑏, 𝜇𝑦) . (26)

So we have the radial eigenfunctions

𝑅𝑛ℓ𝑁

(𝑟) = 𝑟−𝑘ℓ𝑁𝑓 (𝑟)

= 𝐶𝑛ℓ𝑁

𝑒−(𝜇/2)𝑟

2

𝑟(𝜂ℓ𝑁−𝑁/2)

1𝐹1(−𝑛, 𝜂

ℓ𝑁, 𝜇𝑟2

) ,

(27)

where 𝐶𝑛ℓ𝑁

is the normalization constant. It should be notedthat because of the “quantization condition” given by (21), itis possible to write the radial eigenfunctions in a polynomialform 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 constant 𝐶

𝑛ℓ𝑁can be evaluated

from the condition [45]

∫

∞

0

[𝑅𝑛ℓ𝑁

(𝑟)]2

𝑟𝑁−1

𝑑𝑟 = 1. (28)

To evaluate the integration the formula1𝐹1(−𝑞, 𝑝 + 1, 𝑧) =

(𝑞!𝑝!/(𝑞 + 𝑝)!)𝐿𝑝

𝑞(𝑧) is useful here, where 𝐿

𝑝

𝑞(𝑢) are the

Laguerre polynomials. It should be remembered that theformula is applicable only if 𝑞 is a positive integer. Henceidentifying 𝑞 = 𝑛, 𝑝 = 𝜂

ℓ𝑁− 1, and 𝑧 = 𝜇𝑟

2 we have

1𝐹1(−𝑛, 𝜂

ℓ𝑁, 𝜇𝑟2

) =𝑛! (𝜂ℓ𝑁

− 1)!

(𝜂ℓ𝑁

+ 𝑛 − 1)!𝐿(𝜂ℓ𝑁−1)

𝑛(𝜇𝑟2

) . (29)

So using the following formula for the Laguerre polynomials,

∫

∞

0

𝑥𝐴

𝑒−𝑥

[𝐿𝐴

𝐵(𝑥)]2

𝑑𝑥 =Γ (𝐴 + 𝐵 + 1)

𝐵!, (30)

we write the normalization constant

𝐶𝑛ℓ𝑁

= √2𝜇(1/2)𝜂ℓ𝑁√

𝑛!

Γ (𝜂ℓ𝑁

+ 𝑛)

(𝜂ℓ𝑁

+ 𝑛 − 1)!

𝑛! (𝜂ℓ𝑁

− 1)!. (31)

Finally, the energy eigenvalues are obtained from (21) alongwith (17) and (14):

𝐸𝑛ℓ𝑁

=ℏ2

2𝑀𝜖2

+ 𝑎3

= 𝑎3

+ √8ℏ2

𝑎1

𝑀[𝑛 +

1

2+1

4

√(𝑁 + 2ℓ − 2)2

+8𝑀𝑎2

ℏ2] ,

(32)

and we write the corresponding normalized eigenfunctionsas

𝑅𝑛ℓ𝑁

(𝑟) = √2𝜇(1/2)𝜂ℓ𝑁√

𝑛!

Γ (𝜂ℓ𝑁

+ 𝑛)

(𝜂ℓ𝑁

+ 𝑛 − 1)!

𝑛! (𝜂ℓ𝑁

− 1)!

⋅ 𝑒−(𝜇/2)𝑟

2

𝑟(𝜂ℓ𝑁−𝑁/2)

1𝐹1(−𝑛, 𝜂

ℓ𝑁, 𝜇𝑟2

) ,

(33)

or𝑅𝑛ℓ𝑁

(𝑟)

= √2𝜇(1/2)𝜂ℓ𝑁√

𝑛!

Γ (𝜂ℓ𝑁

+ 𝑛)𝑒−𝜇𝑟2/2

𝑟(𝜂ℓ𝑁−𝑁/2)𝐿

(𝜂ℓ𝑁−1)

𝑛(𝜇𝑟2

) .

(34)

Finally, the complete orthonormalized eigenfunctions of the𝑁-dimensional Schrodinger equation with pseudoharmonicpotential can be given by

𝜓 (𝑟, 𝜃1, 𝜃2, . . . , 𝜃

𝑁−2, 𝜙)

= ∑

𝑛,ℓ,𝑚

𝐶𝑛ℓ𝑁

𝑅𝑛ℓ𝑁

(𝑟) 𝑌𝑚

ℓ(𝜃1, 𝜃2, . . . , 𝜃

𝑁−2, 𝜙) .

(35)

We give some numerical results about the variation ofthe energy on the dimensionality 𝑁 obtained from (32) inFigure 1. We summarize the plots for 𝐸

𝑛(𝑁) as a function of

𝑁 for a set of physical parameters (𝑎1, 𝑎2) by taking 𝑎

3= 0,

especially.

4. Results and Discussion

In this section we have shown that the results obtained inSection 3 are very useful in deriving the special cases ofseveral potentials for lower as well as for higher dimensionalwave equation.

4.1. Isotropic Harmonic Oscillator

(1) Three Dimensions (𝑁 = 3). For this case 𝑎1= (1/2)𝑀𝜔

2

and 𝑎2= 𝑎3= 0 which gives from (32)

𝐸𝑛ℓ3

= ℏ𝜔(2𝑛 + 𝑙 +3

2) , (36)

where 𝜔 is the circular frequency of the particle. From (14)and (17) we obtain 𝑘

ℓ3= ℓ + 1. This makes 𝜂

ℓ3= ℓ + 3/2 and

we get radial eigenfunctions from (34).The result agrees withthose obtained in [22].

(2) Arbitrary 𝑁-Dimensions. Here𝑁 is an arbitrary constantand as before 𝑎

1= (1/2)𝑀𝜔

2, 𝑎2= 𝑎3= 0. We have the

energy eigenvalues from (32):

𝐸𝑛ℓ𝑁

= ℏ𝜔(2𝑛 + ℓ +𝑁

2) . (37)

Solving (17) with the help of (14) we have 𝑘ℓ𝑁

= ℓ+𝑁−2 andone can easily obtain the normalization constant from (31):

𝐶𝑛ℓ𝑁

= [2 (𝑀𝜔/ℏ)

(ℓ+𝑁/2)

𝑛!

Γ (ℓ + 𝑁/2 + 𝑛)]

1/2

(𝑛 + ℓ + (𝑁 − 2) /2)!

𝑛! (ℓ + (𝑁 − 2) /2)!,

(38)


2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

0 0.5 1 1.5 2 2.5 3 3.5

E(N

)

N

l = 0

(a) The variation of energy to𝑁 for 𝑛 = 0

5.2

5.4

5.6

5.8

6

6.2

6.4

6.6

0 0.5 1 1.5 2 2.5 3 3.5

E(N

)

N

l = 0l = 1(b) The variation of energy to𝑁 for 𝑛 = 1

8

8.5

9

9.5

10

10.5

11

0 0.5 1 1.5 2 2.5 3 3.5

E(N

)

N

l = 0l = 1

l = 2

(c) The variation of energy to𝑁 for 𝑛 = 2

10.5

11

11.5

12

12.5

13

13.5

14

14.5

15

0 0.5 1 1.5 2 2.5 3 3.5

E(N

)

N

l = 0l = 1

l = 2l = 3

(d) The variation of energy to𝑁 for 𝑛 = 3

Figure 1: The dependence of 𝐸(𝑁) on𝑁. Parameters: 2𝑀 = 1, ℏ = 1, and 𝑎1= 𝑎2= 0.5.

where 𝜂ℓ𝑁

= ℓ + 𝑁/2. The radial eigenfunctions are givenin (34) with the above normalization constant. The resultsobtained here agree with those found in some earlier works[22, 46, 47].

4.2. Isotropic Harmonic Oscillator PlusInverse Quadratic Potential

(1) Two Dimensions (𝑁 = 2). Here we have 𝑎1= (1/2)𝑀𝜔

2,𝑎2

= 0, and 𝑎3= 0, where 𝜔 is the circular frequency of the

particle. So from (31) we obtain

𝐶𝑛ℓ2

= [2 (𝑀𝜔/ℏ)

𝑘ℓ2+1 𝑛!

Γ (𝑘ℓ2+ 𝑛 + 1)

]

1/2

(𝑘ℓ2+ 𝑛)!

𝑛!𝑘ℓ2!

, (39)

where 𝜂ℓ2= 𝑘ℓ2+ 1. 𝑘ℓ2can be obtained from (14) and (17) as

𝑘ℓ2

= √](] + 1) = √ℓ2 + (2𝑀/ℏ2)𝑎2. Hence (32) gives the

energy eigenvalues of the system𝐸𝑛ℓ2

= ℏ𝜔 (2𝑛 + 𝑘ℓ2+ 1) . (40)

This result has already been obtained in [22, 48].

(2) Three Dimensions (𝑁 = 3). Here 𝑎1= (1/2)𝑀𝜔

2, 𝑎2

= 0,and 𝑎3= 0 which gives the energy eigenvalues as

𝐸𝑛ℓ3

=ℏ𝜔

2[4𝑛 + 2 + √(2ℓ + 1)

2

+8𝑀𝑎2

ℏ2] . (41)

Solving (17) we get 𝑘ℓ3

= ] + 1 and hence 𝜂ℓ3

= ] + 3/2. Theradial eigenfunction can hence be obtained from (34) whichalso corresponds to the result obtained in [22].


4.3. 3-Dimensional Schrodinger Equation with Pseudohar-monic Potential. Here (17) gives 𝑘

ℓ3= ] + 1 and this makes

𝜂ℓ3= ] + 3/2. So (31) provides

𝐶𝑛ℓ3

= 𝜇(1/2)(]+3/2)√

2Γ (] + 𝑛 + 3/2)

𝑛![Γ (] +

3

2)]

−1

, (42)

and hence the normalized eigenfunctions become𝑅𝑛ℓ3

(𝑟)

= 𝜇(1/2)(]+3/2)√

2Γ (] + 𝑛 + 3/2)

𝑛![Γ (] +

3

2)]

−1

⋅ 𝑒−(𝜇/2)𝑟

2

𝑟]+11𝐹1(−𝑛, ] +

3

2, 𝜇𝑟2

) ,

(43)

with the energy eigenvalues

𝐸𝑛ℓ3

= 𝑎3+

ℏ2

2𝑀𝜖2

= 𝑎3

+ √8ℏ2

𝑎1

𝑀[𝑛 +

1

2+1

4

√(2ℓ + 1)2

+8𝑀𝑎2

ℏ2] .

(44)

This result corresponds exactly to the ones given in [13].

5. Short Review of Generalized MorsePotential: An Example

The generalizedMorse potential [49] in terms of four param-eters 𝑉

1, 𝑉2, 𝑉3, and 𝜆(> 0) is

𝑉 (𝑟) = 𝑉1𝑒−𝜆(𝑟−𝑟𝑒) + 𝑉

2𝑒−2𝜆(𝑟−𝑟𝑒) + 𝑉

3, (45)

where 𝜆 describes the characteristic range of the potential,𝑟𝑒is the equilibrium molecular separation, and 𝑉

1, 𝑉2, and

𝑉3are related to the potential depth. In this short section

wewill only show Laplace transformable differential equationlike (18) can also be achieved if the exponential and singularterms 1/𝑟 and 1/𝑟2 are properly handled. Looking back to thesolution given by (34) we predetermine the solution for (8)for the potential given by (45) as

𝜓𝑛ℓ𝑚

(𝑟, Ω𝑁) = 𝑟−(𝑁−1)/2

𝑅𝑛ℓ(𝑟) 𝑌𝑚

ℓ(Ω𝑁) . (46)

This substitution facilitates following easier differential equa-tion (without the 1/𝑟 term) similar to (12) of Section 3:

[𝑑2

𝑑𝑟2+2𝑀

ℏ2(𝐸 − 𝑉 (𝑟)) −

𝐴ℓ

𝑁

𝑟2]𝑅𝑛ℓ(𝑟) = 0, (47)

where 𝐴ℓ𝑁= (𝑁 + 2ℓ − 1)(𝑁 + 2ℓ − 3)/4.

Let us introduce a new variable 𝑥 = (𝑟 − 𝑟𝑒)/𝑟𝑒. Hence

inserting (45) into (47) we have

[𝑑2

𝑑𝑥2+2𝑀𝑟2

𝑒

ℏ2{𝐸 − 𝑉

1𝑒−𝛼𝑥

− 𝑉2𝑒−2𝛼𝑥

− 𝑉3}

+𝐴ℓ

𝑁

(1 + 𝑥)2]𝑅 (𝑥) = 0,

(48)

where 𝛼 = 𝜆𝑟𝑒.

It is possible to expand the term 1/(1+𝑥)2 in power series

as within the molecular point of view 𝑥 ≪ 1. So

1

(1 + 𝑥)2≈ 1 − 2𝑥 + 3𝑥

2

− 𝑂 (𝑥3

) . (49)

This expansion can also be rewritten as

1

(1 + 𝑥)2≈ 𝑐0+ 𝑐1𝑒−𝛼𝑥

+ 𝑐2𝑒−2𝛼𝑥

. (50)

By comparing these two it is not hard to justify that 𝑐0= 1 −

3/𝛼 + 3/𝛼2; 𝑐1= 4/𝛼 − 6/𝛼

2; 𝑐2= −1/𝛼 + 3/𝛼

2.Inserting (50) into (48) and changing the variable as 𝑦 =

𝑒−𝛼𝑥 one can easily get

[𝑦2𝑑2

𝑑𝑦2+ 𝑦

𝑑

𝑑𝑦+

𝜖

𝛼2+𝐵1

𝛼2𝑦 +

𝐵2

𝛼2𝑦2

]𝑅 (𝑦) = 0. (51)

Now further assuming the solution of the above differentialequation 𝑅(𝑦) = 𝑦

−𝜔

𝑔(𝑦), as we did in the previous sectionwith proper requirement for bound state scenario, finally wecan construct the following differential equation just like (16)for the perfect platform for Laplace transformation:

𝑦𝑑2

𝑔

𝑑𝑦2+ (1 − 2𝜔)

𝑑𝑔

𝑑𝑦+𝜔2

− 𝜖/𝛼2

𝑦𝑔 + (

𝐵1

𝛼2+𝐵3

𝛼2𝑦)𝑔

= 0,

(52)

where

𝜖 =2𝑀𝑟2

𝑒

ℏ2(𝐸 − 𝑉

3) + 𝑐0𝐴ℓ

𝑁, (53a)

𝐵1= 𝑐1𝐴ℓ

𝑁−2𝑀𝑟2

𝑒

ℏ2𝑉1, (53b)

𝐵2= 𝑐2𝐴ℓ

𝑁−2𝑀𝑟2

𝑒

ℏ2𝑉2. (53c)

Imposing the condition 𝜔2

− 𝜖/𝛼2

= 0 as previously andapproaching the same way as we did in Section 3 one canobtain the energy eigenvalues and bound statewave functionsin terms of confluent hypergeometric function. We haveinvestigated that the results are exact match with [15] for𝑉1= −2𝐷, 𝑉

2= 𝐷, and 𝑉

3= 0, where 𝐷 describes the depth

of the potential.

6. Conclusions

We have investigated some aspects of𝑁-dimensional hyper-radial Schrodinger equation for pseudoharmonic potentialby Laplace transformation approach. It is found that theenergy eigenfunctions and the energy eigenvalues dependon the dimensionality of the problem. In this connection wehave furnished few plots of the energy spectrum 𝐸

𝑛(𝑁) as a

function of𝑁 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 potentialparameters and dimensionality.

The Laplace transform is a powerful, efficient, and accu-rate alternative method of deriving energy eigenvalues andeigenfunctions of some spherically symmetric potentials thatare analytically solvable. It may be hard to predict which kindof potentials is solvable analytically by Laplace transform, butin general prediction of the eigenfunctions with the form like𝑅𝑛ℓ(𝑟) = 𝑟

−𝑘

𝑓(𝑟) always opens the gate of the possibility ofclosed form solutions for a particular potential model. Thiskind of substitution is calledUniversal Laplace transformationscheme. In this connection one might go through [50] tocheck out how Laplace transformation technique behavesover different potentials, specially for Schrodinger equationin lower dimensional domain. It is also true that the techniqueof Laplace transformation is useful if, inserting the potentialinto the Schrodinger equation and using Universal Laplacetransformation scheme via some suitable parametric restric-tions, one is able to get a differential equation with variablecoefficient 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 thesituation [51]. The results are sufficiently accurate for suchspecial potentials at least for practical purpose.

Before concluding we want to mention here that we havenot succeeded in developing the scattering state solution forthe pseudoharmonic potential. If we could develop thosesolutions using the LTA that would have been a remarkableachievement. The main barrier of this success lies on therealization of complex index in (20), because at scatteringstate situation, that is,𝐸 > lim

𝑟→∞𝑉(𝑟), 𝜇 should be replaced

with 𝜇 = 𝑖𝜁, where 𝜁 = √2𝑀𝑎1/ℏ2. Maybe this cumbersome

situation would attract the researchers to study the subjectfurther and we are looking forward to it.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The author Altug Arda thanks Dr. Andreas Fring from CityUniversity London and the Department of Mathematics forhospitality where the final version of this work has beencompleted.

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