dimensional radial schrรถdinger equation with pseudoharmonic
TRANSCRIPT
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Research ArticleExact Analytical Solution of the๐-Dimensional RadialSchrรถdinger 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 ยฉ 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 ๐-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 ๐ธ
๐as a function of dimension๐ 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 [5โ7], 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, ๐-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 ๐-dimensional Schrodinger equations inthe large ๐ limit [19], where a relevant 1/๐ expansion canbe used. Later Yanez et al. have investigated the positionand momentum information entropies of ๐-dimensionalsystem [20]. The quantization of angular momentum in๐-dimensions has been described by Al-Jaber [21]. Otherrecent studies of Schrodinger equation in higher dimensioninclude isotropic harmonic oscillator plus inverse quadraticpotential [22], ๐-dimensional radial Schrodinger equation
Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015, Article ID 137038, 8 pageshttp://dx.doi.org/10.1155/2015/137038
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2 Advances in High Energy Physics
with the Coulomb potential [23]. Some recent works on๐-dimensional Schrodinger equation can be found in thereferences list [24โ31].
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 appropriate๐. The pseudoharmonic potential is expressed in the form[32]
๐ (๐) = ๐ท๐(๐
๐๐
โ๐๐
๐)
2
, (1)
where ๐ท๐= (1/8)๐พ
๐๐2
๐is the dissociation energy with the
force constant ๐พ๐and ๐
๐is 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 ๐-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 equationin๐-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 in๐-dimension.
Motivated by these types of works, in this present paperwe discuss the exact solutions of the ๐-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 transform๐น(๐ ) orL of a function๐(๐ฆ) is definedby [36, 37]
๐น (๐ ) = L {๐ (๐ฆ)} = โซ
โ
0
๐โ๐ ๐ฆ
๐ (๐ฆ) ๐๐ฆ. (2)
If there is some constant ๐ โ R such that |๐โ๐๐ฆ๐(๐ฆ)| โค
M for sufficiently large ๐ฆ, the integral in (2) will exist forRe ๐ > ๐. The Laplace transformmay fail to exist because of asufficiently strong singularity in the function ๐(๐ฆ) as ๐ฆ โ 0.In particular
L [๐ฆ๐ผ
ฮ (๐ผ + 1)] =
1
๐ ๐ผ+1, ๐ผ > โ1. (3)
The Laplace transform has the derivative properties
L {๐(๐)
(๐ฆ)} = ๐ ๐
L {๐ (๐ฆ)} โ
๐โ1
โ
๐=0
๐ ๐โ1โ๐
๐(๐)
(0) ,
L {๐ฆ๐
๐ (๐ฆ)} = (โ1)๐
๐น(๐)
(๐ ) ,
(4)
where the superscript (๐) denotes the ๐th derivative withrespect to ๐ฆ for ๐(๐)(๐ฆ) and with respect to ๐ for ๐น(๐)(๐ ).
The inverse transform is defined as Lโ1{๐น(๐ )} = ๐(๐ฆ).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 ๐(๐ ) = L{๐บ(๐ฆ)} and โ(๐ ) = L{๐ป(๐ฆ)}, thenthe product of these two is the Laplace transform of theconvolution (๐บ โ ๐ป)(๐ฆ), where
(๐บ โ ๐ป) (๐ฆ) = โซ
๐ฆ
0
๐บ (๐ฆ โ ๐)๐ป (๐) ๐๐. (5)
So the convolution theorem yields
L (๐บ โ ๐ป) (๐ฆ) = ๐ (๐ ) โ (๐ ) . (6)
Hence
Lโ1
{๐ (๐ ) โ (๐ )} = โซ
๐ฆ
0
๐บ (๐ฆ โ ๐)๐ป (๐) ๐๐. (7)
If we substitute ๐ค = ๐ฆ โ ๐, then we find the importantconsequence ๐บ โ ๐ป = ๐ป โ ๐บ.
3. Bound State Spectrum
The๐-dimensional time-independent Schrodinger equationfor a particle of mass ๐ with orbital angular momentumquantum number โ is given by [39]
[โ2
๐+2๐
โ2(๐ธ โ ๐ (๐))] ๐
๐โ๐(๐, ฮฉ๐) = 0, (8)
where ๐ธ and ๐(๐) denote the energy eigenvalues and poten-tial. ฮฉ
๐within the argument of the ๐th state eigenfunctions
๐๐โ๐
denotes angular variables ๐1, ๐2, ๐3, . . . , ๐
๐โ2, ๐. The
Laplacian operator in hyperspherical coordinates is writtenas
โ2
๐=
1
๐๐โ1
๐
๐๐(๐๐โ1
๐
๐๐) โ
ฮ2
๐โ1
๐2, (9)
where
ฮ2
๐โ1= โ[
๐โ2
โ
๐=1
1
sin2๐๐+1
sin2๐๐+2
โ โ โ sin2๐๐โ2
sin2๐
โ (1
sin๐โ1๐๐
๐
๐๐๐
sin๐โ1๐๐
๐
๐๐๐
) +1
sin๐โ2๐๐
๐๐
โ sin๐โ2๐ ๐
๐๐] .
(10)
ฮ2
๐โ1is known as the hyperangular momentum operator.
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Advances in High Energy Physics 3
We chose the bound state eigenfunctions ๐๐โ๐
(๐, ฮฉ๐)
that are vanishing for ๐ โ 0 and ๐ โ โ. Applyingthe separation variable method by means of the solution๐๐โ๐
(๐, ฮฉ๐) = ๐
๐โ(๐)๐๐
โ(ฮฉ๐), (8) provides two separated
equations:
ฮ2
๐โ1๐๐
โ(ฮฉ๐) = โ (โ + ๐ โ 2) ๐
๐
โ(ฮฉ๐) , (11)
where ๐๐
โ(ฮฉ๐) is known as the hyperspherical harmonics,
and the hyperradial or in short the โradialโ equation
[๐2
๐๐2+๐ โ 1
๐
๐
๐๐โโ (โ + ๐ โ 2)
๐2
โ2๐
โ2[๐ (๐) โ ๐ธ]]๐
๐โ(๐) = 0,
(12)
where โ(โ+๐โ2)|๐>1
is the separation constant [40, 41] withโ = 0, 1, 2, . . ..
In spite of taking (1) we take the more general form ofpseudoharmonic potential [42]
๐ (๐) = ๐1๐2
+๐2
๐2+ ๐3, (13)
where ๐1, ๐2, and ๐
3are three parameters that can take any
real value. If we set ๐1= ๐ท๐/๐2
๐, ๐2= ๐ท๐๐2
๐, and ๐
3= โ2๐ท
๐,
(13) converts into the special case which we have given in (1).Taking this into (12) and using the abbreviations
] (] + 1) = โ (โ + ๐ โ 2) +2๐
โ2๐2,
๐2
=2๐
โ2๐1,
๐2
=2๐
โ2(๐ธ โ ๐
3) ,
(14)
we obtain
[๐2
๐๐2+๐ โ 1
๐
๐
๐๐โ] (] + 1)
๐2โ ๐2
๐2
+ ๐2
]๐ ๐โ(๐)
= 0.
(15)
We are looking for the bound state solutions for ๐ ๐โ(๐) with
the following properties: ๐ ๐โ(๐ โ 0) = ๐
โ๐ and ๐ ๐โ(๐ โ
โ) = 0. Let us assume a solution of type ๐ ๐โ(๐) = ๐
โ๐
๐(๐)
with ๐ > 0. Here the term ๐โ๐ ensures the fact that ๐
๐โ(๐ โ
โ) = 0 and ๐(๐) is expected to behave like ๐(๐ โ 0) = 0.Now changing the variable as ๐ฆ = ๐
2 and taking ๐(๐) โก
๐(๐ฆ) from (15) we obtain
4๐ฆ๐2
๐
๐๐ฆ2+ 2 (๐ โ 2๐)
๐๐
๐๐ฆ
+ [๐ (๐ + 1) โ ๐ (๐ โ 1) โ ] (] + 1)
๐ฆโ ๐2
๐ฆ + ๐2
] ๐
= 0.
(16)
In order to get an exact solution of the above differentialequation we remove the singular term by imposing thecondition
๐ (๐ + 1) โ ๐ (๐ โ 1) โ ] (] + 1) = 0. (17)
So we have
๐ฆ๐2
๐
๐๐ฆ2โ (๐โ๐
โ๐
2)๐๐
๐๐ฆโ1
4(๐2
๐ฆ โ ๐2
) ๐ = 0, (18)
where ๐โ๐
is taken as the positive solution of ๐, 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 ๐น(๐ ) = L{๐(๐ฆ)} with the boundary condition๐(0) โก ๐(0) = 0 and using (4), (18) can read
(๐ 2
โ๐2
4)๐๐น
๐๐ + (๐โ๐๐ โ
๐2
4)๐น = 0, (19)
where ๐โ๐
= ๐โ๐
โ ๐/2 + 2. This parameter can have integeror noninteger values and there will be integer or nonintegerterm(s) in energy eigenvalue according to values of ๐
โ๐which
can be seen below.The solution of the last equation can be written easily as
๐น (๐ ) = ๐ถ(๐ +๐
2)
โ๐โ๐/2โ๐2/4๐
(๐ โ๐
2)
โ๐โ๐/2+๐2/4๐
, (20)
where ๐ถ is a constant. The term ๐ = โ2๐๐1/โ is a positive
real number aswe restrict ourselves to the choice ๐1> 0. Now,
since ๐ is positive and ๐ > 0, then the second factor of (20)could become negative if ๐/2 > ๐ > 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
๐2
4๐โ1
2๐โ๐
= ๐, ๐ = 0, 1, 2, . . . . (21)
Using (21), we have from (20)
๐น (๐ ) = ๐ถ(๐ +๐
2)
โ๐
(๐ โ๐
2)
โ๐
= ๐ถ๐ (๐ ) โ (๐ ) , (22)
where ๐ = ๐โ๐
+ ๐ and ๐ = โ๐. In order to find ๐(๐ฆ) =
Lโ1{๐น(๐ )}, we find [43]
Lโ1
{(๐ +๐
2)
โ๐
} = ๐บ (๐ฆ) =๐ฆ๐โ1
๐โ(๐/2)๐ฆ
ฮ (๐),
Lโ1
{(๐ โ๐
2)
โ๐
} = ๐ป (๐ฆ) =๐ฆ๐โ1
๐(๐/2)๐ฆ
ฮ (๐).
(23)
Therefore using (7) and (23), we have
๐ (๐ฆ) = Lโ1
{๐น (๐ )} = ๐ถ (๐บ โ ๐ป) (๐ฆ)
= ๐ถโซ
๐ฆ
0
๐บ (๐ฆ โ ๐)๐ป (๐) ๐๐
=๐ถ๐โ(๐/2)๐ฆ
ฮ (๐) ฮ (๐)โซ
๐ฆ
0
(๐ฆ โ ๐)๐โ1
๐๐โ1
๐๐๐
๐๐.
(24)
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4 Advances in High Energy Physics
The integration can be found in [44], which gives
โซ
๐ฆ
0
(๐ฆ โ ๐)๐โ1
๐๐โ1
๐๐๐
๐๐
= ๐ต (๐, ๐) ๐ฆ๐+๐โ1
1๐น1(๐, ๐ + ๐, ๐๐ฆ) ,
(25)
where1๐น1is the confluent hypergeometric functions. Now
using the beta function ๐ต(๐, ๐) = ฮ(๐)ฮ(๐)/ฮ(๐ + ๐), ๐(๐ฆ) canbe written:
๐ (๐ฆ) =๐ถ
ฮ (๐ + ๐)๐โ(๐/2)๐ฆ
๐ฆ๐+๐โ1
1๐น1(๐, ๐ + ๐, ๐๐ฆ) . (26)
So we have the radial eigenfunctions
๐ ๐โ๐
(๐) = ๐โ๐โ๐๐ (๐)
= ๐ถ๐โ๐
๐โ(๐/2)๐
2
๐(๐โ๐โ๐/2)
1๐น1(โ๐, ๐
โ๐, ๐๐2
) ,
(27)
where ๐ถ๐โ๐
is the normalization constant. It should be notedthat because of the โquantization conditionโ given by (21), itis possible to write the radial eigenfunctions in a polynomialform of degree ๐, as
1๐น1(๐, ๐ + ๐, ๐๐ฆ) converges for all finite
๐๐ฆ with ๐ = โ๐ and (๐ + ๐) is not a negative integer or zero.Now the normalization constant ๐ถ
๐โ๐can be evaluated
from the condition [45]
โซ
โ
0
[๐ ๐โ๐
(๐)]2
๐๐โ1
๐๐ = 1. (28)
To evaluate the integration the formula1๐น1(โ๐, ๐ + 1, ๐ง) =
(๐!๐!/(๐ + ๐)!)๐ฟ๐
๐(๐ง) is useful here, where ๐ฟ
๐
๐(๐ข) are the
Laguerre polynomials. It should be remembered that theformula is applicable only if ๐ is a positive integer. Henceidentifying ๐ = ๐, ๐ = ๐
โ๐โ 1, and ๐ง = ๐๐
2 we have
1๐น1(โ๐, ๐
โ๐, ๐๐2
) =๐! (๐โ๐
โ 1)!
(๐โ๐
+ ๐ โ 1)!๐ฟ(๐โ๐โ1)
๐(๐๐2
) . (29)
So using the following formula for the Laguerre polynomials,
โซ
โ
0
๐ฅ๐ด
๐โ๐ฅ
[๐ฟ๐ด
๐ต(๐ฅ)]2
๐๐ฅ =ฮ (๐ด + ๐ต + 1)
๐ต!, (30)
we write the normalization constant
๐ถ๐โ๐
= โ2๐(1/2)๐โ๐โ
๐!
ฮ (๐โ๐
+ ๐)
(๐โ๐
+ ๐ โ 1)!
๐! (๐โ๐
โ 1)!. (31)
Finally, the energy eigenvalues are obtained from (21) alongwith (17) and (14):
๐ธ๐โ๐
=โ2
2๐๐2
+ ๐3
= ๐3
+ โ8โ2
๐1
๐[๐ +
1
2+1
4
โ(๐ + 2โ โ 2)2
+8๐๐2
โ2] ,
(32)
and we write the corresponding normalized eigenfunctionsas
๐ ๐โ๐
(๐) = โ2๐(1/2)๐โ๐โ
๐!
ฮ (๐โ๐
+ ๐)
(๐โ๐
+ ๐ โ 1)!
๐! (๐โ๐
โ 1)!
โ ๐โ(๐/2)๐
2
๐(๐โ๐โ๐/2)
1๐น1(โ๐, ๐
โ๐, ๐๐2
) ,
(33)
or๐ ๐โ๐
(๐)
= โ2๐(1/2)๐โ๐โ
๐!
ฮ (๐โ๐
+ ๐)๐โ๐๐2/2
๐(๐โ๐โ๐/2)๐ฟ
(๐โ๐โ1)
๐(๐๐2
) .
(34)
Finally, the complete orthonormalized eigenfunctions of the๐-dimensional Schrodinger equation with pseudoharmonicpotential can be given by
๐ (๐, ๐1, ๐2, . . . , ๐
๐โ2, ๐)
= โ
๐,โ,๐
๐ถ๐โ๐
๐ ๐โ๐
(๐) ๐๐
โ(๐1, ๐2, . . . , ๐
๐โ2, ๐) .
(35)
We give some numerical results about the variation ofthe energy on the dimensionality ๐ obtained from (32) inFigure 1. We summarize the plots for ๐ธ
๐(๐) as a function of
๐ for a set of physical parameters (๐1, ๐2) by taking ๐
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 (๐ = 3). For this case ๐1= (1/2)๐๐
2
and ๐2= ๐3= 0 which gives from (32)
๐ธ๐โ3
= โ๐(2๐ + ๐ +3
2) , (36)
where ๐ is the circular frequency of the particle. From (14)and (17) we obtain ๐
โ3= โ + 1. This makes ๐
โ3= โ + 3/2 and
we get radial eigenfunctions from (34).The result agrees withthose obtained in [22].
(2) Arbitrary ๐-Dimensions. Here๐ is an arbitrary constantand as before ๐
1= (1/2)๐๐
2, ๐2= ๐3= 0. We have the
energy eigenvalues from (32):
๐ธ๐โ๐
= โ๐(2๐ + โ +๐
2) . (37)
Solving (17) with the help of (14) we have ๐โ๐
= โ+๐โ2 andone can easily obtain the normalization constant from (31):
๐ถ๐โ๐
= [2 (๐๐/โ)
(โ+๐/2)
๐!
ฮ (โ + ๐/2 + ๐)]
1/2
(๐ + โ + (๐ โ 2) /2)!
๐! (โ + (๐ โ 2) /2)!,
(38)
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Advances in High Energy Physics 5
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 to๐ for ๐ = 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 to๐ for ๐ = 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 to๐ for ๐ = 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 to๐ for ๐ = 3
Figure 1: The dependence of ๐ธ(๐) on๐. Parameters: 2๐ = 1, โ = 1, and ๐1= ๐2= 0.5.
where ๐โ๐
= โ + ๐/2. 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 (๐ = 2). Here we have ๐1= (1/2)๐๐
2,๐2
= 0, and ๐3= 0, where ๐ is the circular frequency of the
particle. So from (31) we obtain
๐ถ๐โ2
= [2 (๐๐/โ)
๐โ2+1 ๐!
ฮ (๐โ2+ ๐ + 1)
]
1/2
(๐โ2+ ๐)!
๐!๐โ2!
, (39)
where ๐โ2= ๐โ2+ 1. ๐โ2can be obtained from (14) and (17) as
๐โ2
= โ](] + 1) = โโ2 + (2๐/โ2)๐2. Hence (32) gives the
energy eigenvalues of the system๐ธ๐โ2
= โ๐ (2๐ + ๐โ2+ 1) . (40)
This result has already been obtained in [22, 48].
(2) Three Dimensions (๐ = 3). Here ๐1= (1/2)๐๐
2, ๐2
= 0,and ๐3= 0 which gives the energy eigenvalues as
๐ธ๐โ3
=โ๐
2[4๐ + 2 + โ(2โ + 1)
2
+8๐๐2
โ2] . (41)
Solving (17) we get ๐โ3
= ] + 1 and hence ๐โ3
= ] + 3/2. Theradial eigenfunction can hence be obtained from (34) whichalso corresponds to the result obtained in [22].
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6 Advances in High Energy Physics
4.3. 3-Dimensional Schrodinger Equation with Pseudohar-monic Potential. Here (17) gives ๐
โ3= ] + 1 and this makes
๐โ3= ] + 3/2. So (31) provides
๐ถ๐โ3
= ๐(1/2)(]+3/2)โ
2ฮ (] + ๐ + 3/2)
๐![ฮ (] +
3
2)]
โ1
, (42)
and hence the normalized eigenfunctions become๐ ๐โ3
(๐)
= ๐(1/2)(]+3/2)โ
2ฮ (] + ๐ + 3/2)
๐![ฮ (] +
3
2)]
โ1
โ ๐โ(๐/2)๐
2
๐]+11๐น1(โ๐, ] +
3
2, ๐๐2
) ,
(43)
with the energy eigenvalues
๐ธ๐โ3
= ๐3+
โ2
2๐๐2
= ๐3
+ โ8โ2
๐1
๐[๐ +
1
2+1
4
โ(2โ + 1)2
+8๐๐2
โ2] .
(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 ๐
1, ๐2, ๐3, and ๐(> 0) is
๐ (๐) = ๐1๐โ๐(๐โ๐๐) + ๐
2๐โ2๐(๐โ๐๐) + ๐
3, (45)
where ๐ describes the characteristic range of the potential,๐๐is the equilibrium molecular separation, and ๐
1, ๐2, and
๐3are 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 1/๐ and 1/๐2 are properly handled. Looking back to thesolution given by (34) we predetermine the solution for (8)for the potential given by (45) as
๐๐โ๐
(๐, ฮฉ๐) = ๐โ(๐โ1)/2
๐ ๐โ(๐) ๐๐
โ(ฮฉ๐) . (46)
This substitution facilitates following easier differential equa-tion (without the 1/๐ term) similar to (12) of Section 3:
[๐2
๐๐2+2๐
โ2(๐ธ โ ๐ (๐)) โ
๐ดโ
๐
๐2]๐ ๐โ(๐) = 0, (47)
where ๐ดโ๐= (๐ + 2โ โ 1)(๐ + 2โ โ 3)/4.
Let us introduce a new variable ๐ฅ = (๐ โ ๐๐)/๐๐. Hence
inserting (45) into (47) we have
[๐2
๐๐ฅ2+2๐๐2
๐
โ2{๐ธ โ ๐
1๐โ๐ผ๐ฅ
โ ๐2๐โ2๐ผ๐ฅ
โ ๐3}
+๐ดโ
๐
(1 + ๐ฅ)2]๐ (๐ฅ) = 0,
(48)
where ๐ผ = ๐๐๐.
It is possible to expand the term 1/(1+๐ฅ)2 in power series
as within the molecular point of view ๐ฅ โช 1. So
1
(1 + ๐ฅ)2โ 1 โ 2๐ฅ + 3๐ฅ
2
โ ๐ (๐ฅ3
) . (49)
This expansion can also be rewritten as
1
(1 + ๐ฅ)2โ ๐0+ ๐1๐โ๐ผ๐ฅ
+ ๐2๐โ2๐ผ๐ฅ
. (50)
By comparing these two it is not hard to justify that ๐0= 1 โ
3/๐ผ + 3/๐ผ2; ๐1= 4/๐ผ โ 6/๐ผ
2; ๐2= โ1/๐ผ + 3/๐ผ
2.Inserting (50) into (48) and changing the variable as ๐ฆ =
๐โ๐ผ๐ฅ one can easily get
[๐ฆ2๐2
๐๐ฆ2+ ๐ฆ
๐
๐๐ฆ+
๐
๐ผ2+๐ต1
๐ผ2๐ฆ +
๐ต2
๐ผ2๐ฆ2
]๐ (๐ฆ) = 0. (51)
Now further assuming the solution of the above differentialequation ๐ (๐ฆ) = ๐ฆ
โ๐
๐(๐ฆ), 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:
๐ฆ๐2
๐
๐๐ฆ2+ (1 โ 2๐)
๐๐
๐๐ฆ+๐2
โ ๐/๐ผ2
๐ฆ๐ + (
๐ต1
๐ผ2+๐ต3
๐ผ2๐ฆ)๐
= 0,
(52)
where
๐ =2๐๐2
๐
โ2(๐ธ โ ๐
3) + ๐0๐ดโ
๐, (53a)
๐ต1= ๐1๐ดโ
๐โ2๐๐2
๐
โ2๐1, (53b)
๐ต2= ๐2๐ดโ
๐โ2๐๐2
๐
โ2๐2. (53c)
Imposing the condition ๐2
โ ๐/๐ผ2
= 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] for๐1= โ2๐ท, ๐
2= ๐ท, and ๐
3= 0, where ๐ท describes the depth
of the potential.
6. Conclusions
We have investigated some aspects of๐-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 ๐ธ
๐(๐) as a
function of๐ for a given set of physical parameters ๐1, ๐2, โ,
and ๐ = 0, 1, 2, 3 keeping ๐3= 0. The general results obtained
in this paper have been verified with earlier reported results,
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Advances in High Energy Physics 7
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 like๐ ๐โ(๐) = ๐
โ๐
๐(๐) 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 ๐๐ (๐ > 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,๐ธ > lim
๐โโ๐(๐), ๐ should be replaced
with ๐ = ๐๐, where ๐ = โ2๐๐1/โ2. 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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