0323_0331

FRACTIONAL DIMENSIONAL HARMONIC OSCILLATOR

R. EID1, SAMI I. MUSLIH2, DUMITRU BALEANU3,∗, E. RABEI4

1Department of Mathematics, Atilim University, 06836 Incek-Ankara, Turkey2Department of Mechanical Engineering and Energy Processes,

Southern Illinois University, Carbondale, IL, 62901, USAEmail: [email protected]

3Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences,Cankaya University- 06530, Ankara, Turkey

Emails: [email protected], [email protected] of Physics-Mutah University, Karak-Jordan

Received October 19, 2009

The fractional Schrodinger equation corresponding to the fractional oscillatorwas investigated. The regular singular points and the exact solutions of the correspond-ing radial Schrodinger equation were reported.

Key words: fractional space, fractional oscillator, Schrodinger equation.

PACS: 04.60.Kz

1. INTRODUCTION

Fractional dimensional space has successfully been applied as an effectivephysical description of confinement in low-dimensional system. This method re-places the real confining structure with an effective space, where the measure of theanisotropy or confinement is given by the non-integer dimension (see for example[1]-[4] and the references therein).

As it is known, many of the investigations into low-dimensional semiconduc-tors structures are based on a mathematical basis introduced in [5] in which he de-scribed integration on a space of α dimensions and provided a generalization of theLaplace operator on this space.

Recent progress includes the description of a single coordinate momentum ope-rator in this fractional dimensional space based on generalized Wigner relations [6,7]presenting realization of parastatics [8].

The fractional dimensions appears as an explicit parameter when the physicalproblem is formulated in α dimensions such that α maybe extended to non-integervalues, as in Wilson’s study of quantum field theory models in less than four dimen-

∗On leave of absence from Institute of Space Sciences, P.O. Box MG-23, RO-077125, Magurele-Bucharest, Romania

(c) RJP 56(3-4) 323–331 2011Rom. Journ. Phys., Vol. 56, Nos. 3-4, P. 323–331, Bucharest, 2011

324 R. Eid et al. 2

sions [9], or in the approach to quantum mechanics mentioned in [5]. On the otherhand the experimental measurement of the dimensional α of our real world is givenby α = (3± 10−6) [5, 9]. The fractional value of α agrees with the experimentalphysical observations that in general relativity, gravitational fields are understood tobe geometric perturbations (curvatures) in our space-time [10], rather than entitiesresiding within a flat space-time. On the other hand, Zeilinger and Svozil [11] re-ported that the current discrepancy between theoretical and experimental values ofthe anomalous magnetic moment of the electron could be resolved if the dimensio-nality of space α is α= 3− (5.3±2.5)×10−7.

The simple harmonic oscillator is one of the most important topics in quantummechanics. It is applicable in different physical situations and has the great advantagethat it has closed solutions for the energy eigenvalues and eigenfunctions [12]. In thecontext of connecting the Coulomb and oscillator potential in arbitrary dimensionsobtain the generalized KS transformation: from five-dimensional hydrogen to eight-dimensional isotropic oscillator [13].

The system of a fractional-dimensional Bose-like oscillator of one degree offreedom whose canonical variables satisfy the general Wigner commutation rela-tions and the momentum-position uncertainty relations are obtained in a fractional-dimensional space [14]. Besides in [15], the authors formulated an algebra approachto quantum mechanics in fractional dimensions in which the momentum and the posi-tion operators satisfy the R-deformed Heisenberg relations which lead to the parabo-son operators which are used to solve harmonic oscillator in d dimensional fractionalspace. The main aim of this paper is to find the exact solutions of the Schrodingerequation of the fractional dimensional oscillator. The plan of the paper is as follows:In Section 2 the model is presented. In Section 3 we have presented the exact solu-tions corresponding to several values of β. Finally, Section 4 presents our conclu-sions.

2. THE MODEL

The Schrodinger equation to start with is given below

(~2

2µrα−1∂

∂r(rα−1

∂

∂r) +

Λ2α(Ω)

2µr2+

1

2kβr

β−1)

Ψ(r,Ω) = EΨ(r,Ω), (2.1)

where Λ2α(Ω)Ψ(r,Ω) =−l(l+α−2)Ψ(r,Ω), for l = 0,1,2,3, kβ is the spring con-

stant with units Joule(meter)β−1 , α and β are dimensional fractional space.

The next step is to make use of the change of variable Ψ(r,Ω) = R(r)Y (Ω).

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3 Fractional dimensional harmonic oscillator 325

As a result the radial equation part becomes

R′′(r) +

α−1

rR′(r) +

[2µ

~2(E− 1

2kβr

β−1)− l(l+α−2)

r2

]R(r) = 0, (2.2)

where kβ|β=3 = ω2µ and the convergent solution should be for (r ∈ (0,∞)).

By making the second change of variable, namely R(r) = rl[e−µω2r2

2~ ]y(r) weobtain the differential equation corresponding to y(r) as follows

y′′(r) +

1

r[(2l+α−1)− 2µω2r

~]y′(r)

+ [(−µω~

(2l+α) +2µ

~2(E− 1

2kβr

β−1) +µ2

~2ω2r2]y(r) = 0. (2.3)

Finally, by using the new variables x = µωr2

~ and k2 = 2µE~2 the equation (2.3)

becomes

xd2y

dx2+ (A−x)

dy

dx+

(B− A

2− x

4− C

4xβ−12

)y(x) = 0 , (2.4)

where A= 2l+α2 , B = k2~2

4µω and C =kβ~ω ( ~

µω )β−12 .

3. EXACT SOLUTIONS OF THE RADIAL DIFFERENTIAL EQUATION

Let us take y(x) = ex/2 F (x). Substituting y(x), y′(x) and y′′(x) into thedifferential equation (2.4), we may obtain

xd2F

dx2+A

dF

dx+

(B− C

4xβ−12

)F (x) = 0 . (3.5)

Our aim is to solve (3.5).The first step is to find the regular singular points forequation (3.5).

Lemma. x0 = 0 is a regular singular point for equation (3.5) if and only ifβ ≥−1 .

Proof. Sinceα0 = limx→0

xA

x=A and β0 = lim

x→0x2B− C

4 xβ−12

x=−C

4limx→0

xβ+12

are both finite if and only if β ≥−1. Thus, x0 = 0 is a regular singular point if andonly if β ≥−1.

Case 1. For β =−1, the Equation (3.5) becomes

x2d2F

dx2+Ax

dF

dx+

(Bx− C

4

)F (x) = 0. (3.6)

The next step is to use the substitution F (x) = x1−A2 G(x) which leads to the

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326 R. Eid et al. 4

following equation

x2G′′(x) +xG′(x) +

(Bx− A

2−2A+ 1 +C

4

)G(x) = 0. (3.7)

This equation can be written in the form

x2G′′(x) +xG′(x) +1

4

[(2√B√x)2−(√

(A−1)2 +C)2]

G(x) = 0,

which is the well-known Bessel’s equation of the first kind and it has the generalsolution

G(x) = C1Jν

(2√B√x)

+C2Yν

(2√B√x),

where ν =√

(A−1)2 +C and

Jν(t) =

∞∑m=0

(−1)mt2m+ν

22m+νm!Γ(ν+m+ 1), Yν(t) =

Jν(t)cosνπ−J−ν(t)

sinνπ.

Therefore, corresponding to β = −1 , equation (2.4) has a general solution ofthe form

y(x) = ex/2 x1−A2

C1 Jν

(2√B x

)+C2 Yν

(2√B x

).

Case 2. For β =−12 , the equation (3.5) becomes

x2d2F

dx2+A x

dF

dx+

(B x− C

4x1/4

)F (x) = 0 . (3.8)

For this equation, we use the following substitution t= 2√−Bx.

After doing some calculations, we arrive at the differential equation

t2d2F

dt2+ (2At− t) dF

dt−

(t2 +

C

(−4B)1/4

√t

)F (t) = 0 (3.9)

By using the transformation F (t) = et G(t) , the above differential equation maytake the new form

t2d2G

dt2+(2t2 + 2At− t

) dGdt

+

(2At− t− C

(−4B)1/4

√t

)G(t) = 0 .

The next step is to use the following new variable w =√t. Therefore, the

obtained differential equation can be written as

d2G

dw2+

4w2 + 4A−3

w

dG

dw+

8Aw−4w−4 C

(−4B)1/4

wG(w) = 0.

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Finally we use the linear transformation w =i√2z where i=

√−1.

Now, in terms of the new variable z, the previous differential equation can bewritten as

d2G

dz2− 2z2−1 + 4−4A

z

dG

dz− 1

2

(8A−4) z+ 4 C

(−B)1/4

zG(z) = 0. (3.10)

This is the well-known HeunB equation with α = 4A− 4, β = 0, γ = 0 and δ =4C

(−B)1/4. Hence, corresponding to β = −1

2, the general solution of equation (2.4)

is the expression

y(x) = e(x+4√−Bx)/2

C1HeunB

(4A−4,0,0,− 4iC

(−B)1/4,2i(−Bx)1/4

)+

C2HeunB

(4−4A,0,0,

4iC

(−B)1/4,2i(−Bx)1/4

)x1−A

.

Case 3. For β = 0 the equation (3.5) becomes the differential equation;

xd2F

dx2+A

dF

dx+

(B− C

4x−1/2

)F (x) = 0 . (3.11)

For this equation, we use the same substitution which is used in case 1, that is,t=−2i

√Bx, where i=

√−1.

After substituting these expressions into equation (2.5) and doing some simpli-fications, one may arrive at the differential equation

td2F

dt2+ (2A−1)

dF

dt−(t+

iC

2√B

)F (t) = 0.

By using the transformation F (t) = et G(t) , the above differential equation maytake the new form

t

2

d2G

dt2+

(t+

2A−1

2

)dG

dt−(

1−2A

2+

iC

4√B

)G(t) = 0. (3.12)

By using the following change of variable z =−2t, one may write the equation(3.12) reduced to the following form

zd2G

dz2+ (−z+ 2A−1)

dG

dz−(

1−2A

2+ i

C

4√B

)G(z) = 0. (3.13)

This is the well-known Hypergeometric equation in the general form. Thus, the

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328 R. Eid et al. 6

general solution of (2.4) corresponding to β = 0, is the expression

y(x) =e(x−4i√Bx)/2

C1 Hypergeom

([2A−1

2− i C

4√B

], [2A−1] ,4i

√Bx

)+

C2 Hypergeom

([3−2A

2− i C

4√B

], [3−2A] ,4i

√Bx

)x1−A

.

Case 4. In the following we consider the case β = 1. Therefore, equation (3.5)becomes the differential equation

x2d2F

dx2+A x

dF

dx+

(B x− C

4

)F (x) = 0 . (3.14)

If we use the substitution F (x) = x1−A2 G(x) and after doing some simplifications,

we arrive at the differential equation

x2 G′′(x) +x G′(x) +1

4

[(√4B−C

√x)2− (A−1)2

]G(x) = 0, (3.15)

which is the well-known Bessel’s equation of the first kind and it has the generalsolution as given below

G(x) = C1 Jν

(√4B−C

√x)

+C2 Yν

(√4B−C

√x),

where ν =A−1,

Jν(t) =∞∑m=0

(−1)m t2m+ν

22m+ν m! Γ(ν+m+ 1), Yν(t) =

Jν(t) cosνπ−J−ν(t)

sinνπ.

Therefore, corresponding to β = 1 , equation (2.4) has a general solution of theform

y(x) = ex/2 x1−A2

C1 Jν

(√(4B−C) x

)+C2 Yν

(√(4B−C) x

).

Case 5. For β = 3 , the equation (3.5) becomes the differential equation

xd2F

dx2+A

dF

dx+

(B− C

4x

)F (x) = 0 . (3.16)

For this equation, we use the substitution F (x) = x−A2 G(x) .

After substituting F (x), F ′(x) and F ′′(x) into equation (3.16) and doing somesimplifications, we arrive at the differential equation

x2d2G

dx2+

(2A−A2

4+Bx− C

4x2)G(x) = 0 .

By using the linear transformation t = x√C, the above differential equation may

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take the new form

G′′(t) +

(−1

4+

B√C

t+

14 −[A−12

]2t2

)G(t) = 0 . (3.17)

Since the well-known Whittaker’s equation

G′′(z) +

(−1

4+a

z+

14 − b

2

z2

)G(z) = 0 (3.18)

has general solution

G(z) = C1 WhittakerM (a,b,z) +C2 WhittakerW (a,b,z) (3.19)

it implies that equation (2.8) has general solution

G(x) = C1 WhittakerM

(B√C,A−1

2,√C x

)+C2 WhittakerW

(B√C,A−1

2,√C x

).

(3.20)

Therefore, the solution of equation (3.17) becomes

F (x) = x−A2

C1 WhittakerM

(B√C,A−1

2,√C x

)+

C2 WhittakerW

(B√C,A−1

2,√C x

).

Hence, corresponding to β = 3 , the general solution of equation (2.4) is the expres-sion

y(x) = ex/2 x−A2

C1 WhittakerM

(B√C,A−1

2,√C x

)+

C2 WhittakerW

(B√C,A−1

2,√C x

).

Case 6. For β = 7 , the equation (2.4) becomes

xd2F

dx2+A

dF

dx+

(B− C

4x3)F (x) = 0. (3.21)

Using the substitution t=

√C

4x2 we obtain the differential equation

2t2d2F

dt2+ (At+ t)

dF

dt−(

2t2− B

C1/4

√t

)F (t) = 0. (3.22)

Now, the simple transformation F (t) = et G(t) , reduces the above differential equa-tion to the form

2t2d2G

dt2+(4t2 +At+ t

) dGdt

+

(At+ t+

B

C1/4

√t

)G(t) = 0. (3.23)

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330 R. Eid et al. 8

Taking w =√t and after using some simplifications, the last differential equa-

tion can be written in the form

wd2G

dw2+(4w2 +A

) dGdw

+

(2w+ 2Aw+ 2

B

C1/4

)G(w) = 0. (3.24)

Finally, the linear transformation w =i√2z, where i =

√−1 gives, in terms

of the new variable z , the following differential equation

d2G

dz2− 2z2−1 + 1−A

z

dG

dz− 1

2

(2A+ 2) z− 2√2Bi

C1/4

zG(z) = 0. (3.25)

This is the well-known HeunB equation with α = A− 1, β = 0, γ = 0 and

δ =−2B√

2i

C1/4. Hence, corresponding to β = 7, the general solution of equation (2.4)

is the expression

y(x) = e(2x+√C x2)/4

C1HeunB

(A−1,0,0,

2√

2Bi

C1/4,

√2i

2C1/4x

)

+C2 HeunB

(1−A,0,0, 2

√2Bi

C1/4,

√2i

2C1/4x

)x1−A

.

4. CONCLUSIONS

Fractional dynamics started to play an important role in several branches ofscience and engineering. In this paper, by using suitable change of variables, wehave obtained the exact solutions of the Schrodinger equation corresponding to thefractional dimensional harmonic oscillator.

It was proved that x0 = 0 is a regular singular point for equation (3.5) if andonly if β ≥−1.

An interesting point to be specified here is that for the special case α= 1, β ≤ 1we obtained the case which is called spiked harmonic oscillator.

Acknowledgements. One of the authors (S.M.) would like to thank Fulbright foundation forfinancial support. This work is partially supported by the Scientific and Technical Research Council ofTurkey.

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REFERENCES

1. X. He, Solid State Commun. 75, 111 (1990).2. X. He, Phys. Rev. B 43, 2063 (1991).3. X. He, Phys. Rev. B 42, 751 (1990).4. S. I. Muslih and D. Baleanu, Nonl. Anal.: Real World Appl. 8, 198 (2007).5. F. H. Stillinger, J. Math. Phys. 18, 1224 (1977).6. A Matos-Abiague, J. Phys. A : Math. Gen. 34, 3125 (2001).7. A Matos-Abiague, J. Phys. A : Math. Gen. 34, 11059 (2001).8. S. Jing, J. Phys. A : Math. Gen. 31, 6347 (1998).9. K. G. Willson, Phys. Rev. D 7, 2911 (1973).

10. C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973).11. A. Zeilinger and K. Svozil, Phys. Rev. Lett. 54, 2553 (1985).12. M. Moshinsky and Y. F. Sminov, The Harmonic Oscillator on Modern Physics (Harwood Aca-

demic, Amsterdam, 1996).13. Davtyan et al., J. Phys. A. : Math. Gen. 20, 6121 (1987).14. A. Matos-Abiague, J. Math. Phys. A:Math. Gen. 34, 3125 (2001).15. M. A. Lohe and A. Thilaagam, J. Phys. A: Math. Gen. 37, 6181 (2004).

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