The -Deformed Calculus and Some Physical Applicationsm-hikari.com/astp/astp2017/astp1-4-2017/p/chungASTP1-4... · 2017. 1. 4. · The -deformed calculus and some physical applications

Advanced Studies in Theoretical PhysicsVol. 11, 2017, no. 2, 37 - 75

HIKARI Ltd, www.m-hikari.comhttps://doi.org/10.12988/astp.2017.61033

The α-Deformed Calculus and

Some Physical Applications

Jae Yoon Kim, Hyung Ok Yoon, Eun Ji Jang, Won Sang Chung∗

Department of Physics and Research Institute of Natural ScienceCollege of Natural Science

Gyeongsang National University, Jinju 660-701, Korea∗ Corresponding author

Hassan Hassanabadi

Physics Department, Shahrood University of Technology, Shahrood, Iran

Copyright c© 2016 Jae Yoon Kim et al. This article is distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduc-

tion in any medium, provided the original work is properly cited.

Abstract

In this paper we propose a new deformed calculus called a α-deformedcalculus. We use this new derivative to formulate a new calculus theory.We also discuss the α-deformed exponential, α-deformed logarithm andα-deformed trigonometry. We discuss the α-deformed Laplace transfor-mation. As applications we discuss the α-deformed chemical reactiondynamics, α-deformed Euler equation, α-deformed mechanics and α-deformed quantum mechanics.

1 Introduction

Fractional calculus [1, 2, 3] has attracted much interest among of manyresearchers either theoretically or in different fields of applications [4, 5]. Thetheory of q-deformed fractional calculus was initiated in early of fifties of lastcentury [6, 7, 8, 9, 10]. Then, this theory has started to be developed in thelast decade or so [11]-[16].

38 Jae Yoon Kim et al.

Recently, a new kind of fractional derivative called a conformable fractionalderivative was proposed by Khalil, Al Horani, Yousef and Sababheh [17]. Thisderivative depends just on the basic limit definition of the derivative. Namely.for a function f : (0,∞)→ R the (conformable) fractional derivative of order0 < α ≤ 1 of f at x > 0 was defined by

Dαxf(x) = lim

ε→0

f(x+ εx1−α)− f(x)

ε(1)

and the fractional derivative at 0 is defined as (Dαxf)(0) = limx→0(D

αxf)(x).

Some application of this type of fractional derivative is given in some literatures[18-30].

From the definition of the conformable fractional derivative, we can easilyobtain the α-deformed exponential function eα(x) obeying Dα

xeα(x) = eα(x).The α-deformed exponential function can be expressed in terms of the ordinaryexponential function as follows: eα(x) = e

1αxα . For this deformed exponential,

we have eα(x)eα(y) 6= eα(x + y), instead, we have eα(x)eα(y) = eα(x ⊕ y),where the α-deformed addition is defined as x ⊕ y = (xα + yα)1/α. Similarly,the α-deformed subtraction is defined as xy = (xα−yα)1/α for x > y. Usingthe α-deformed deformed subtraction, we can define a new kind of deformedderivative called a α-deformed derivative as

DαxF (x) = lim

y→x

F (y)− F (x)1α

(y x)α(2)

where we assume y > x. This definition seems somewhat unnatural becausethe ordinary subtraction is used in numerator while the α-deformed subtractionis used in denominator. This fact impels us to define a new kind of deformedderivative called a α-deformed derivative in a natural way.

In this paper we propose a new deformed calculus called a α-deformed cal-culus where the conformable fractional derivative given in the eq.(3) is replacedwith

DαxF (x) = lim

y→x

F (y) F (x)

y x(3)

We use this new derivative called a α-deformed derivative to formulate a newcalculus theory. This paper is organized as follows: In section II we discussthe α-deformed addition and α-deformed subtraction, α-deformed number,α-deformed derivative and α-deformed integral. In section III we find theα-deformed exponential, α-deformed logarithm and α-deformed trigonometryand investigate their properties. In section IV we discuss the α-deformedLaplace transformation and investigate its properties. In section V we dealwith some application such as the α-deformed chemical reaction dynamics,α-deformed Euler equation, α-deformed mechanics and α-deformed quantummechanics.

The α-deformed calculus and some physical applications 39

2 α-deformed calculus

In this section we discuss the algebraic structure related to the α-deformedcalculus.

2.1 α-deformed addition and subtraction

Now let us modify the definition of the α-deformed addition and α-deformedsubtraction so that they may work well for negative numbers.

Definition 2.1 The α-deformed addition and α-deformed subtraction are de-fined as follows:

x⊕ y = ||x|α−1x+ |y|α−1y|1/α−1(|x|α−1x+ |y|α−1y)

x y = ||x|α−1x− |y|α−1y|1/α−1(|x|α−1x− |y|α−1y) (4)

where we assume that α > 0.

When α = 1 is taken, the α-deformed addition ( or α-deformed subtraction)reduces to an ordinary addition ( or ordinary subtraction ). We can easilycheck that the α-deformed addition is commutative and associative, so we canobtain the following formula:

N⊕k=0

xk =

∣∣∣∣∣N∑k=1

|xk|α−1xk

∣∣∣∣∣1/α−1 N∑

k=1

|xk|α−1xk (5)

whereN⊕k=0

ak = a0 ⊕ a1 ⊕ a2 ⊕ · · · ⊕ aN (6)

When xk ≥ 0 for k = 1, 2, · · · , N , the relation (5) reduces to

N⊕k=0

xk =

(N∑k=1

xαk

)1/α

(7)

For the α-deformed addition, the identity is still 0; indeed we have

x⊕ 0 = 0⊕ x = x (8)

For the α-deformed addition, the inverse of x denoted by x is defined by

x⊕ (x) = (x)⊕ x = 0 (9)


For x > 0, y > 0, the eq.(4) can be written as

x⊕ y = (xα + yα)1/α

x y =

{(xα − yα)1/α (x > y)

−(yα − xα)1/α (x < y)(10)

For x > 0, y < 0, the eq.(4) can be written as

x⊕ y =

{(xα − (−y)α)1/α (x > −y)

−((−y)α − xα)1/α (x < −y)

x y = (xα + (−y)α)1/α (11)

For x < 0, y > 0, the eq.(4) can be written as

x⊕ y =

{(yα − (−x)α)1/α (−x < y)

−((−x)α − yα)1/α (−x > y)

x y = −(yα + (−x)α)1/α (12)

For x < 0, y < 0, the eq.(4) can be written as

x⊕ y = −((−x)α + (−y)α)1/α

x y =

{((−y)α − (−x)α)1/α (x > y)

−((−x)α − (−y)α)1/α (x < y)(13)

The above relations can be unified into the following ones:

x⊕ y =

{(|x|α−1x+ |y|α−1y)1/α if x+ y > 0

−(−|x|α−1x− |y|α−1y)1/α if x+ y < 0(14)

x y =

{(|x|α−1x− |y|α−1y)1/α if x− y > 0

−(−|x|α−1x+ |y|α−1y)1/α if x− y < 0(15)

From the above relations we can easily check x y = x⊕ (−y), which impliesx = −x.

Proposition 2.1 For the α-deformed addition and α-deformed subtraction,we have the following property:1. Distributivity

(kx)⊕ (ky) = k(x⊕ y), (kx) (ky) = k(x y), k ∈ R (16)

2. Expansion

(A⊕B)(C ⊕D) = AC ⊕BC ⊕ AD ⊕BD (17)


Proof. The proof is as follows:

(A⊕B)(C ⊕D) = (A⊕B)C ⊕ (A⊕B)D

= (AC ⊕BC)⊕ (AD ⊕BD)

= AC ⊕BC ⊕ AD ⊕BD (18)

which completes the proof of the eq.(17). The proof of the eq.(16) is similar.�

From the eq.(17) we have the following identities:

(x⊕ y)(x y) = x2 y2 (19)

(x⊕ y)2 = x2 ⊕ 21/αxy ⊕ y2 (20)

(x y)2 = x2 21/αxy ⊕ y2 (21)

(x⊕ y)(x2 xy ⊕ y2) = x3 ⊕ y3 (22)

(x y)(x2 ⊕ xy ⊕ y2) = x3 y3 (23)

2.2 α-deformed number ( shortly α-number)

From the associativity of ⊕, we have the following formula :

1⊕ 1⊕ 1⊕ · · · ⊕ 1︸︷︷︸n times

= (n)α = n1/α (24)

Here we call (n)α a α-number of n, where nα reduces to n when α goes to 1.For real number x, we can define the α-number (x)α as follows:

(x)α = |x|1/α−1x =

{x1/α (x > 0)

−(−x)1/α (x < 0)(25)

Here we have the following:

0α = 0, 1α = 1, (−1)α = −1 (26)

For the α-number, the following holds:(mn

)α

=(m)α(n)α

(27)

Proposition 2.2 For two α-numbers (x)α and (y)α, the following holds:

(x)α ⊕ (y)α = (x+ y)α (28)

(x)α (y)α = (x− y)α (29)


Proof. For x > 0, y > 0, the eq.(28) can be written as

(x)α ⊕ (y)α = x1/α ⊕ y1/α

=((x1/α)α + (y1/α)α

)1/α= (x+ y)α (30)

For x > 0, y < 0, x+ y > 0, the eq.(28) can be written as

(x)α ⊕ (y)α = x1/α ⊕ (−(−y)1/α)

= x1/α (−y)1/α

=((x1/α)α − ((−y)1/α)α

)1/α= (x+ y)α (31)

For x > 0, y < 0, x+ y < 0, the eq.(28) can be written as

(x)α ⊕ (y)α = x1/α ⊕ (−(−y)1/α)

= x1/α (−y)1/α

= −(−(x1/α)α + ((−y)1/α)α

)1/α= (x+ y)α (32)

For x < 0, y > 0, x+ y > 0, the eq.(28) can be written as

(x)α ⊕ (y)α = −(−x)1/α ⊕ y1/α

= y1/α (−x)1/α

=((y1/α)α − ((−x)1/α)α

)1/α= (x+ y)α (33)

For x < 0, y > 0, x+ y < 0, the eq.(28) can be written as

(x)α ⊕ (y)α = −(−x)1/α ⊕ y1/α

= y1/α (−x)1/α

= −(−(y1/α)α + ((−x)1/α)α

)1/α= (x+ y)α (34)

For x < 0, y < 0, the eq.(28) can be written as

(x)α ⊕ (y)α = −(−x)1/α ⊕ (−(−y)1/α)

= −((−x)1/α ⊕ (−y)1/α)

= −(((−x)1/α)α + ((−y)1/α)α

)1/α= (x+ y)α (35)


which completes the proof. �For α-deformed addition, we have the identity (0)α = 0 obeying

(x)α ⊕ (0)α = (x)α (36)

The inverse of (x)α is given by (−x)α because

(x)α ⊕ (−x)α = (0)α (37)

Proposition 2.3 For n α-numbers (x1)α, · · · , (xn)α, the following holds:

n⊕k=1

(xk)α =

(n∑k=1

xk

)α

(38)

Proof. It is simple. �

2.3 α-deformed derivative

Now let us define the α-deformed derivative with a help of the α-deformedaddition and α-deformed subtraction.

Definition 2.2 The α-deformed derivative ( or α-derivative ) is defined asfollows:

DαxF (x) = lim

h→0

F (x⊕ h) F (x)

h(39)

or

DαxF (x) = lim

y→x

F (y) F (x)

y x(40)

Proposition 2.4 If F (x) is increasing in the neighborhood of x, we have

DαxF (x) =

(|x|1−α|F (x)|α−1F ′(x)

)1/α(41)

If F (x) is decreasing in the neighborhood of x, we have

DαxF (x) = −

(−|x|1−α|F (x)|α−1F ′(x)

)1/α(42)

Proof. Let us assume y > x in the definition (40). First let us consider thecase that F (x) is increasing near x. In this case we know F ′(x) > 0. ForF (y) > F (x) > 0, y > x > 0, we have

DαxF (x) =

(limy→x

[F (y)]α − [F (x)]α

yα − xα

)1/α

=

(limy→x

α[F (y)]α−1F ′(y)

αyα−1

)1/α

=(x1−α[F (x)]α−1F ′(x)

)1/α(43)


For F (x) < F (y) < 0, y > x > 0, we have

DαxF (x) =

(limy→x

[−F (x)]α − [−F (y)]α

yα − xα

)1/α

=

(limy→x

α[−F (y)]α−1F ′(y)

αyα−1

)1/α

=(x1−α[−F (x)]α−1F ′(x)

)1/α(44)

For F (y) > F (x) > 0, x < y < 0, we have

DαxF (x) =

(limy→x

[F (y)]α − [F (x)]α

(−x)α − (−y)α

)1/α

=

(limy→x

α[F (y)]α−1F ′(y)

α(−y)α−1

)1/α

=((−x)1−α[F (x)]α−1F ′(x)

)1/α(45)

For F (x) < F (y) < 0, x < y < 0, we have

DαxF (x) =

(limy→x

[−F (x)]α − [−F (y)]α

(−x)α − (−y)α

)1/α

=

(limy→x

α[−F (y)]α−1F ′(y)

α(−y)α−1

)1/α

=((−x)1−α[−F (x)]α−1F ′(x)

)1/α(46)

which completes the proof of the eq.(41).

Now let us consider the case that F (x) is decreasing near x. In this casewe know F ′(x) < 0. For F (x) > F (y) > 0, y > x > 0, we have

DαxF (x) = − lim

y→x

F (x) F (y)

y x

= −(

limy→x

[F (x)]α − [F (y)]α

yα − xα

)1/α

= −(

limy→x

−α[F (y)]α−1F ′(y)

αyα−1

)1/α

= −(−x1−α[F (x)]α−1F ′(x)

)1/α(47)


For F (x) > F (y) > 0, x < y < 0, we have

DαxF (x) = − lim

y→x

F (x) F (y)

(−x) (−y)

= −(

limy→x

[F (x)]α − [F (y)]α

(−x)α − (−y)α

)1/α

= −(

limy→x

−α[F (y)]α−1F ′(y)

α(−y)α−1

)1/α

= −(−(−x)1−α[F (x)]α−1F ′(x)

)1/α(48)

For F (y) < F (x) < 0, y > x > 0, we have

DαxF (x) = − lim

y→x

(−F (y)) (−F (x))

y x

= −(

limy→x

[−F (y)]α − [−F (x)]α

yα − xα

)1/α

= −(

limy→x

−α[−F (y)]α−1F ′(y)

αyα−1

)1/α

= −(−x1−α[−F (x)]α−1F ′(x)

)1/α(49)

For F (y) < F (x) < 0, x < y < 0, we have

DαxF (x) = − lim

y→x

(−F (y)) (−F (x))

(−x) (−y)

= −(

limy→x

[−F (y)]α − [−F (x)]α

(−x)α − (−y)α

)1/α

= −(

limy→x

−α[−F (y)]α−1F ′(y)

α(−y)α−1

)1/α

= −(−(−x)1−α[−F (x)]α−1F ′(x)

)1/α(50)

which completes the proof. �The α-derivative is not linear because Dα

x [F (x) + G(x)] 6= DαxF (x) +

DαxG(x). Instead, it is α-linear because

Dαx (F (x)⊕G(x)) = Dα

xF (x)⊕DαxG(x) (51)

For any real number a ∈ R, we have

Dαx (aF (x)) = aDα

xF (x) (52)


Proposition 2.5 For the α -derivative, the following Leibniz rule holds:

Dαx [F (x)G(x)] = (Dα

xF (x))G(x)⊕ F (x)(DαxG(x)) (53)

Proof. Let us assume that y > x > 0 and F (y) > F (x) > 0, G(y) > G(x) > 0.Then, form the definition of α-derivative, we get

Dαx [F (x)G(x)] =

(limy→x

[F (y)]α[G(y)]α − [F (x)]α[G(x)]α

yα − xα

)1/α

=(x1−α[F (x)]α−1F ′(x)[G(x)]α + x1−α[F (x)]α[G(x)]α−1G′(x)

)1/α= ([Dα

xF (x)]α[G(x)]α + [F (x)]α[DαxG(x)]α)1/α

= (DαxF (x))G(x)⊕ F (x)(Dα

xG(x)) (54)

For another cases, one can prove the Leibniz rule in a similar way. Thus, wecompleted the proof. �

From the Leibniz rule, we have the following commutation relation:

Dαxx xDα

x = 1 (55)

orDαxx = 1⊕ xDα

x (56)

Proposition 2.6 For the α -derivative, the following holds:

Dαxx

n = nαxn−1, n = 0, 1, 2, · · · (57)

Proof. Let us assume that the eq.(57) holds for n− 1. Then, for n we have

Dαxx

n = (1⊕ xDαx )xn−1

= xn−1 ⊕ x(n− 1)αxn−2

= (1⊕ (n− 1)α)xn−1

= nαxn−1 (58)

which completes the proof. We can also prove this proposition from the defi-nition of the fractional derivative. For x > 0, xn is increasing, so we have

Dαxx

n = (|x|1−α|xn|α−1nxn−1)1/α

= nαxn−1 (59)

For x < 0, we should be careful in applying the definition of fractional deriva-tive because x2m+1 is increasing but x2m is decreasing. For x2m+1, we get

Dαxx

2m+1 = (|x|1−α|x2m+1|α−1(2m+ 1)x2m)1/α

= (2m+ 1)α|x|2m

= (2m+ 1)αx2m (60)


For x2m, we get

Dαxx

2m = −(−|x|1−α|x2m|α−1(2m)x2m−1)1/α

= −(2m)α|x|2m−1

= (2m)αx2m−1 (61)

which completes the proof. �

Proposition 2.7 For the α-derivative, the following holds:

Dαx (xnG(x)) = (Dα

xxn)G(x)⊕ xnDα

xG(x) (62)

Proof. Let us assume that the eq.(62) holds for n. Then, for n+ 1 we have

Dαx (xn+1G(x)) = (1⊕ xDα

x )xnG(x)

= xnG(x)⊕ x(Dαxx

n)G(x)⊕ xn+1DαxG(x)

= (1⊕ nα)xnG(x)⊕ xn+1DαxG(x)

= (n+ 1)αxnG(x)⊕ xn+1Dα

xG(x)

= (Dαxx

n)G(x)⊕ xnDαxG(x) (63)


Proposition 2.8 For the α-derivative, the following Leibnitz rule holds:

Dαx [F (x)G(x)] = (Dα

xF (x))G(x)⊕ F (x)(DαxG(x)) (64)

Dαx [

1

G(x)] =Dα

xG(x)

G(x)2(65)

Dαx [F (x)

G(x)] =

(DαxF (x))G(x) F (x)Dα

xG(x)

G(x)2(66)


Proposition 2.9 For the α-derivative, the following chain rule holds:

Dαx [F (x)]n = (n)α[F (x)]n−1Dα

xF (x) (67)

Proof. It is easy. �


2.4 α-deformed Integral

In this subsection we find the α-deformed integral as an inverse operation of theα-deformed derivative and investigate some properties of fractional integral.

Definition 2.3 The α-deformed integral from 0 to x ( x > 0 ) is defined asfollows: When F (x) > 0, the α-deformed integral from 0 to x ( x > 0 ) isgiven by

Iα0|xF (x) =

(∫ x

0

αdx|x|α−1|F (x)|α)1/α

(68)

When F (x) < 0, the fractional integral from 0 to x ( x > 0 ) is given by

Iα0|xF (x) = −(−∫ x

0


(69)

Proposition 2.10 For the α-deformed integral the following holds:

DαxI

α0|xF (x) = F (x) (70)

where x > 0

Proof. For F > 0, F ′ > 0, we have

DαxI

α0|xF (x) = [|x|1−α|Iα0|xF (x)|α−1(Iα0|xF (x))′]1/α

= (F (x)α)1/α = F (x) (71)

For F > 0, F ′ < 0, we know (Iα0|xF (x))′ > 0, so we have

DαxI

α0|xF (x) = [|x|1−α|Iα0|xF (x)|α−1(Iα0|xF (x))′]1/α

= (F (x)α)1/α = F (x) (72)

For F < 0, F ′ > 0, we know (Iα0|xF (x))′ < 0, so we have

DαxI

α0|xF (x) = −[−|x|1−α|Iα0|xF (x)|α−1(Iα0|xF (x))′]1/α

= −[(−F (x))α]1/α = F (x) (73)

For F < 0, F ′ < 0, we know (Iα0|xF (x))′ < 0, so we have

DαxI

α0|xF (x) = −[−|x|1−α|Iα0|xF (x)|α−1(Iα0|xF (x))′]1/α

= −[(−F (x))α]1/α = F (x) (74)




Iα0|xDαxF (x) = F (x) F (0) (75)

The above relation is written as follows:1. For F > 0, F ′ > 0, we have

Iα0|xDαxF (x) = [(F (x))α − (F (0))α]1/α (76)

2. For F > 0, F ′ < 0, we have

Iα0|xDαxF (x) = − [(F (0))α − (F (x))α]1/α (77)

3. For F < 0, F ′ > 0, we have

Iα0|xDαxF (x) = [(−F (0))α − (−F (x))α]1/α (78)

4. For F < 0, F ′ < 0, we have

Iα0|xDαxF (x) = − [(−F (x))α − (−F (0))α]1/α (79)

Proof.1. For F > 0, F ′ > 0, we have

Iα0|xDαxF (x) =

(∫ x

0

αdx|x|α−1|DαxF (x)|α

)1/α

=

(∫ x

0

αdx|F (x)|α−1F ′(x)

)1/α

= [(F (x))α − (F (0))α]1/α (80)

2. For F > 0, F ′ < 0, we know DαxF (x) < 0, so we have

Iα0|xDαxF (x) = −

(∫ x

0


)1/α

= −(−∫ x

0


)1/α

= − [(F (0))α − (F (x))α]1/α (81)

3. For F < 0, F ′ > 0, we know DαxF (x) > 0, so we have

Iα0|xDαxF (x) =

(∫ x

0


)1/α

=

(−∫ x

0

αdx(−F (x))α−1(−F ′(x))

)1/α

= [(−F (0))α − (−F (x))α]1/α (82)


4. For F < 0, F ′ < 0, we know DαxF (x) < 0, so we have

Iα0|xDαxF (x) = −

(∫ x

0


)1/α

= −(−∫ x

0


)1/α

= − [(−F (x))α − (−F (0))α]1/α (83)



Iα0|x(aF (x)) = |a|[Iα0|xF (x)] (84)

where x > 0 and a ∈ R.


Proposition 2.13 The α-deformed integral is α-linear; For two functionsF (x), G(x), we have

Iα0|x(F (x)⊕G(x)) = [Iα0|xF (x)]⊕ [Iα0|xG(x)] (85)

where x > 0.


Proposition 2.14 When a < b < c and F (x) ≥ 0 or F (x) ≤ 0 in [a, c], forthe α-deformed integral the following holds:

Iαa|cF (x) = [Iαa|bF (x)]⊕ [Iαb|cF (x)] (86)

Proof. For F (x) ≥ 0 in [a, c] we have

[Iαa|bF (x)]⊕ [Iαb|cF (x)] =([Iαa|bF (x)]α + [Iαb|cF (x)]α

)1/α=

(∫ b

a

αdx|x|α−1|F (x)|α +

∫ c

b


=

(∫ c

a


= [Iαa|bF (x)]⊕ [Iαb|cF (x)] (87)


For F (x) ≤ 0 in [a, c] we have

[Iαa|bF (x)]⊕ [Iαb|cF (x)] = −([−Iαa|bF (x)]α + [−Iαb|cF (x)]α

)1/α= −

(−∫ b

a

αdx|x|α−1|F (x)|α −∫ c

b


= −(−∫ c

a


= [Iαa|bF (x)]⊕ [Iαb|cF (x)] (88)


Proposition 2.15 When a > 0, for the α-deformed integral the followingholds:

Iα−a|aF (x) = Iα0|a[F (x)⊕ F (−x)] (89)

If F (−x)⊕ F (x) = 0 in [−a, a], we have

Iα−a|aF (x) = 0 (90)

In this case F (x) is called a α-odd function. If F (−x) = −F (x) in [−a, a], wehave

Iα−a|aF (x) = (2α)Iα0|aF (x) (91)

In this case F (x) is called a α-even function.


Proposition 2.16 When x > 0 and n = 0, 1, 2, · · · , for the α-deformed inte-gral the following holds:

Iα0|xxn =

xn+1

(n+ 1)α(92)

Iα−x|0x2n =

x2n+1

(2n+ 1)α(93)

Iα−x|0x2n+1 = − x2n+2

(2n+ 2)α(94)


3 α-deformed exponential, α-deformed loga-

rithm and α-deformed trigonometry

In this section we find the α-deformed exponential, α-deformed logarithm andα-deformed trigonometry and investigate their properties.


3.1 α-deformed exponential

Definition 3.1 The α-exponential function is defined from the following rela-tion:

Dαxeα(x) = eα(x) (95)

Proposition 3.1 The α-Taylor expansion is given by

F (x) =∞⊕k=0

anxn (96)

where

an =1

nα!(Dα

x )nF (x)|x=0 (97)


Proposition 3.2 The α-exponential has the following expression:

eα(x) =∞⊕n=0

xn

(n)α!(98)

Proof. Let us consider the case that x > 0. If we set

eα(x) =∞⊕n=0

anxn (99)

From Dαxeα(x) = eα(x), we get

(n)αan = an−1 (100)


Proposition 3.3 The α-exponential can also be written as

eα(x) = e1α|x|α−1x (101)

or

eα(x) =

{exp

(1αxα)

(x > 0)

exp(− 1α

(−x)α)

(x < 0)(102)

Proof. Let us consider the case that x > 0. From the eq.(99), we get

[eα(x)]α =∞∑n=0

(xn

nα!

)α=∞∑n=0

xαn

n!= ex

α

(103)


Let us consider the case that x < 0. From the eq.(99), we get

eα(x) =∞⊕n=0

xn

nα!

=∞⊕n=0

(−1)n(−x)n

nα!

= 1 (−x)⊕ (−x)2

2α! (−x)3

3α!⊕ · · · (104)

Thus, we have

[eα(x)]α = 1− (−x)α +

((−x)2

2α!

)α−(

(−x)3

3α!

)α−+ · · ·

= 1− (−x)α +(−x)2α

2!− (−x)3α

3!−+ · · ·

= e−(−x)α

(105)


Proposition 3.4 The inverse of a α- exponential called a α-logarithm is givenby

lnα(x) =

{(α lnx)1/α (x > 1)

− (−α lnx)1/α (0 < x < 1)(106)

Indeed we knoweα(lnα(x)) = lnα(eα(x)) = x (107)

Proof. It is simple. �The α-exponential function obeys the following relations:

eα(x)eα(y) = eα(x⊕ y)

eα(x)/eα(y) = eα(x y) (108)

The α-logarithm obeys the following relations:

lnα(xy) = lnα x⊕ lnα y

lnα

(x

y

)= lnα x lnα y (109)

Proposition 3.5 For the α-exponential, w have

Dαxeα(ax) = aeα(ax) (110)

Proof. It can be easily proved by using ∂x(1α|x|α−1x

)= |x|α−1. �


3.2 α-deformed trigonometric function

Definition 3.2 The α-deformed sine and cosine function is defined as follows:

cα(x) = | cos(|x|α−1x)|1/α−1 cos(|x|α−1x), sα(x) = | sin(|x|α−1x)|1/α−1 sin(|x|α−1x)(111)

The eq.(111) can be written as follows:

cα(x) =

{[cos(|x|α−1x)]1/α (if cos(|x|α−1x) > 0)

−[− cos(|x|α−1x)]1/α (if cos(|x|α−1x) < 0)(112)

sα(x) =

{[sin(|x|α−1x)]1/α (if sin(|x|α−1x) > 0)

−[− sin(|x|α−1x)]1/α (if sin(|x|α−1x) < 0)(113)

Proposition 3.6 For the α-deformed sine and cosine function, the followingholds:

|sα(x)|2α + |cα(x)|2α = 1 (114)

From the eq.(114) we have the following :

sα(0) = 0, cα(0) = 1

sα

[(π2

)1/α]= 1, cα

[(π2

)1/α]= 0

sα

[(π)1/α

]= 0, cα

[(π)1/α

]= −1

sα

[(3π

2

)1/α]

= −1, cα

[(3π

2

)1/α]

= 0

sα

[(2π)1/α

]= 0, cα

[(2π)1/α

]= 1 (115)

Proof. It is easy. �The α-deformed trigonometric functions obey the following addition for-

mulas:sα(x⊕ y) = sα(x)cα(y)⊕ cα(x)sα(y)

sα(x y) = sα(x)cα(y) cα(x)sα(y)

cα(x⊕ y) = cα(x)cα(y) sα(x)sα(y)

cα(x y) = cα(x)cα(y)⊕ sα(x)sα(y) (116)

Proposition 3.7 For the α-deformed trigonometric functions, the followingholds;

Dαxcα(x) = −sα(x) (117)

Dαxsα(x) = cα(x) (118)


Proof. It is easy. �From the definition of the α-deformed exponential we have the following

relations:

eα(ix) = e1α|ix|α−1ix = ei

|x|α−1xα , eα(−iαx) = e

1α|−ix|α−1(−ix) = e−i

|x|α−1xα (119)

The eq.(119) can be written as

[eα(ix)]α = ei|x|α−1x = cos |x|α−1x+ i sin |x|α−1x (120)

[eα(−ix)]α = e−i|x|α−1x = cos |x|α−1x− i sin |x|α−1x (121)

Proposition 3.8 The relation between the α-deformed exponential and α-deformed trigonometric functions are given by

eα(ix) = cα(x)⊕ isα(x), eα(−ix) = cα(x) isα(x) (122)

where

(a⊕ bi)α = (a)α + (b)αi, (a, b ∈ R) (123)

The α-series expansion of α-deformed trigonometric functions are given by

cα(x) =∞⊕n=0

(−1)nx2n

(2n)α!, sα(x) =

∞⊕n=0

(−1)nx2n+1

(2n+ 1)α!(124)


3.3 α-deformed polar coordinate

Now let us find the α-deformed polar coordinate. Let us denote the α-deformedradial coordinate and the α-deformed angular coordinate by rα and θα, respec-tively. Then, we have

x = rαcα(θα), y = rαsα(θα) (125)

where

rα = (|x|2α + |y|2α)1/2α, θ = t−1α

(yx

)(126)

and tα(x) = sα(x)/cα(x).We remark that the α-length rα is different from a ordinary length r unless

α = 1, instead, we have the following relation

rα = r(| sin θ|2α + | cos θ|2α)1/2α (127)


so the α-length varies with polar angle. For 0 < θα <(π2

)1/α, the relation

between the α-deformed polar angle and polar angle is

θα =(tan−1(tan θ)α

)1/α(128)

Using Cartesian coordinates, an α-deformed infinitesimal area element can becalculated as dAα = dxαdyα. The substitution rule for multiple α-integralsstates that, when using the α-deformed polar coordinate, the α-deformed Ja-cobian determinant of the coordinate conversion formula has to be considered:

Jα =

∣∣∣∣Drαx DrαyDθαx Dθαy

∣∣∣∣α

= rα (129)

where α-determinant is defined as∣∣∣∣a bc d

∣∣∣∣α

= ad bc (130)

Thus, the α-deformed infinitesimal area element in the α-deformed polar co-ordinate is given by

dAα = rαdrαdθα (131)

Let us consider the equation of the α-deformed circle:

|x|2α + |y|2α = R2αα , Rα > 0, (132)

The α-deformed area of the α-deformed circle with α-deformed radius Rα isthen given by

Aα = π1/αR2α (133)

which reduces to πR21 in the limit α→ 1.

3.4 α-deformed complex number

Now let us discuss the α-deformed complex numbers. We can define the α-deformed complex number z and its conjugate as follows;

z = x⊕ iy = rαeα(iθα) = rαcα(θα)⊕ irαsα(θα), (134)

z∗ = x iy = rαeα(−iθα) = rαcα(θα) irαsα(θα), (135)

The norm |z|α of the α-deformed complex number z is given by

|z|2α = zz∗ = r2α = x2 ⊕ y2 (136)

which is a α-length in two dimension.


Let us introduce the α-deformed complex derivatives

Dαz =

1

(2)α(Dα

x iDαy ), Dα

z∗ =1

2α(Dα

x ⊕ iDαy ) (137)

Then, we have

Dαz z = 1, Dα

z z∗ = 0, Dα

z∗z = 0, Dαz∗z∗ = 1 (138)

D∗zzn = 0, Dzz

n = (n)αzn−1 (139)

3.5 α-deformed rotation

Now let us the α-deformed rotation in xy plane which makes rα invariant. Fortwo n× n square matrix A,B, let us define the α-deformed product of A andB by

(AB)αij =n⊕k=1

AikBkj (140)

Then, the α-deformed length squared is described by

r2α = (XTX)α (141)

where

X =

(xy

)(142)

and XT means the transpose of X. The α-deformed length invariance implies

(r′α)2 = rα2 (143)

or(x′)2 ⊕ (y′)2 = x2 ⊕ y2 (144)

The α-deformed rotation is given by(x′

y′

)=

(cα(θα) sα(θα)−sα(θα) cα(θα)

)(xy

), (145)

where the α-deformed rotation matrix forms the SOα(2) group.

4 α-deformed Laplace transformation

In this section we introduce α-deformed Laplace transformation and investigateits properties.


Definition 4.1 α-deformed Laplace transform is defined by

Lα(F (x)) = Iα0|∞eα(−sx)F (x), (s > 0) (146)

where s is assumed to be sufficiently large.

Limiting α→ 1, the eq.(146) reduces to an ordinary Laplace transform.

Proposition 4.1 For the α-deformed Laplace transform, the following holds:

Lα(aF (x)) = |a|Lα(F (x)) (147)

Lα(F (x)⊕G(x)) = Lα(F (x))⊕ Lα(G(x)) (148)


Proposition 4.2 For F (x) = xN , (N = 0, 1, 2, · · · ) , we have the followingresult:

Lα(xN) =Nα!

sn+1(149)


Proposition 4.3 For sufficiently large s , the following holds:

Lα(eα(ax)) =1

s a(150)

Proof. From the definition we have

Lα(eα(ax)) =

(∫ ∞0

αdxxα−1|eα(sx)eα(ax)|α)1/α

=

(∫ ∞0

αdxxα−1[eα(−(s a)x)]α)1/α

=

(∫ ∞0

αdxxα−1e−(sa)αxα)1/α

=1

s a(151)


Proposition 4.4 When a > 0, for the α-deformed trigonometric functions,we have

Lα(cα(ax)) =s

s2 a2=

(sα

s2α + a2α

)1/α

(152)

Lα(sα(ax)) =a

s2 a2=

(aα

s2α + a2α

)1/α

(153)


Proof. We have

Lα(cα(ax)) = Lα

(eα(iαx)⊕ eα(−iαx)

2α

)=

1

2α

(1

s iαa⊕ 1

s⊕ iαa

)=

1

2α

(s

s2 a2

)(154)

The proof of the eq.(153) is similar. �

5 Applications

5.1 α-deformed chemical reaction equation

In chemical kinetics, relaxation methods are used for the measurement of veryfast reaction rates. The ordinary relaxation function I(t) obeying the relax-ation equation

d

dtI(t) = − 1

τ0I(t), (155)

which gives

I(t) = I(0) exp

[− t

τ0

](156)

The above decay function was used by Becquerel in the course of study ontime evolution of luminescence.

With a help of the α-deformed calculus, the relaxation equation is turnedinto

Dαt I(t) = − 1

τ0I(t) (157)

which gives

I(t) = I(0)eα

(t

τ0

)= exp

[− 1

α

(t

τ0

)α], (158)

5.2 α-deformed version of the Euler-Lagrange equation

To derive the α-deformed version of the Euler-Lagrange equation let us intro-duce the following α-deformed functional

Jα[y] = Iα0|x0f(y(x), Dαxy(x)) (159)

We will find the condition that Jα[y] has a local minimum. To do so, weconsider the new α-deformed functional depending on the parameter ε

Jα[ε] = Iα0|x0f(Y (x, ε), DαxY (x, ε)), (160)


whereY (x, ε) = y(x)⊕ εη(x) (161)

DαxY (x, ε) = Dxy(x)⊕ εDα

xη(x) (162)

andη(0) = η(x0) = 0 (163)

Then, the condition for an extreme value is that

[Dαε J

α[ε]]ε=0 = 0, (164)

From the chain rule of the α-deformed derivative, we have

Dαε J [ε] = Iα0|x0 [D

αY fD

αε Y ⊕Dα

DαxYfDα

ε (DαxY )]

= Iα0|x0 [DαY fη ⊕Dα

DαxYfDα

xη] = 0 (165)

Using the integration by parts, we get

I0|x0 [(DαY f Dα

x (DαDαxY

))η(x)] = 0, (166)

which gives the α-deformed Euler-Lagrange equation

Dαy f Dα

x (DαDαx y

) = 0 (167)

5.3 α-deformed version of Newton’s equation

Now let us apply the α-deformed calculus of variations to the α-deformedversion of the classical mechanics which we call the α-deformed mechanics. Inthis case we consider the α-deformed action

S = Iα0|tL(x(t), Dαt x(t)), (168)

where L is the α-deformed Lagrangian. The minimum condition of the α-deformed action gives

DαxL = Dα

t

(∂L

∂Dαt x

)(169)

In the similar way as the ordinary mechanics, we introduce the following La-grangian

L =1

2m(Dα

t x)2 V (x), (170)

where m is a mass and V (x) is a potential energy. From the eq.(169) and theeq.(170), we have the α-deformed equation of motion

m(Dαt )2x = −Dα

xV (x) (171)


For free motion, we have V (x) = 0; so we know that mDαt x is a constant of

motion. Generally, we can define the α-deformed canonical momentum p as

p =∂L

∂Dαt x

(172)

Thus, for the Lagrangian of the type (170), we have

p =∂L

∂Dαt x

= mDαt x, (173)

which we will call the α-deformed linear momentum. The α-deformed equationof motion then reads

F = Dαt p = −Dα

xV (x) (174)

The α-deformed hamiltonian H is obtained from the Legendre transformationas follows:

H(p, x) = pDαt x L (175)

For the Lagrangian of the type (170), we have

H(p, x) =p2

2m⊕ V (x) (176)

From the definition of the α-deformed linear momentum, we can define theα-deformed velocity v as

v(t) = Dαt x(t) (177)

and the α-deformed acceleration a as

a(t) = Dαt v(t) = (Dα

t )2x(t), (178)

Thus, the α-deformed version of the Newton’s law reads

F = ma = mDαt v = m(Dα

t )2x(t) (179)

5.3.1 α-deformed harmonic oscillator

Now let us introduce the α-deformed harmonic oscillator problem whose La-grangian is given by

L =1

2m(Dα

t x)2 − 1

2mw2

0x2 (180)

The α-deformed equation of motion then reads

m(Dαt )2x = −mw2

0x (181)

with the following initial condition

x(0) = A, Dαt x(0) = 0 (182)


Solving the eq.(181), we get

x(t) = Acα (w0t) , (183)

If we denote by T the α-deformed period defined by x(t⊕ T ) = x(t), we have

w0 =(2π)1/α

T(184)

5.4 α-deformed quantum mechanics

The α-deformed quantum mechanics starts with the following representation

x⇔ x, p⇔ ~iDαx , H ⇔ −~

iDαt (185)

where H is a α-deformed Hamiltonian operator. In the α-deformed quantummechanics, the commutator of the α-position operator x and α-momentumoperator p is replaced with the following α-commutator

[p, x] =~i

(186)

where[A,B] = AB BA (187)

The eq.(185) gives the following fractional Schrodinger equation

i~Dαt ψ =

[− ~2

2m(Dα

x )2 ⊕ Vα(x)

]ψ (188)

where we assume that the α-Hamiltonian is defined by the α-sum of the kineticenergy and potential energy. From the eq.(188) we can obtain the followingrelation

Dαt ρα(x, t)⊕Dα

x jα(x, t) = 0, (189)

where the α-deformed probability density ρα(x, t) is given by

ρα(x, t) = ψ∗ψ (190)

and the α-deformed probability flux jα(x, t) is given by

jα(x, t) =~

2mi(ψ∗Dα

xψ ψDαxψ∗) (191)

If we set ψ(x, t) = eα(i~Et

)u(x), we have the time-independent Schrodinger

equation as follows; [− ~2

2m(Dα

x )2 ⊕ Vα(x)

]u = Eu (192)


In the Hilbert space related to one-dimensional α-deformed quantum mechan-ics, the α-deformed inner product is given by

〈f |g〉α = I∞−∞g∗(x)f(x) (193)

The α-deformed expectation value of a physical operator O with respect tothe state u(x) is defined by

〈O〉α = 〈u|Ou〉α = I∞−∞u∗(x, t)Ou(x, t) (194)

and O is a Hermitian operator if it obeys

〈Ou|u〉α = 〈u|Ou〉α (195)

We can easily check that both α-deformed position operator and α-deformedmomentum operator is Hermitian.

5.4.1 Infinite potential well

The infinite potential well describes a particle free to move in a small spacesurrounded by impenetrable barriers. The potential energy in this model isgiven as

V (x) =

{0 (0 < x < L)

∞ (x < 0, x > L)(196)

The wave function u(x) can be found by solving the generalized conformablefractional time-independent Schrodinger equation for the system:

− ~2

2m(Dα

x )2u = Eu (197)

Solving the eq.(197), we get

u(x) = Acα

(√2mE

~2x

)⊕Bsα

(√2mE

~2x

)(198)

From u(0) = 0, we have A = 0, so we have

u(x) = Bsα

(√2mE

~2x

)(199)

From u(L) = 0 we get√2mE

~2L = (nπ)1/α, n = 1, 2, 3, · · · (200)


Thus, the energy levels are

(E)n =~2(nπ)2/α

2mL2, n = 1, 2, 3, · · · (201)

and the normalized wave functions are

un =

√21/α

Lsα

((nπ)1/α

Lx

)(202)

Here, the probability density is given by

Pn =21/α

L

∣∣∣∣sα((nπ)1/α

Lx

)∣∣∣∣2 (203)

Fig.1 and Fig.2 shows the plot of P1 and P2 for α = 1 (red), α = 0.9 (green),α = 0.8 (blue), α = 1.1 (yellow) and α = 1.2 (brown) when L = 1. We knowthe following:

1. When 0 < α < 1 the peak position of the probability moves left and thevalues of peaks increases as the value of α decreases.

2. When α > 1 the peak position of the probability moves right and thevalues of peaks decreases as the value of α increases.Now let us compute the expectation values. The expectation values of positionis

〈x〉n =1

21/αL (204)

and

〈x2〉n =

(1

3− 1

2n2π2

)1/α

L2 (205)

The expectation value of momentum is

〈p〉n = 0, 〈p2〉n =~2(nπ)2/α

L2(206)

If we define the α-deformed uncertainty in x and p as

∆αx =√〈x2〉〈x〉2, ∆αp =

√〈p2〉〈p〉2 (207)

we have the following uncertainty relation:

∆αx∆αp = ~(n2π2

12− 1

2

)1/2α

(208)

Fig.3 shows the plot of ∆αx∆αp/~ versus n for α = 1 (red), α = 0.9 (green),α = 0.8 (blue), α = 1.1 (yellow) and α = 1.2 (brown) when ~ = 1. Thisproduct increases with increasing n, having a minimum value for n = 1. Thevalue of this product for n=1 is about equal to 0.568~ for α = 1, 0.493~ forα = 0.8, 0.533~ for α = 0.9, 0.598~ for α = 1.1 and 0.624~ for α = 1.2.Thus, we know that the value of ∆αx∆αp for the ground state increases withincreasing α. Especially, for α ≈ 0.8164, the value of ∆αx∆αp is nearly thesame as 1

2~.


5.4.2 Harmonic oscillator

The Schrodinger equation for the α-deformed harmonic oscillator is then givenby [

− ~2

2m(Dα

x )2 ⊕ 1

2mw2x2

]u = Eu (209)

or [(Dα

x )2 (mw

~

)x2 ⊕ ε

]u = 0 (210)

where

ε =2mE

~2(211)

If we setu(x) = eα(−ax2)v(x) (a > 0) (212)

we getDαxu(x) = eα(−ax2) [−(2)αaxv(x)⊕Dα

xv(x)] (213)

and

(Dαx )2u(x) = eα(−ax2)

[(Dα

x )2v(x) (2)2αaxDαxv(x)⊕ ((2)2αa

2x2 (2)αa)v]

(214)Using the eq.(213) and eq.(214) and demanding that the x2 terms vanish, wehave

a =mw

(2)α~(215)

Then, the Schrodinger equation for v(x) is[(Dα

x )2 (2)2αaxDαx ⊕ (ε (2)αa)

]v = 0 (216)

Introducing the variable y through

y =√

(2)αax =

√mw

~x (217)

we get [(Dα

y )2 (2)αyDαy ⊕

(~εmw 1

)]v = 0 (218)

If we set~εmw 1 = (2n)α, n = 0, 1, 2, · · · (219)

we have the following energy level

En = 21/α−1~w(n+

1

2

)1/α

, n = 0, 1, 2, . . . (220)


The first few energies are

E0 =1

2~w

E1 =(3)α

2~w

E2 =(5)α

2~w

E3 =(7)α

2~w (221)

which shows that the energy is not equidistant. Indeed, we have

En+1 − En = 21/α−1~w(n)α

[1F0

(− 1

α; ;− 3

2n

)− 1F0

(− 1

α; ;− 1

2n

)](222)

If we consider the α-difference of two successive energies, we have

En+1 En = 21/α−1~w (223)

which shows that energies are α-equidistant.For the choice (219), the eq.(218) implies the α-deformed Hermite equation:[

(Dαy )2 (2)αyD

αy ⊕ (2n)α

]v = 0 (224)

where v(y) = Hαn (y) is a α-deformed Hermite polynomial. The α-deformed

Hermite polynomial is generated from the following relation:

g(x, t) = eα(−t2)eα((2)αxt) =∞⊕n=0

Hαn (y)

(nα)!tn (225)

From the generating function we have the following two recurrence relations:

DαyH

αn = (2n)αH

αn−1(y) (226)

Hαn+1 (2)αxH

αn ⊕ (2n)αH

αn−1 = 0 (227)

One can easily check that above two relations give the α-deformed Hermiteequation. From the expansion of the generating function we have

Hαn (y) =

∣∣∣∣∣∣∣bn2c∑

m=0

(−1)m(n)α!

(m)α!(n− 2m)α!((2)αx)n−2m

∣∣∣∣∣∣∣1/α−1 b

n2c∑

m=0

(−1)m(n)α!

(m)α!(n− 2m)α!((2)αx)n−2m

(228)


where bxc is a floor function. The first few α-deformed Hermite equations are

Hα0 (y) = 1

Hα1 (y) = |2y|1/α−12y

Hα2 (y) = |4y2 − 2|1/α−1(4x2 − 2)

Hα3 (y) = |8y3 − 12y|1/α−1(8y3 − 12y)

Hα4 (y) = |16y4 − 48y2 + 12|1/α−1(16y4 − 48y2 + 12) (229)

Fig.4,5,6,7 show the plots of Hα1 (y), Hα

2 (y), Hα3 (y), Hα

4 (y), respectively, for α =1 (red), α = 0.9 (green), α = 0.8 (blue), α = 1.1 (yellow) and α = 1.2 (brown).

Thus, we have the following eigenfunction corresponding to the energy En:

un(x) =1

(2)nα(n)α!

(mw

(π)α~

)1/4

eα

(−mwx

2

(2)α~

)Hαn

(mw~x), n = 0, 1, 2, . . .

(230)where we used

Iα−∞|∞eα(−x2) =√

(π)α (231)

Thus, the probabilities are given by

Pn = |un|2 (232)

Fig.8, 9, 10 shows P0, P1, P2, respectively, for α = 1 (red), α = 0.9 (green),α = 0.8 (blue), α = 1.1 (yellow) and α = 1.2 (brown).

Now let us compute the expectation values for the position and momentum.For the momentum operator, we have

〈p〉1/αn = 0, 〈p2〉1/αn = m~w(n+

1

2

)1/α

(233)

For the position operator, we have

〈x〉n = 0, 〈x2〉n =~mw

(n+

1

2

)1/α

(234)

Therefore we have the following uncertainty relation

(∆x)n(∆p)n = ~(n+

1

2

)1/α

(235)

For ground state, we have

(∆x)0(∆p)0 = ~(

1

2

)1/α

(236)

Thus, for 0 < α < 1, we have ∆x∆p < ~2

while for α > 1, we have ∆x∆p > ~2.


6 Conclusion

In this paper we proposed a new deformed calculus called a α-deformed calculuswhere the α-deformed derivative is replaced with

DαxF (x) = lim

y→x

F (y) F (x)

y x(237)

We used this new derivative to formulate a new calculus theory. Startingwith the definition of the α-deformed addition and α-deformed subtraction,we formulate the theory of α-deformed number, α-deformed derivative andα-deformed integral. We found many properties related to the α-deformedderivative and α-deformed integral. We also proposed the α-deformed expo-nential, α-deformed logarithm and α-deformed trigonometry and investigatetheir properties. We also found the relation between the α-deformed trigono-metric function and α-deformed exponential function. We also discussed theα-deformed polar coordinate. We discussed the α-deformed Laplace transfor-mation and investigated its properties. As applications we discussed the α-deformed chemical reaction dynamics, α-deformed Euler equation, α-deformedmechanics and α-deformed quantum mechanics. In the α-deformed mechan-ics, the energy was shown to be expressed in terms of the α-deformed addi-tion of the kinetic energy and potential energy. In the α-deformed quantummechanics, we discussed two examples; Infinite potential well and harmonicoscillator problem. For the infinite potential well we found that for 0 < α < 1the peak position of the probability moves left and the values of peaks in-creases as the value of α decreases while for α > 1 the peak position of theprobability moves right and the values of peaks decreases as the value of αincreases. We also found that the value of ∆αx∆αp for the ground state in-creases with increasing α. Especially, for α ≈ 0.8164, the value of ∆αx∆αp isnearly the same as 1

2~. For the harmonic oscillator, we obtained the energy

level En = 21/α−1~w(n+ 1

2

)1/α, n = 0, 1, 2, . . . and the wave equation by

using the α-deformed Hermite equation. From computation of the expectationvalues for the position and momentum, we found that for 0 < α < 1, we have∆x∆p < ~

2while for α > 1, we have ∆x∆p > ~

2.

Acknowledgements. This work was supported by the National Re-search Foundation of Korea Grant funded by the Korean Government (NRF-2015R1D1A1A01057792) and by the Gyeongsang National University Fund forProfessors on Sabbatical Leave, 2016.


Figure 1: Plot of P1 for α = 1 (red), α = 0.9 (green), α = 0.8 (blue), α = 1.1(yellow) and α = 1.2 (brown) when L = 1.

Figure 2: Plot of P2 for α = 1 (red), α = 0.9 (green), α = 0.8 (blue), α = 1.1(yellow) and α = 1.2 (brown) when L = 1.

Figure 3: Plot of ∆αx∆αp/~ versus n for α = 1 (red), α = 0.9 (green), α = 0.8(blue), α = 1.1 (yellow) and α = 1.2 (brown) when ~ = 1 .


Figure 4: Plot of Hα1 (y) for α = 1 (red), α = 0.9 (green), α = 0.8 (blue),

α = 1.1 (yellow) and α = 1.2 (brown).








Figure 8: Plot of P0 for α = 1 (red), α = 0.9 (green), α = 0.8 (blue), α = 1.1(yellow) and α = 1.2 (brown).




References

[1] I. Podlubny, Fractional Differential Equations, Academic Press, 1999.https://doi.org/10.1016/s0076-5392(99)x8001-5

[2] S. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Deriva-tives: Theory and Applications, Gordon and Breach, Yverdon, 1993.

[3] A. A. Kilbas, M. H. Srivastava, J. J. Trujillo, Theory and Applicationof Fractional Differential Equations, North Holland Mathematics Studies, Vol.204, Elsevier, 2006. https://doi.org/10.1016/s0304-0208(06)x8001-5

[4] R. L. Magin, Fractional Calculus in Bioengineering, Begell House, 2006.

[5] Fatma Bozkurt, T. Abdeljawad, M. A. Hajji, Stability analysis of a frac-tional order differential equation model of a brain tumor growth depending onthe density, Applied and Computational Mathematics, 14 (2015), no. 1, 50-62.

[6] W. Hahn, Beitrage zur Theorie der Heineschen Reihen. Die 24 Integraleder Hypergeometrischen q-Differenzengleichung. Das q-Analogon der Laplace-Transformation, Math. Nachr., 2 (1949), no. 6, 340-379.https://doi.org/10.1002/mana.19490020604

[7] R. P. Agrawal, Certain fractional q-integrals and q-derivatives, Mathe-matical Proc. Camb. Phil. Soc., 66 (1969), 365-370.https://doi.org/10.1017/s0305004100045060

[8] W.A. Al-Salam, Some fractional q-integrals and q-derivatives, Proc.


Edin. Math. Soc., 15 (1969), 135-140.https://doi.org/10.1017/s0013091500011469

[9] W.A. Al-Salam, A. Verma, A fractional Leibniz q-formula, Pacific Jour-nal of Mathematics, 60 (1975), 1-9.

[10] W.A. Al-Salam, q-Analogues of Cauchy’s formula, Proc. Amer. Math.Soc., 17 (1966), 616-621. https://doi.org/10.1090/s0002-9939-1966-0197637-4

[11] M. R. Predrag, D. M. Sladana, S. S. Miomir, Fractional Integrals andDerivatives in q-calculus, Applicable Analysis and Discrete Mathematics, 1(2007), 311-323.

[12] F. M. Atici, P.W. Eloe, Fractional q-calculus on a time scale, Journalof Non- Linear Mathematical Physics, 14 (2007), no. 3, 341-352.https://doi.org/10.2991/jnmp.2007.14.3.4

[13] T. Abdeljawad, D. Baleanu, Caputo q-Fractional Initial Value Prob-lems and a q-Analogue Mittag-Leffler Function, Communications in NonlinearScience and Numerical Simulations, 16 (2011), no. 12, 4682-4688.https://doi.org/10.1016/j.cnsns.2011.01.026

[14] Thabet Abdeljawad, J. O. Alzabut, The q-fractional analogue forGronwall-type inequality, 2013 (2013), Article ID 543839, 1-7.https://doi.org/10.1155/2013/543839

[15] Thabet Abdeljawad, Betul Benli, Dumitru Baleanu, A generalized q-Mittag- Leffler function by q-Caputo fractional linear equations, Abstract andApplied Analysis, 2012 (2012), Article ID 546062, 1-11.https://doi.org/10.1155/2012/546062

[16] Fahd Jarad, Thabet Abdeljawad, Dumitru Baleanu, Stability of q-farctional non-autonomous systems, Nonlinear Analysis: Real World Applica-tions, 14 (2012), 780-784. https://doi.org/10.1016/j.nonrwa.2012.08.001

[17] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new defini-tion of Fractional Derivative, J. Comput. Appl. Math., 264 (2014), 65-70.https://doi.org/10.1016/j.cam.2014.01.002

[18] T. Abdeljawad, M. Al Horani, R. Khalil, Conformable fractional semi-groups of operators, J. Semigroup Theory Appl., 2015 (2015), Article ID 7.


[19] I. Abu Hammad and R. Khalil, Fractional Fourier series with applica-tions, Amer. J. Comput. Appl. Math., 4 (2014), no. 6, 187-191.

[20] M. Abu Hammad and R. Khalil, Abel’s formula and Wronskian forconformable fractional differential equations, Internat. J. Diff. Equ. Appl.,13 (2014), no. 3, 177-183.

[21] M. Abu Hammad and R. Khalil, Conformable fractional heat differ-ential equations, Internat. J. Pure Appl. Math., 94 (2014), no. 2, 215-221.https://doi.org/10.12732/ijpam.v94i2.8

[22] M. Abu Hammad and R. Khalil, Legendre fractional differential equa-tion and Legendre fractional polynomials, Internat. J. Appl. Math. Research,3 (2014), no. 3, 214-219. https://doi.org/10.14419/ijamr.v3i3.2747

[23] D. R. Anderson and D. J. Ulness, Properties of the Katugampola frac-tional derivative with potential application in quantum mechanics, J. Math.Phys., 56 (2015), no. 6, 063502. https://doi.org/10.1063/1.4922018

[24] W. S. Chung, Fractional Newton mechanics with conformable frac-tional derivative, J. Comput. Appl. Math., 290 (2015), 150-158.https://doi.org/10.1016/j.cam.2015.04.049

[25] E. Hesameddini and E. Asadollahifard, Numerical solution of multi-order fractional differential equations via the sinc collocation method, Iran. J.Numer. Anal. Optim., 5 (2015), no. 1, 37-48.

[26] W. Kelley and A. Peterson, The Theory of Differential Equations Clas-sical and Qualitative, Pearson Prentice Hall, Upper Saddle River, NJ, 2004.

[27] R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, A new defi-nition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.https://doi.org/10.1016/j.cam.2014.01.002

[28] Y. Li, K. H. Ang and G. C. Y. Chong, PID control system analysisand design, IEEE Control Syst. Mag., 26 (2006), no. 1, 32-41.https://doi.org/10.1109/mcs.2006.1580152

[29] M. D. Ortigueira and J. A. Tenreiro Machado, What is a fractionalderivative?, J. Comput. Phys., 293 (2015), 4-13.https://doi.org/10.1016/j.jcp.2014.07.019


[30] K. R. Prasad, B. M. B. Krushna, Existence of multiple positive solu-tions for a coupled system of iterative type fractional order boundary valueproblems, J. Nonlinear Funct. Anal., 2015 (2015), Article ID 11.

Received: October 30, 2016; Published: January 6, 2017

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