265 
 

American Scientific Research Journal for Engineering, Technology,  and Sciences  (ASRJETS) 

ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

© Global Society of Scientific Research and Researchers  

http://asrjetsjournal.org/ 
 

How Fractional Charge on an Electron in the Momentum 

Space is Quantized?  

Saleem Iqbal 
a
*, Farhana Sarwar

 b
, Syed Mohsin Raza

 c 
, Abdul Rehman

 d
 

a,d
 Department of Mathematics, University of Balochistan, Quetta 87300, Pakistan  

b
 Department of Mathematics F.G.Girls Degree College, Madrissa Road , Quetta, Cantt, 87300, Pakistan. 

c 
Department of Physics, University of Balochistan, Quetta 87300, Pakistan 

    a
 Email: saleemiqbal81@yahoo.com 

b
 Email: f_saleem10@yahoo.com 

c
 Email: smraza7@yahoo.com 

 d
 Email: abdul_maths@yahoo.com 

 

Abstract  

With our conjecture on charge quantization (quantum dipole moment in a momentum space) and using 

Fractional Fourier Transform (FRFT) analysis on Hermite Polynomials (usually used for quantum oscillators), 

we obtained energy profiles (eigenfunctions) for fractional quantum states on the continuously changing surface 

of the electron. The charge on an electron as a physical constant and a single entity is degenerate because it 

always resides on the surface. The charge is fractionally quantized in momentum space. The continuous 

charging surface of the electron is due to competition between the centrifugal and electodynamic potentials. The 

fractional quantized states of charges in the momentum space are the manifestations of gyroscopic constants, 

𝑔2

ℏ𝑐
(0.2 − 0.8);  twisting and twigging of energy profiles (quantum electrodynamic behavior), oscillatory 

behavior of energy associated with degeneracy and indeed the position of fractional quanta in terms of rotational 

vector, 𝛼(𝑡, 𝜔) in complex plane. 

Keywords: Fractional Fourier Transform, Fractional Charge Quantization, Hermite polynomials 

1. Introduction  

We attempted to decipher a new idea based on fractional charge quantization on an electron by using Fractional 

Fourier Transform. The charge on an electron, being a physical constant and single entity, is fractionally 

quantized due to momentum impact by photons. 

------------------------------------------------------- 

* Corresponding author. 

http://asrjetsjournal.org/
mailto:saleemiqbal81@yahoo.com
mailto:f_saleem10@yahoo.com
mailto:smraza7@yahoo.com


American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No 2, pp 265-272 

 

266 
 

When a photon collides with an electron, the morphology of a bounded electron is changed due to inelastic 

collision. The electron quanta is stretched, as a consequence of which, the wave length, 𝜆  increases and the     

frequency, 𝜈 decreases, by maintaining, 𝑐 = 𝜆𝜈. The oscillatory frequency of the bounded electron decreases on 

the hypothetical wall of the electron string. This hypothetical wall behaves like an adiabatic wall. Due to 

overwhelming centrifugal potential as compare to electrodynamic potential, the electron quanta string is self ----

-twisted and twigged (swirling effect). The charge on electron quanta is distributed on twigs (sub quanta). These 

twigs are beaded sub-quanta on an electron string. The charge on electron, which is a single entity and constant, 

is distributed on these sub quanta (twigs) and hence the fractional charge quantization. Remember that each of 

these sub-quanta on an electron has an integrated oscillatory effect (discovered in this paper), .i.e.,2𝑛𝑓 , where 

0.1 ≤ 𝑛𝑓 ≤ 0.9, and are beaded on an electron quanta string. The momentum impact of a photon on a bounded 

electron causes stretching. This stretching is a manifestation of quantum mechanical scattering (inelastic 

scattering) which hold true for Compton and photoelectric effects, too. The stretching, twisting and twigging 

holds true for quantized particles, but not for free particles as the case is for Compton and photoelectric effects. 

That is why Eisenstein of Caltech (USA), on the basis of experimental results, considered quasi particle  nature 

of bounded electron due to its morphological changes.  

2. Theory  

 The quantum dipole moments lead to charge quantization [1, 2, 3, 4]  

𝑥 = ℎ𝑞                                                       (1) 

Where 𝑥 is the quantum dipole moment, 𝑞 the charge and ℎ is the Planck’s constant (quantum action) 

The matter energy such as of an electron exists in the form of transverse wave. This energy is oscillatory 

(quantum action). and configures  a space called a wave packet or “quanta”. We consider that the charge of an 

electron is treated as its density which is not only smeared on the surface but also inside the volume despite the 

fact that charges always reside on the surface. With momentum impact, the electron quanta is first stretched, 

twisted and then twigged. We envisage the electron like a flexible ball, the surface of which would vary 

continuously due to competing centrifugal and electrodynamic potential. The coupling constant 
𝑒2

ℏ𝑐
~

1

137
on an 

electron is overwhelmed by the gyroscopic constant, 
𝑔2

ℏ𝑐
(0.2 − 0.8) due to pronounced centrifugal potential. 

This causes the charge on an electron to become degenerate and fractionally quantized on its surface and hence 

the charge quantization. The depth of the quantum well of an electron is equivalent to its radius. With fractional 

charge quantization, the envelope of energy associated with an electron (electron quanta) is twisted and twigged 

to smear the density (charge) of energy in its fractional components, as a consequences of which, the fractional 

charges float on their respective broken “quanta” only on the surface. Each of the broken sub-quanta is woven in 

a string due to whirling and swirling effects (electro weak interaction) on an electron. These broken sub-quanta 

are degenerate fractional charged quantized states in the momentum space. Each of sub-quanta would have the 

oscillatory behavior. 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No 2, pp 265-272 

 

267 
 

Almeida [5] defines the fractional Fourier transform (FRFT) of a function 𝑥(𝑡), with angle 𝛼 (𝑡 is the time and 𝑢 

is the frequency) as 

ℱ𝛼[𝑥(𝑡)] = 𝑋𝛼(𝑢) = ∫ 𝑥(𝑡)

∞

−∞

𝐾𝛼(𝑡, 𝑢)𝑑𝑡 

=

{
 

 √
1−𝑗 cot𝛼

2𝜋
𝑒𝑗

𝑢2

2
cot𝛼

∫ 𝑥(𝑡)𝑒𝑗
𝑡2

2
cot𝛼−𝑗𝑢𝑡 csc𝛼𝑑𝑡  𝑖𝑓 𝛼   𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 𝜋  

∞

−∞

𝑥(𝑡)                                                                            𝑖𝑓 𝛼   𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 2𝜋

𝑥(−𝑡)                                                                    𝑖𝑓 𝛼 + 𝜋   𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 2𝜋

              (2) 

 

In our case we consider a function 𝑓(𝑥) and frequency 𝑢 = 𝜔 

𝑓(𝑥) = 𝐻𝑛𝑓(𝑥)𝑒𝑥𝑝 (−
𝑥2

2
)                                                                                                       (3) 

With equation (1)               𝑥 ≡ ℎ𝑞,    0.1 ≤ 𝑛𝑓 ≤ 0.9                                                             (4)                                                                                           

We know that the FRFT of 𝑓(𝑥) is given as follows [5]  

ℱ𝛼[𝑓(𝑥)] = 𝑒
𝑖𝑛𝑓𝛼𝐻𝑛𝑓(𝑥)exp (−

𝑥2

2
)                                                                                          (5) 

Using eq (4) on eq (3), we have 

𝑓(𝑥) = 𝐻𝑛𝑓(ℎ𝑞)𝑒𝑥𝑝 [(−
ℎ𝑞

√2
)
2

]                                                                                                  (6) 

Using eq (5) 

ℱ𝛼 [𝐻𝑛𝑓(ℎ𝑞)𝑒𝑥𝑝 [(−
ℎ𝑞

√2
)
2

]] = 𝑒𝑖𝑛𝑓𝛼𝐻𝑛𝑓(ℎ𝑞)exp [(−
ℎ𝑞

√2
)
2

]                                                    (7) 

Where 𝛼  is the angle of rotation in the complex plane (𝑡, 𝜔) With changing surface and indeed the shape of an 

electron, 𝛼  is also changed in (𝑡, 𝜔) coordinates , 𝜔 = 2𝜋𝜈. We are dealing with fractional quantum oscillators 

and hence with fractional charge distributions so that Fourier transform (FT) should not enter in our analysis. 

For this purpose we set   𝛼 ≠
𝜋

2
. With 𝛼 = 1, (𝛼 =

𝑎𝜋

2
) we get the FRFT to change into FT. The FRFT analysis 

is a time frequency distribution and an extension of the classical FT. Considering the Schrodinger’s equation for 

oscillatory quanta of fractional charges on the continuous changing surface of the electron [6, 7, 8]. 

−
ℎ2

2𝜇

𝑑2𝜓(𝑥)

𝑑𝑥2
+

1

2
𝑘𝑥2𝜓(𝑥) = 𝐸𝜓(𝑥)                                                                                            (8)           



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No 2, pp 265-272 

 

268 
 

Where  𝜔 = (
𝑘

𝜇
)

1

2
 ,  𝑘 the restoring constant 𝜇 the reduced mass of an electron. Using the dimensionless variable  

𝜉 = 𝛼𝑥, 𝛼 = (
𝜇𝑘

ℏ2
)

1

4
= (

𝜇𝜔

ℏ
)

1

2
                                                                                  (9) 

We shall replace dimensionless, α with rotation vector in coordinates, (𝑡, 𝜔).  Substitution of eq (9) in eq (8), we 

have 

 𝑑2𝜓(𝜉)

𝑑𝜉2
+ (𝜆 − 𝜉2)𝜓(𝜉) = 0                                                                                    (10) 

Where   𝜆  is also dimensionless, but considered as binding energy of the quantum system. For large   |𝜉|, it is 

readily verified that the eigenfunctions   𝜓(𝜉) = 𝜉𝑃𝑒±𝜉
2  exists. The asymptotic analysis provides us an 

indication for valid solutions to eq (10) and for all 𝜉 having the form 𝜉𝑃 where P is the polynomial. Thus    

𝜓(𝜉) = 𝑒−
𝜉2

2 𝐻(𝜉)                                                                                                     (11) 

Using eq (11) in eq (10), we have Hermite equation 

    𝑑2𝐻

𝑑𝜉2
− 2𝜉

𝑑𝐻

𝑑𝜉
+ (𝜆 − 1)𝐻 = 0                                                                                  (12) 

We assume a solution to eq (12) in the form of a finite polynomial 

𝐻(𝜉) = ∑ 𝑎𝑠
𝑁
𝑠=0 𝜉2𝑠; 𝑠 ≥ 0                                                                                              (13)        

Using eq (13) in eq (12), we obtain a recursion formula to reproducing the shape of the energy profiles 

oscillating within the momentum quantized space 

𝜎
𝑠+2=

2𝑠+1−𝜆
(𝑠+2)(𝑠+1)

𝑎𝑠 ,   𝑠≥0
                                                                                                        (14) 

For an upper cut off and the coefficients so that the polynomial equation (14) is not an infinite series, we have to 

insert a condition 

𝜆 = 2𝑛𝑓 + 1, 0.1 ≤ 𝑛𝑓 ≤ 0.9  ⟹ 1.2 ≤ 𝜆 ≤ 2.8                                                         (15) 

With dimensionless eigen value, 𝜆 =
2𝐸

ℏ𝜔
   and measuring the variation of electron radius, .i.e., depth of the 

quantum well in units of   (
ℏ

𝜇𝜔
)

1

2
  where  𝜔2 =

𝑘

𝜔
, we can ascertain that  

  𝐸𝑛𝑓 = (𝑛𝑓 +
1

2
)ℏ𝜔 = (𝑛𝑓 +

1

2
)ℏ𝜈;   ℏ =

ℎ

2𝜋
, 𝑛 = 0                                                     (16) 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No 2, pp 265-272 

 

269 
 

Using eq (16) with the collection of even and odd cases, the physically acceptable solution of eq (10) 

corresponding to eigenvalues (eq(16)) are given by  

𝜓𝑛𝑓(𝜉) = 𝑒
−
𝜉2

2 𝐻𝑛𝑓(𝜉);     0.1 ≤ 𝑛𝑓 ≤ 0.9                                                    (17)    

where the function 𝐻𝑛𝑓(𝜉)  are  polynomials of order 𝑛𝑓 . Moreover, the polynomials 𝐻𝑛𝑓(𝜉)  satisfy   the 

Hermite equation (eq(12)) with 𝜆 = 2𝑛𝑓 + 1, [eq(15)]. 

 𝑑2𝐻𝑛𝑓

𝑑𝜉2
− 2𝜉

𝑑𝐻𝑛𝑓

𝑑𝜉
+ 2𝑛𝐻𝑛𝑓 = 0,    0.1 ≤ 𝑛𝑓 ≤ 0.9                                          (18) 

Where the function 𝐻𝑛𝑓(𝜉) are Hermite polynomial. Their constant is traditionally chosen so that the highest 

power of 𝜉 appear with the coefficients of 2𝑛𝑓   in 𝐻𝑛𝑓(𝜉). 

     𝐻𝑛𝑓(𝜉) = 2
𝑛𝑓  ; 0.1 ≤ 𝑛𝑓 ≤ 0.9                                                                     (19) 

Eq (19) is consist with the following definition of Hermite polynomials 

𝐻𝑛𝑓(𝜉) = (−1)
𝑛𝑓𝑒𝜉

2 𝑑
𝑛𝑓

𝑑𝜉
𝑛𝑓
𝑒−𝜉

2
                                                                           (20) 

𝐻𝑛𝑓(𝜉) = 𝑒
𝜉2

2 (𝜉 −
𝑑

𝑑𝜉
)𝑛𝑓𝑒−

𝜉2

2                                                                                 (21) 

Eq (20) and (21) show fractional exponents of 
𝑑

𝑑𝜉
 and (𝜉 −

𝑑

𝑑𝜉
) and can be dealt either with Heaviside 

approximation or Lypanov exponents for attractors. The Lypanov exponents for equations (20) and (21) will 

preferably show the behavior of attracting the fractional quantum states in the momentum space with a string, 

.i.e., a quantum wire. 

We calculate the values of the Hermite polynomials from eqs (19), (20), and (21) as shown below 

𝐻𝜊(ℎ𝑞) = 1 for 𝑛 = 0 and 𝐻1(ℎ𝑞) = 2 for 𝑛 = 1, 𝜉 ≡ ℎ𝑞 now for 0.1 ≤ 𝑛𝑓 ≤ 0.9 

𝐻0.1(ℎ𝑞) = 1.072, 𝐻0.2(ℎ𝑞) = 1.149, 𝐻0.3(ℎ𝑞) = 1.231, 𝐻0.4(ℎ𝑞) = 1.319, 𝐻0.5(ℎ𝑞) = 1.414, 

𝐻0.6(ℎ𝑞) = 1.516, 𝐻0.7(ℎ𝑞) = 1.624, 𝐻0.8(ℎ𝑞) = 1.741, 𝐻0.9(ℎ𝑞) = 1.8666.              (22)                          

The generating function is given by the following relations  

𝐺(ℎ𝑞, 𝑠) = 𝑒−𝑠
2+2𝑠ℎ𝑞 ∑

𝐻𝑛𝑓
(ℎ𝑞)𝑠𝑛

𝑛!

∞
𝑛=0                                                                                  (23) 

 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No 2, pp 265-272 

 

270 
 

The above relation says that if the function 𝑒−𝑠
2+2𝑠ℎ𝑞   is expanded in a power series in 𝑠, the coefficients of 

successive powers of 𝑠 are just  
1

𝑛 !
  times the Hermite polynomial, 𝐻𝑛𝑓    . By using eq (23) and  𝑛 = 0, we can 

prove that the Hermite polynomials satisfy the recursion relations by the following relations: 

𝐻𝑛𝑓+1(ℎ𝑞) − 2ℎ𝑞𝐻𝑛𝑓(ℎ𝑞) + 2𝑛𝑓𝐻𝑛𝑓−1(ℎ𝑞) = 0 

 𝑑𝐻𝑛𝑓
(ℎ𝑞)

𝑑ℎ𝑞
= 2𝑛𝑓𝐻𝑛𝑓−1(ℎ𝑞) = 0                                                                        (24) 

Using eq (24) for each of the fractional discrete and distinct values of 𝐸𝑛𝑓   given by  𝜆𝑓 = 2𝑛𝑓 + 1, 0.1 ≤ 𝑛𝑓 ≤

0.9 , there is only one physically acceptable solution for eigenfunctions having oscillatory behavior in 

fractionally quantized momentum space, .i.e., 

 𝜓𝑛𝑓(ℎ𝑞) = 𝑁𝑛𝑓𝑒
−𝛼2(

ℎ𝑞

√2
)
2

𝐻𝑛𝑓(𝛼ℎ𝑞)                                                                                         (25) 

Considering the Hermite generating function and equate the coefficient of equal parts of 𝑠 and 𝑡, the normalized 

eigenfunctions  are given by                                                                 

𝜓(𝑥) =  (
𝛼

√𝜋2𝑛𝑛!
)
1
2
𝑒−𝛼

2𝑥2

2 𝐻𝑛(𝛼𝑥)                                                                                               (26) 

For our case  𝑥 ≡ ℎ𝑞,   0.17 ≤ 𝛼 ≤ 1.53 where 𝛼 =
𝑎𝜋

2
≡ 𝜆 as 𝑎 ≠ 1. The rotation vector will show  that the 

position of the fractional quantized momentum space for charge quantization on the varying surface of the 

electron in terms of radius. Put 𝑛! = 0! = 1, the exponent 𝑛 in terms of 𝑛𝑓 (fractional exponents) and the 

subscript 𝑛 in 𝐻 with  𝑛𝑓, where 0.1 ≤ 𝑛𝑓 ≤ 0.9 in eq (26), we have 

 𝜓𝑛𝑓(ℎ𝑞) = (
0.17≤𝛼≤1.53

√𝜋2
𝑛𝑓 .1

)
1
2
𝑒
−𝛼2(

ℎ𝑞

√2
)
2

𝐻𝑛𝑓(𝛼ℎ𝑞)                                                                         (27) 

Using eq (22), .i.e., 𝐻𝑛𝑓(ℎ𝑞) = 𝐻𝑛𝑓(𝛼ℎ𝑞)  in eq (27) and putting 𝑛𝑓 = 0.1 (𝛼 = 0.17), 𝑛𝑓 = 0.2 (𝛼 = 0.34), 

…,𝑛𝑓 = 0.9 (𝛼 = 1.53). We can reproduce the distribution of the fractional quantized states for charges on the 

surface of the electron. In other words, we can either reproduce the shape of the fractional charge distribution 

which are beaded in a string on the surface of the electron or the shape of the garland with beads with fractional 

charge quantization. The garland with beads (fractional charge quantization with sub quanta) could be envisaged 

like a quanta wire. For FT representation of eigenfunctions, the eq (26), we put 𝛼 =
𝜋

2
, 𝑛 = 0 and 1, we shall 

then  have the asymptotic variation of  𝜓𝑛(ℎ𝑞)  with 𝑥 ≡ ℎ𝑞. 

With arbitrary values of  𝑥 starting from zero (depth of the quantum well of electron) to radius of an electron, 

𝑥 = 𝑟𝑒(10 − 15𝜇𝑚) and using eq (26) with 𝑛! = 0! = 1, 20.1≤𝑛𝑓≤0.9, 𝐻𝑛𝑓(𝛼𝑥) = 𝐻𝑛𝑓(ℎ𝑞) and 0.17 ≤ 𝛼 ≤ 1.53, 

we can reproduce the energy profile for each of the fractional states inside the quantum well.  



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No 2, pp 265-272 

 

271 
 

At 𝑥 = 𝑟𝑒 we have the brim of the quantum well. Each of the eigenfunctions for 0.1 ≤ 𝑛𝑓 ≤ 0.9 will show the 

whirling profile for energy whereas the fractional change quantization on the surface of an electron is a swirling 

phenomenon. 

On comparison of eq (7) and (27) with condition that 𝐻𝑛𝑓(ℎ𝑞) = 𝐻𝑛𝑓(𝛼ℎ𝑞), we have  𝑒
−𝛼2(

ℎ𝑞

√2
)
2

≡ 1  

 we find  

𝑒𝑖𝑛𝑓𝛼 = (
𝛼

√𝜋2
𝑛𝑓
)
1
2
𝑒−𝛼

2
                                                                 (28) 

 Considering eq (27) with unitary operator 

  𝐻𝑛𝑓(ℎ𝑞) = 𝐻𝑛𝑓(𝛼ℎ𝑞) ≡ 𝑈𝑜𝑝,                        

  𝑈𝐻𝑛𝑓(𝛼ℎ𝑞)𝑈
𝑇 = 𝐼𝑜𝑝                              With this unitary operator        𝐻𝑛𝑓(ℎ𝑞) = 𝐻𝑛𝑓(𝛼ℎ𝑞) ≡ 𝐼𝑜𝑝 = 1   

𝐻𝑛𝑓(ℎ𝑞) Converges to unity because a new space is configured for  an electron quanta with twisting and 

twigging effects. Gaussian like function, .i.e., 𝑒−
𝛼2(

ℎ𝑞

√2
)
2

≡ 𝑒−𝛼
2
= 1 also converges to unity for a new 

configured space. The quanta of electron initially existed in the Wiener space, but with twisting and twigging a 

new space, .i.e., a Wigner space is configured. Wiener space is transformed in to a Wigner space which is a 

reciprocal space. The reciprocity is a manifestation of hyperbolic space which depends only on operators. Thus  

𝑒𝑖𝑛𝑓𝛼 = (
𝛼

√𝜋2
𝑛𝑓
)
1
2
                                                                     (29) 

This reciprocal space is a manifestation of reflection under inversion symmetry  (orthogonality is maintained). 

3. Conclusion 

We presented a new thesis about the morphology of a bounded electron which suffers momentum impact, as a 

consequence of which the electron quanta is first stretched twisted and then twigged. This behavior different 

from, Compton and photoelectric effects, respectively. Such morphology of a bounded electron quanta is termed 

as quasi particle. This morphology of an electron can explained Fractional Charge Quantization, Quantum Hall 

Effect, Giant Magneto Resistance and Quantum Capacitance.  

Reference 

[1] M. Gormani, Fazlur Rehman, Syed Mohsin Raza, and M.A. Ahmed,“Quantum Behaviour of Dielectric in 

Dolomite of Balochistan, Pakistan,” Jr.Chem.Soc.Pak, vol. 28, no. 5, pp. 414–416, 2006. 

 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 14, No 2, pp 265-272 

 

272 
 

[2] T. Samina Yousaf, Syed Mohsin Raza , Masoom Yasinzi, and Mujeeb ur Rehman, “Absorption of Radiant  

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