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We developed energy profiles for the fractional quantized states both on the surface of electron due to overwhelming centrifugal potentials and inside the electron at different locations of the quantum well due to overwhelming attractive electrodynamic potentials. The charge as a physical constant and single entity is taken as density and segments on their respective sub-quanta (floats on sub quanta) and hence the fractional charge quantiz at in. There is an integrated oscillatory effect which ties all fractional quantized states both on the surface and in the interior of the volume of an electron. The eigenfunctions, i.e., the energy profiles for the electron show the shape of a string or a quantum wire in which fractional quantized states are beaded. We followed an entirely different approach and indeed thesis to reproducing the eigenfunctions for the fractional quantized states for a single electron. We produced very fascinating mathematical formulas for all such cases by using Hermite and Laguerre polynomials, spherical based and Neumann functions and indeed asymptotic behavior of Bessel and Neumann functions. Our quantization theory is dealt in the momentum space.

The inside of the electron is treated like a potential well with its depth equivalent to radius

string. This coupling, in fact, is due to gyroscopic behaviour of an electron. The coupling constant

constant entity of charge to become degenerate and fractionally quantized above its surface (of course due to segmented mass distribution but coupled with a string at different locations just above the surface).

The radial eigen function

For

where

quantum number (determines the shape of the whirling strips in the quantum well). We roughly estimated the radius or depth of the quantum well of an electron by considering the charge as density of electron

Each of the broken sub quanta are woven in a string due to whirling and swirling effects (electro weak interactions) on an electron and is manifestation of gyroscopic behaviour. Equation (1) can be rewritten as

where

When broken sub quanta above the surface of an electron are woven in a string, they should have an integrated oscillatory effects.

Writing

The exponent n of 2 will have fractional quantization,

where

The matter energy such as of an electron exists in the form of transverse wave.This is oscillatory (quantum action) and configures a space called a wave packet or “quanta”. We consider that the charge on 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. This is the reason that fractional charges float on their respective segmented masses above the surface of an electron. Considering the symmetry of the harmonic oscillator potential,solution for eigen function exists in the form

in which the differential equation for the radial eigen function (on the surface of an electron)

Modifying Equation (4) with Equation (5) for fractional charge quantized states having an integrated oscillatory effect at the surface of an electron.

We have calculated

where

found

tends to unity and is valid for Equation (8). The reason for the convergence in the above expression is explained in [

shows the energy profile as a straight line with fractional quantum states and that the profile is a tangent on the surface of an electron. The fractional quantum states

Writing the series for Equation (10), i.e..,

Using this series Equation (11) and making it to terminate with asymptotic condition, we have the recurrence relation (to reproduce the more appropriate shape of the string on the surface of an electron).

and Ultimately,we get the series in the form

When centrifugal potential dominates over the attractive electrodynamic potential in the quantum well with its depth equal to radius of an electron swirling causes the strips of the whirlpool to change into elliptic orbits with enhanced Eulerian angles. Thus we have

With Equation (14),

and

With Equation (15) and

The particular solutions of Equation (16) are spherical Bessel functions

and the spherical Neumann functions

Just as in the case of the free particle the condition that

where A is complex constant and can be determined by normalizing the eigen functions

by using momentum operator for fractional charge quantization

Putting Equation (20) in Equation (19), we have

Equation (21) represents the eigenfunction of the swirling strips of the whirl pool in the quantum well. Using Equation (15) in Equation (17) and modifying Equation (21), we have

For

The value of K expressed in Equation (3) will further be modified for Equation (23)

For fractional charge quantization,

we have

We consider

Thus the logarithmic derivative

Equation (1) with

The asymptotic behaviour of the spherical Hankel functions of first and second kind with

With

where B is a complex constant and can be determined by normalizing the eigenfunction

with theoretical eigenvalues

The complex constant B in Equation (28) at

with

(for a free particle).

Remember

Equation (32) is the transcendental equation for arbitrary

and

Equation (34) refers to odd parity solution to quantum potential well. Thus, setting

with

where

(by ignoring the negative sign because it shows attractive electrodynamic potential) in

Equation (38) shows the strength of the quantum well at different depths for whirling strips corresponding to fractional quantized states inside the electron.

The energy profile (eigenfunctions) of the quasi particle nature of electron quanta is obtained. These energy profiles of stretched electron quanta with sub-quanta (twiggs) on the lateral surface in the momentum space due to twisting (gyroscopic behaviour) will pave a new dimension in theoretical physics to let us know about how particles and sub particles settle in the lowest energy state. With our conjuncture of fractional charge quantization, the shape of the energy profile and energy eigenvalues above the surface of quasi particle nature of an electron quanta in chromotized water sample (confirmed with absorption and Fourier transform inferred (FTIR) spectroscopies) is verified. The other results of energy profiles such as on the surface and inside the surface of quasi particle behaviour of an electron quanta will decipher new physics for understanding the complex nature of exchange energy fields, may be the intermediatory nature of strong forces in the nucleus of the atom due to quarks.

SaleemIqbal,FarhanaSarwar,Syed MohsinRaza, (2016) Eigenfunctions for a Quantum Wire on a Single Electron at Its Surface and in the Quantum Well with Beaded Fractional Quantized States for the Fractional Charges. Journal of Applied Mathematics and Physics,04,320-327. doi: 10.4236/jamp.2016.42039