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The quantum electrodynamic (QED) behaviour is studied for quantum Hall effect (QHE). Quantum theory with conjecture of fractional charge quantization (quantum dipole moment), eigenfunctions for fractional charge quantization at the surface of a twisted and twigged electron quanta and above its surface, fractional Fourier transform and Hermite function for fractional charge quantization is developed. With energy eigen value equation for QHE and with energy operator on an eigenfunction of a twisted and twigged electron quanta, the corresponding eigenfunctions are normalized with Schrodinger’s quantum wave mechanical equation for electric scalar and magnetic potentials, respectively (QED behavior). The fractional electric and magnetic fields with their corresponding potentials for the quantized fractional states in semiconducting hereto structures are theoretically calculated. Such mathematical expressions are in good agreement with experimental results of Nobel Prize winning scientists Klitzing, Haroche, Peter and Gruebber. Our results can also explain the hybridized states of orbits with emphasis on sigma and pi bonding and their corresponding antibonding orbitals as a manifestation of electrophilic and nucleophilic chemical reactions.

Von Klitzing in the year 1987 won the Nobel Prize for physics on deciphering quantum Hall effect in heteoro- structures (sandwich like) semi conductors [

A new theory describing how charge being a constant physical entity on an electron in the momentum space is fractionally quantized, is presented [

The magnetoresistance in quantum Hall effect (QHE) is of two types, one is longitudinal and the other is transverse. The longitudinal magnetoresistance is associated with magnetic field parallel to the current. The excitonic quantized Hall state becomes maximum at the total Landau level with vanishing longitudinal component and overwhelming transverse magnetoresistance [

Saleem Iqbal et al. [

Equation (1) represents plane wave for a rotation vector

The Hermite function for the fractional quantum states [

The normalized eigenfunction for a twisted and twigged electron quanta above its surface [

Equation (4) yields theoretical eigenvalues

Rewriting the equation of QHE [

where

we know the values of x and y and change the crystal momentum with their corresponding momentum operators. With Hermitian Hamiltonian operator, energy eigen values can be determined. The QED behaviour of a quantized twisted and twigged electron quanta in the semi conductor heteorostructurescan be envisaged with electric fields (fractionally quantized in QHE). Writing the electric filed,

where

where

where

matter, i.e.,

Equation(8) for dipole radiation yields

The last term in Equation (10) is oscillatory with natural frequency

which is neglected in our calculations. The part of some energy above the surface of the twisted and twigged electron quanta with its equivalent frequency coincides with natural frequency of the sample (resonance or damped oscillation), as a consequence of which, the energy associated with resonance is dissipated in free space where as the remaining energy is used to quantize the twisted and twigged electron quanta [Equation (5) is applicable with Equation (4)]. Equation (10) becomes

In our case, we have fractional electric fields appearing due to fractional charge quantization on twigs (sub- quanta) which are at the lateral surface of an electron string quanta.

Changing the x- and y-components of crystal momentum,

On comparison of Equations (13) and (14), we have

Equation (15) on rearrangement for

Equation (17) on simplification yields

Subtracting Equation (16) from itself for both LHS and RHS, we have

Putting Equation (19) in Equation (18) for determining the electric scalar potential due to a quantized twisted and twigged electron quanta in semiconductor heteorostructures, we get on simplification

Equation (20) shows that the scalar electric filed potential is zero for fractional charge quantization, i.e., for QHE

Adding Equation (16) from its self for both LHS and RHS, we get on simplification

which is similar to Equation (19). Thus, the validity of

On comparison of RHS of Equations (13) and (14), we can put the equality of RHS to the equality of the above expression:

Using Parseval’ formulas,. i.e.,

on Equation (22) where

The

Integrating this expression with respect to time

For oscillatory effect in semi conductor hetreostructure the energy is dissipated, i.e.,

becomes negligible. Equation (25) becomes

Equation (26) shows that the magnetic vector potential for fractional quantized states depends on twisting angle

where

[from Equation (26)]

Equation (27) shows the quantization of fractional electric field for

where

Putting Equation (27) in Equation (29), we have

The negative sign shows induced magnetic fields due to twisting and corresponding phase changes (energy changes) on twigs. Neglecting the negative sign, we have

Equation (30) shows that

follows the helical pattern for dipole radiations due to twisted magnetic fields for each of the twigs on the lateral surface of an electron string quanta in semiconducting hetreostructures for QHE.

With conjecture of fractional charge quantization, eigenfunctions for a bounded, stretched, twisted and twigged electron quanta are obtained. The fractional electric and magnetic fields at each of the twigs of a bounded, stretched, and twisted electron quanta, and the corresponding scalar electric and magnetic vector potentials for the quantized fractional states in semiconducting heterostructures are theoretically calculated and explained with obtained mathematical relations. Our results can also explain the hybridized states of orbits with emphasis on sigma and pi bonding and their corresponding antibonding orbitals as a manifestation of electrophilic and nucleophilic chemical reactions.

Saleem Iqbal,Muhammad Zafar,Farhana Sarwar,Syed Mohsin Raza,Muhammad Afzal Rana,1 1, (2016) Application of Electrodynamic Theory on Quantum Hall Effect. World Journal of Condensed Matter Physics,06,87-94. doi: 10.4236/wjcmp.2016.62012