Journal of Applied Mathematics and Physics, 2016, 4, 320-327 Published Online February 2016 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.4236/jamp.2016.42039 How to cite this paper: Iqbal, S., Sarwar, F. and Raza, S.M. (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 Ap- plied Mathematics and Physics, 4, 320-327. http://dx.doi.org/10.4236/jamp.2016.42039 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 Saleem Iqbal 1 , Farhana Sarwar 1,2 , Syed Mohsin Raza 3 1 Department of Mathematics, University of Balochistan, Quetta, Pakistan 2 Department of Mathematics, F. G. Girls Degree College, Quetta, Pakistan 3 Department of Physics, University of Balochistan, Quetta, Pakistan Received 10 January 2016; accepted 22 February 2016; published 25 February 2016 Copyright © 2016 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract 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 ef- fect 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 quan- tized 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. Keywords Fractional Charge Quantization, Fractional Fourier Transform, Hermite Polynomials, Sub Quanta of Electron, Spherical Bessel and Neumann Functions, Lagueree Polynomials