Abstract—The hierarchy of states is developed with increasing the value of l, the orbital angular momentum, is now called "Shrivastava's hierarchy of fractional states". We have examined the odd denominator states as extended to a hierarchy of continued fraction which describes a very large abundance of fractional states. The heirarchy developed by random numbers m and pi (i=1,2,…,4) by using the continued fraction is known as the Haldane's hierarchy. We find that the predictions of the Dirac equation agree with the idea of fractional charges. We have introduced the combination of spin and orbital quantum numbers, including the negative sign for spin, in such a way that there occur fractional charges through the Bohr magneton. This leads to doubling of eigen values so that we define an additional matrix the properties of which are important when magnetic field is present. There is a spin-charge coupling so that spin ½ particle can have the zero or one charge. The Dirac equation can accommodate not only charges of 0 and ±e but also fractional values such as 1/3 and 2/3.

Index Terms—Dirac equation, fractional charges, hierarchy, angular momentum.

Keshav N. Shrivastava is with the University of Malaya, Kuala Lumpur, and with the University of Hyderabad, Hyderabad 500046, India (e-mail: keshav1001@yahoo.com).

Cite:Keshav N. Shrivastava, "Hierarchy of States and the Dirac Equation in the Quantum Hall Effect," International Journal of Applied Physics and Mathematics vol. 3, no. 4, pp. 264-269, 2013.