The Quantum Hall Effect and Relativistic Hydrogen Atom

In two-dimensional electron gas when a large magnetic field is applied in one direction and an electric field perpendicular to it, there is a current in a direction perpendicular to both. This current is called the Hall effect. It remained without quantization until 1980 when it was found that the quantization leads to the correct measurement of h/e. Therefore, the quantized Hall effect was further studied at high magnetic fields where fractional quantization is found. The fractional charge can arise from the “incompressibility” in the flux quantization. Laughlin wrote a wave function, the excitations of which are fractionally charged quasiparticles. This wave function comes in competition with charge density waves but for a few fractions it does not give the ground state. If incompressibility is not considered, and it is allowed to be compressible, the fractional charge can arise from the angular momentum which appears in the Bohr magneton in the form of g values. Usually the positive spin is considered but we consider the positive as well as the negative values so that there is spin-charge coupling. The values thus calculated for the fractional charge agree with the experimental data on the quantum Hall effect. The fractional charge requires both the positive as well as negative sign for the spin. When this idea is applied to the hydrogen atom, it is necessary to perform the calculation using the Dirac equation, so that in a relativistically correct treatment, the Bohr’s -1/n type eigen values are seriously affected. Multiple values for the charge of the electron emerge and the eigen values of the hydrogen atom then depend on the effective electron charge.