Common physical mechanism for integer and fractional quantum Hall effects

Integer and fractional quantum Hall effects were studied with different physics models and explained by different physical mechanisms. In this paper, the common physical mechanism for integer and fractional quantum Hall effects is studied, where a new unified formulation of integer and fractional quantum Hall effect is presented. Firstly, we introduce a 2-dimensional ideal electron gas model in the presence of strong magnetic field with symmetry gauge, and the transverse electric filed $\varepsilon_2$ is also introduced to balance Lorentz force. Secondly, the Pauli equation is solved where the wave function and energy levels is given explicitly. Thirdly, after the calculation of the degeneracy density for 2-dimensional ideal electron gas system, the Hall resistance of the system is obtained, where the quantum Hall number $\nu$ is introduced. It is found that the new defined $\nu$, called filling factor in the literature, is related to radial quantum number n and angular quantum number $|m|$, the different $n$ and $|m|$ correspond to different $\nu$. This provides unification explaination for integer and fractional quantum Hall effects. It is predicated that more new cases exist of fractional quantum Hall effects without the concept of fractional charge.