Abstract: An equation of motion satisfied by free two-component Pauli spinors is derived by seeking the square root of the Hamiltonian operator of a free electron,starting from Schrdinger equation in the non-relativistic framework. Then,using the minimum coupling principle,an equation of motion for an electron in the external electromagnetic fields,the so-called Pauli equation,is obtained. Based on it,an inconsistency between classical electrodynamics and quantum mechanics is explored,and the analysis for relativistic spin and non-relativistic spin is carried out from the representation theory of groups and the quantization.

Key words: non-relativity, Schrdinger equation, Hamiltonian operator, minimal coupling principle, Paulie-quation