Application of Clifford Algebra in Solving the Eigen Equations of Quantum Mechanics

Clifford algebra is unified language and efficient tool for geometry and physics. In this paper, we introduce this algebra to derive the integrable conditions for Dirac and Pauli equations. This algebra shows the standard operation procedure and deep insights into the structure of the equations. Usually, the integrable condition is related to the special symmetry of transformation group, which involves some advanced mathematical tools and its availability is limited. In this paper, the integrable conditions are only regarded as algebraic conditions. The commutators expressed by Clifford algebra have a neat and covariant form, which are naturally valid in curvilinear coordinate system and curved space-time. For Pauli and Schr\"odinger equation, many solutions in axisymmetric potential and magnetic field are also integrable. We get the scalar eigen equation in dipole magnetic field. By the virtue of Clifford algebra, the physical researches may be greatly promoted.