We revisit the topic of two-state quantum systems using the Clifford Algebra in three dimensions \(Cl_3\). In this description, both the quantum states and Hermitian operators are written as elements of \(Cl_3\). By writing the quantum states as elements of the minimal left ideals of this algebra, we compute the energy eigenvalues and eigenvectors for the Hamiltonian of an arbitrary two-state system. The geometric interpretation of the Hermitian operators enables us to introduce an algebraic method to diagonalize these operators in \(Cl_3\). We then use this approach to revisit the problem of a spin-1/2 particle interacting with an external arbitrary constant magnetic field, obtaining the same results as in the conventional theory. However, Clifford algebra reveals the underlying geometry of these systems, which reduces to the Larmor precession in an arbitrary plane of \(Cl_3\).

我们将使用Clifford代数在三个维度\（Cl_3 \）中重新讨论二态量子系统的主题。在此描述中，量子态和厄米算子都被写为\（Cl_3 \）的元素。通过将量子态写为该代数的最小左理想元素，我们可以计算任意二态系统的哈密顿量的能量本征值和本征向量。Hermitian算子的几何解释使我们能够引入代数方法以对\（Cl_3 \）中的这些算子进行对角化。然后，我们使用这种方法重新研究自旋1/2粒子与外部任意恒定磁场相互作用的问题，获得与传统理论相同的结果。但是，克利福德代数揭示了这些系统的基本几何形状，从而简化为\（Cl_3 \）的任意平面中的拉莫尔进动。