论文标题
重新审视的两国量子系统：几何代数方法
Two-State Quantum Systems Revisited: a Geometric Algebra Approach
论文作者
论文摘要
我们使用几何代数（GA）在三个维度$ \ Mathcal G_3 $中使用几何代数（GA）重新审视两国量子系统的主题。在此描述中，量子状态和遗传操作员均写为$ \ Mathcal g_3 $的元素。通过将量子状态作为该代数的最小左左左元素的要素，我们计算了任意两国系统的哈密顿量的能量特征值和特征值。 Hermitian操作员的几何解释使我们能够引入一种代数方法，以将这些操作员对角线化。然后，我们使用这种方法来重新审视与外部任意恒定磁场相互作用的自旋$ 1/2 $粒子的问题，从而获得与常规理论相同的结果。但是，GA揭示了这些系统的基本几何形状，这些几何形状将$ \ Mathcal G_3 $的任意平面减少到Larmor进动。
We revisit the topic of two-state quantum systems using Geometric Algebra (GA) in three dimensions $\mathcal G_3$. In this description, both the quantum states and Hermitian operators are written as elements of $\mathcal G_3$. 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 GA. 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, GA reveals the underlying geometry of these systems, which reduces to the Larmor precession in an arbitrary plane of $\mathcal G_3$.
