两态量子系统的几何代数研究
Two-State Quantum Systems Revisited: a Geometric Algebra Approach
摘要：
我们使用三维几何代数（GA）$\mathcal G_3$重新讨论了两态量子系统的话题。在本说明书中，量子态和厄米算符都被写为$\mathcal G_3$的元素。通过把量子态写成这个代数的最小左理想的元素，我们计算了任意两态系统的哈密顿量的能量本征值和本征向量。厄米特算子的几何解释使我们能够引入一种代数方法来对角化这些运营商在GA。然后，我们使用这种方法重新审视的问题的自旋$1/2$粒子与外部任意恒定磁场相互作用，获得相同的结果，在传统的理论。然而，遗传算法揭示了这些系统的基本几何结构，这减少到拉莫尔进动在任意平面的$\mathcal G_3$。
英文摘要：
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 $\mathcal G_3$.
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