Two-State Quantum Systems Revisited: a Geometric Algebra Approach

TL;DR

This paper employs Geometric Algebra to analyze two-state quantum systems, providing a geometric perspective that simplifies calculations and reveals underlying structures, including the behavior of spin-$1/2$ particles in magnetic fields.

Contribution

It introduces a novel geometric algebra framework for two-state quantum systems, enabling algebraic diagonalization and geometric interpretation of quantum states and operators.

Findings

Successfully computes energy eigenvalues and eigenvectors using GA

Reproduces known results for spin-$1/2$ in magnetic fields

Reveals geometric structure underlying quantum dynamics

Abstract

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 $\mathcal G_3$.