Applying Classical Geometry Intuition to Quantum Spin

TL;DR

This paper demonstrates how classical geometric intuition can be applied to derive and understand the properties of quantum spin-1/2 systems, highlighting the connection between classical geometry and quantum angular momentum.

Contribution

It introduces a non-rigorous, geometry-based approach to derive Pauli matrices and basis relationships, emphasizing classical-quantum correspondence.

Findings

Geometric orthogonality guides the form of Pauli matrices

Classical intuition helps understand quantum basis relationships

Illustrates differences and connections between geometric space and Hilbert space

Abstract

Using concepts of geometric orthogonality and linear independence, we logically deduce the form of the Pauli spin matrices and the relationships between the three spatially orthogonal basis sets of the spin-1/2 system. Rather than a mathematically rigorous derivation, the relationships are found by forcing expectation values of the different basis states to have the properties we expect of a classical, geometric coordinate system. The process highlights the correspondence of quantum angular momentum with classical notions of geometric orthogonality, even for the inherently non-classical spin-1/2 system. In the process, differences in and connections between geometrical space and Hilbert space are illustrated.