Introduction: The Mathematics of Spin
Spin-1/2 particles (electrons, protons, neutrons) live in a 2-dimensional Hilbert space. The spin operators are represented by the Pauli matrices.
The Spin States
\[|+\rangle = |\uparrow\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad |-\rangle = |\downarrow\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\]The Pauli Matrices
\[\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\]Spin operators: \(\hat{S}_i = \frac{\hbar}{2}\sigma_i\)
Properties
- \(\sigma_i^2 = I\) (each squares to identity)
- \(\sigma_i\sigma_j = i\epsilon_{ijk}\sigma_k + \delta_{ij}I\)
- \([\sigma_i, \sigma_j] = 2i\epsilon_{ijk}\sigma_k\)
- All trace zero, all Hermitian
The Quantum Connection
The Pauli matrices are the fundamental building blocks of quantum information. A single qubit is a spin-1/2 system; quantum gates are operations on Pauli matrices. Every 2×2 Hermitian matrix can be written as a combination of Pauli matrices plus identity—they form a basis for the observable algebra of the qubit.