Pauli Matricies and Quantum Measurements

Pauli spin matrices are operators that describe the action of quantum measurements in the context of spin, particularly for spin-½ particles. These matrices are used to represent measurements along the x, y, and z axes in quantum mechanics.

\[\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\]

Let’s focus on \(\sigma_x\). When transforming from either \(y\)- or \(z\)-space to \(x\)-space, some vectors in \(x\)-space will carry over, meaning they will remain eigenvectors of \(\sigma_x\). The key difference is that they will point in the directions corresponding to both \(\pm \ket{\psi}\), with a proportionality factor \(\lambda\).

\[\sigma_x \ket{\psi} = \pm \lambda \ket{\psi}\]

What makes the Pauli spin matrices special is that they transform the space while leaving certain vectors unchanged, except for being scaled by a factor of ±1.

For a vector \(\ket{\psi}\), if it is an eigenvector of a Pauli matrix \(\sigma_i\) (where \(i = x, y, z\) ), it satisfies the following equation:

\[\sigma_i \ket{\psi} = \pm \ket{\psi}\]

This means that applying the Pauli matrix \(\sigma_i\) to the vector \(\ket{\psi}\) leaves it unchanged, except for a scaling factor of \(\pm 1\).