The Pauli matrices are not introduced as isolated famous matrices. In the logic of the chapter, they are built from the spectral decomposition of the Stern-Gerlach spin observables. The two eigenvalues are \(\pm1\); the eigenvectors are the outgoing spin states selected by the apparatus.
For a magnet aligned along \(z\), the eigenstates are \(|+\rangle\) and \(|-\rangle\). The spectral form gives
Using the \(z\)-basis vectors, the outer products are
Therefore
For a magnet aligned along \(x\), the eigenstates are \(|+\rangle_x\) and \(|-\rangle_x\). The spectral form begins as
Substitute the basis relations \(|\pm\rangle_x=(|+\rangle\pm|-\rangle)/\sqrt2\). The diagonal contributions cancel and the off-diagonal contributions remain:
In the \(z\)-basis, this becomes
The matrices must reproduce the Stern-Gerlach outcomes. For \(z\), the check is immediate:
For \(x\), the matrix is not diagonal in the \(z\)-basis, but it is diagonal in its own eigenbasis:
In the \(z\)-basis, \(\hat\sigma_z\) is diagonal because the basis vectors are already the possible results of SG(Z). The operator \(\hat\sigma_x\) is not diagonal in that same basis because an SG(X) device asks a different question. Its eigenvectors are balanced superpositions of \(|+\rangle\) and \(|-\rangle\).
This is the first algebraic sign of measurement incompatibility. The apparatuses SG(Z) and SG(X) do not simply reveal two preexisting labels at once; they correspond to different bases.
The chapter does not construct \(\hat\sigma_y\) at this point because the \(y\)-basis vectors have not yet been introduced. The full angular-momentum algebra will be developed later. For now, \(\hat\sigma_z\) and \(\hat\sigma_x\) are enough to show how matrix mechanics encodes measurements and non-commuting observables.
The Pauli matrix does not draw the apparatus, but it contains what the apparatus can output. Its eigenvalues label the separated beams, while its eigenvectors describe the states carried by those beams after the measurement.
The Pauli matrices encode spin measurements along different axes. In the \(z\)-basis, \(\hat\sigma_z\) is diagonal because \(|+\rangle\) and \(|-\rangle\) are already its eigenstates:
The \(x\)-spin operator is not diagonal in that same basis, but its eigenvectors are the \(x\)-basis states:
This prepares students for Stern-Gerlach exercises: choose the apparatus axis, choose the matching operator, then use the appropriate eigenbasis.
This page is designed to support short guided exercises on: Spin-1/2 operators, the z and x bases, and the Pauli matrices as the working example of matrix mechanics.
Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.