Canonical quantization replaces classical Poisson brackets with commutators via $\{f,g\}_{\mathrm{PB}} \to \frac{1}{i\hbar}[\hat f,\hat g]$. The canonical pair satisfies $[\hat x,\hat p]=i\hbar$.

When classical expressions are nonlinear in $x$ and $p$, quantization is not unique because operator ordering matters. This is one reason modern treatments often begin from symmetries, algebras, or path integrals.

Angular momentum operators obey $[\hat J_i,\hat J_j]=i\hbar\,\varepsilon_{ijk}\hat J_k$. The ladder operator method yields eigenvalues $J^2=j(j+1)\hbar^2$ and $J_z=m\hbar$.

Spin is not literal spatial rotation of a point particle; it is an irreducible representation of the rotation group and its double cover. Spin-$\frac{1}{2}$ systems are described by Pauli matrices and two-component spinors.