 Noncommuting observables in two dimensions

For two dimensionless observables on a two-level system, takeBoth are Hermitian matrices: $A$ has eigenvalues $1,0$ and $B$ has eigenvalues $1,-1$. YetFor instance, $AB|0⟩=0$ whereas $BA|0⟩=|1⟩$. In $AB$, $B$ acts first. Neither product is self-adjoint, so the product of two observables is not automatically another observable. The commutator records this dependence on order. These operator products by themselves do not specify the outcomes of consecutive measurements; that additionally requires measurement in quantum mechanics.