= Noncommuting observables in two dimensions
{title2=$[A,B]=AB-BA$}
For two dimensionless <observables> on a <two-level system>, take
$$
A=\begin{pmatrix}1&0\\0&0\end{pmatrix},\qquad B=\begin{pmatrix}0&1\\1&0\end{pmatrix}.
$$
Both are <Hermitian matrices>: $A$ has <eigenvalues> $1,0$ and $B$ has <eigenvalues> $1,-1$. Yet
$$
AB=\begin{pmatrix}0&1\\0&0\end{pmatrix},\qquad BA=\begin{pmatrix}0&0\\1&0\end{pmatrix},\qquad [A,B]=\begin{pmatrix}0&1\\-1&0\end{pmatrix}\ne0.
$$
For instance, $AB|0\rangle=0$ whereas $BA|0\rangle=|1\rangle$. 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>.
Back to article page
