Commutator with powers of an eigenoperator
= Commutator with powers of an eigenoperator
{title2=$[A,B]=cB\ \Rightarrow\ [A,B^n]=ncB^n$}
Suppose <linear operators> $A,B$ obey $[A,B]=cB$, where $c$ is a scalar, on a common invariant <operator domain>. The <commutator product rule> gives
$$
[A,B^n]=\sum_{r=0}^{n-1}B^r[A,B]B^{n-1-r}=ncB^n,\qquad n\geq1.
$$
For $n=0$ both sides vanish. This identity explains how powers of a <creation operator> shift the <number operator> <eigenvalue>.