I am going through Groenewold's theorem and in his book: On The Principles of Elementary Quantum Mechanics, page 45, eq. 4.11:
$\frac{1}{6}\left[\left(\mathbf{p}^{3}+3 c_{1} \mathbf{p}+d_{1}\right),\left(\mathbf{q}^{2}+c_{2}\right)\right]=\frac{1}{2}\left(\mathbf{p}^{2} \mathbf{q}+\mathbf{q} \mathbf{p}^{2}\right)+c_{1} \mathbf{q}$
where $\mathbf{p}$ and $\mathbf{q}$ are the canonical operators.
I cannot get the right-hand side. What I get is $\frac{1}{6}([\mathbf{p}^3,\mathbf{q}^2]+3c_1[\mathbf{p},\mathbf{q}^2])$. Help?
From $[p,\,q]=1$, we can prove $[p,\,q^n]=nq^{n-1}$ by induction, using $[a,\,bc]=[a,\,b]c+b[a,\,c]$. For our purposes, we don't need to realize or prove this general result; we only need to verify the $n=2$ case, then the $n=3$ case. So$$[p^3,\,q^2]=[p^3,\,q]q+q[p^3,\,q]=3(p^2q+qp^2),\,[p,\,q^2]=2q.$$