Let $H=0.5(p^2+q^2)$ be the Hamiltonian in natural units. Let $f_{n}$ be the eigenfunctions of H. Show that $\langle f_{n},f_{m}\rangle=1$ if n=m, and equal to 0 otherwise. Do this by using the creation\annihaltor operators: $a_{+}$ , $a_{-}$
we have that $f_{n}=\frac{a_{+}^{n}}{(n!)^{0.5}}f_{0}$ so
$\langle f_{n},f_{m}\rangle=\frac{\langle f_{0},a_{-}^n(a_{+})^mf_{0}\rangle}{(n!m!)^{0.5}}$
$a_{-}f_{0}=0$ so $[a_{-}^n,a_{+}^m]=a_{-}^na_{+}^m$ and
$[a_{-},a_{+}^m] =ma_{+}^{m+1}$.
How does $[a_{-},a_{+}^m]$ relate to $[a_{-}^n,a_{+}^m]$? When I know this can I consider cases n>m, n
Thanks
Is this what you are looking for?
\begin{align*} [a_-^n,a_+^m]=a_-a_-^{n-1}a_+^m-a_+^ma_-^{n-1}a_-&=a_-[a_-^{n-1},a_+^{m}]a_-\\ &=\cdots\\ &=a_-^{n-1}[a_-,a_+^{m}]a_-^{n-1} \end{align*}