 Solution

For the quantum harmonic oscillator, using $[\hat{x},\hat{p}]=i\hslash$, direct multiplication givesThe commutators areThus each ladder operation shifts energy by one quantum. The ladder operator norms $\Vert a|n⟩{\Vert }^2=n$ and $\Vert a^{†}|n⟩{\Vert }^2=n+1$, with conventional phases, giveLet $\ell =\sqrt{\hslash /(m\omega )}$. In position space $\hat{p}=-i\hslash {\partial }_x$, so $a{\psi }_0=0$ becomes $(x+{\ell }^2{\partial }_x){\psi }_0=0$. Normalizing and then applying $a^{†}$ givesTheir overlap vanishes because ${\psi }_0^{\ast }{\psi }_1$ is odd on the full real line; the Gaussian second moment verifies $\int |{\psi }_1{|}^{2}dx=1$.