Harmonic-oscillator wavefunction inflections (${\psi }_n^{\prime \prime }=b^{-2}({\xi }^2-2n-1){\psi }_n$)

For a one-dimensional quantum harmonic oscillator, let $b=\sqrt{\hslash /(m\omega )}$ and $\xi =x/b$. The stationary Schrödinger equation gives the displayed curvature relation. Inflections occur at classical turning points $\xi =\pm \sqrt{2n+1}$ and at simple wavefunction wavefunction nodes, provided the curvature changes sign. Thus the first excited state has an inflection at its central wavefunction node as well as at both turning points. Restricting attention to where $V=E$ misses wavefunction node inflections.