= Harmonic-oscillator wavefunction inflections
{title2=$\psi_n''=b^{-2}(\xi^2-2n-1)\psi_n$}
For a one-dimensional <quantum harmonic oscillator>, let $b=\sqrt{\hbar/(m\omega)}$ 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.
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