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Let $\omega =\sqrt{k/m}$, $b=\sqrt{\hslash /(m\omega )}$ and $\xi =x/b$. With ${\psi }_0={\pi }^{-1/4}{b}^{-1/2}{e}^{-{\xi }^2/2}$, the next two normalized harmonic-oscillator wavefunctions are ${\psi }_1=\sqrt{2}\xi {\psi }_0$ and ${\psi }_2=(2{\xi }^2-1){\psi }_0/\sqrt{2}$. Their energies are $E_n=(n+1/2)\hslash \omega$, with respectively zero, one and two wavefunction nodes, alternating even and odd parity. The Schrödinger equation givesFor harmonic-oscillator wavefunction inflections, the inflections include both classical turning points and simple wavefunction wavefunction nodes. In dimensionless coordinates the complete sets areThe curvature changes sign at every listed point; none of these states has a wavefunction node coinciding with its turning points.

For the quantum harmonic oscillator, put ${\ell }_0=\sqrt{\hslash /(m\omega )}$ and $\xi =x/{\ell }_0$. The normalized energy eigenstates arewhere $H_n$ is a Hermite polynomial. Successive energies are equally spaced; the ground state retains zero-point energy $\hslash \omega /2$. The $n$th wavefunction has $n$ nodes of a wavefunction and parity $(-1{)}^n$. In particular, the first three shapes are proportional to $e^{-{\xi }^2/2}$, $\xi e^{-{\xi }^2/2}$, and $(2{\xi }^2-1)e^{-{\xi }^2/2}$. The classical turning points satisfy $x=\pm {\ell }_0\sqrt{2n+1}$; the wavefunctions extend beyond them and decay, rather than ending abruptly.

The wavefunctions are plotted with arbitrary vertical scale about their energy levels. Their signed excursions are amplitudes, not additional energies or probability densities.

A quantum particle above a hard reflecting wall in a linearly increasing potential $V(x)=Fx$ for $x>0$. The Schrödinger equation reduces to the Airy equation. Decay at infinity selects $\mathrm{Ai}$, while the wall requires $\mathrm{Ai}(-z_n)=0$. Thus ${\psi }_n\propto \mathrm{Ai}(x/\ell -z_n)$ and the energies are the displayed values. The $n$th state has $n-1$ interior nodes and a decaying tail beyond its classical turning point. A gravitational realization has $F=mg$; gravitational neutron bound-state transmission tests such confinement experimentally.

For two simple turning points $a(E)<b(E)$ surrounding one classically allowed well,

A point or surface at which a wavefunction vanishes. In one dimension a simple node of a real stationary wavefunction changes its sign. A node is not generally the same as a classical turning point; for a smooth finite potential, the stationary Schrödinger equation also makes the second derivative vanish at a wavefunction node.

For a smooth symmetric confining potential energy with four separated simple classical turning points $-x_2<-x_1<x_1<x_2$, defineThe Airy turning-point connection formula applied to a decaying exterior wavefunction gives a central WKB approximation proportional to ${\chi }^{-1/2}[2\cos \theta e^K+\sin \theta e^{-K}]$, where $\chi =\sqrt{2m(V-E)}$ and $K(x)={\hslash }^{-1}{\int }_x^{x_1}\chi dx$. Since ${\chi }^{\prime }(0)=0$ and $2K(0)=\varphi$, the even Neumann boundary condition gives $\tan \theta =2e^{\varphi }$, while the odd Dirichlet boundary condition gives $\tan \theta =-2e^{\varphi }$. For $\varphi \gg 1$, $\theta =(n+\frac{1}{2})\pi \mp \frac{1}{2}{e}^{-\varphi }+O(e^{-3\varphi })$, with the minus correction for even parity. Since ${\theta }^{\prime }=\mathcal{T}/(2\hslash )$ for the single-well classical oscillation period $\mathcal{T}$, the leading double-well tunneling splitting is $2\hslash e^{-\varphi }/\mathcal{T}$.