Inverse construction of a quantum potential ($V=E+{\hslash }^2{\psi }^{\prime \prime }/(2m\psi )$)

For a real, sufficiently smooth, nonvanishing wavefunction of a particle with constant mass, the time-independent Schrödinger equation determines the potential energy through the displayed logarithmic-curvature formula. An additive choice of energy origin fixes the remaining constant. For $\psi =e^{-a{x}^4}$ with $a>0$ and $V(0)=0$, it gives $E=0$ and $V=({\hslash }^2/2m)(16a^2{x}^6-12a{x}^2)$. At a node the quotient must instead be interpreted by a limit; an arbitrary trial wavefunction can produce singularities rather than a regular potential.