= Inverse construction of a quantum potential
{title2=$V=E+\hbar^2\psi''/(2m\psi)$}
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 $\psi=e^{-ax^4}$ with $a>0$ and $V(0)=0$, it gives $E=0$ and $V=(\hbar^2/2m)(16a^2x^6-12ax^2)$. At a node the quotient must instead be interpreted by a limit; an arbitrary trial <wavefunction> can produce singularities rather than a regular potential.
Back to article page
