A Characteristic Particle Length

It is argued that there are characteristic intervals associated with any particle that can be derived without reference to the speed of light $c$. Such intervals are inferred from zeros of wavefunctions which are solutions to the Schr\"odinger equation. The characteristic length is $\ell=\beta^2\hbar^2/(8Gm^3)$, where $\beta=3.8\dots$; this length might lead to observational effects on objects the size of a virus.