  We show in spatially one dimensional Madelung fluid that a simple requirement
on local stability of the maximum of quantum probability density will, if
combined with the global scale invariance of quantum potential, lead to a class
of quantum probability densities globally being self-trapped by their own
self-generated quantum potentials, possessing only a finite-size spatial
support. It turns out to belong to a class of the most probable wave function
given its energy through the maximum entropy principle. We proceed to show that
there is a limiting case in which the quantum probability density becomes the
stationary-moving soliton-like solution of the Schr\"odinger equation.
