Abstract
Every spatially spherical, continuous, and scalable probability distribution of a particle is shown to have a general upper bound for its local densities that varies as the inverse cube of the distance from its origin, the coefficient of proportionality depending only on the form of the distribution function. For such a quantum distribution, a rigorous upper bound is derived for the coefficient that is simply the square root of its Heisenberg product divided by 2π. This enables the bound to be evaluated in terms of spectral properties of the particle. A similar orientationally dependent bound is derived for certain nonspherical distributions.