  We show that the intriguing localization of a free particle wave-packet is
possible due to a hidden scale present in the system. Self-adjoint extensions
(SAE) is responsible for introducing this scale in quantum mechanical models
through the nontrivial boundary conditions. We discuss a couple of classically
scale invariant free particle systems to illustrate the issue. In this context
it has been shown that a free quantum particle moving on a full line may have
localized wave-packet around the origin. As a generalization, it has also been
shown that particles moving on a portion of a plane or on a portion of a three
dimensional space can have unusual localized wave-packet.
