Hidden scale in quantum mechanics

TL;DR

This paper demonstrates that hidden scales introduced via self-adjoint extensions enable localization of free particle wave-packets in quantum mechanics, challenging classical scale invariance and revealing new localized states.

Contribution

It reveals how self-adjoint extensions create hidden scales in quantum systems, leading to localized wave-packets in classically scale-invariant free particle models.

Findings

Localized wave-packets can exist due to hidden scales.

Self-adjoint extensions induce nontrivial boundary conditions.

Localization occurs in systems on partial spaces like half-lines or sectors.

Abstract

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.