Renormalization in Quantum Mechanics

TL;DR

This paper applies Wilson renormalization to quantum mechanical systems, revealing fixed points and crossover phenomena, and explores broader implications including potential analogues of the $C$ function.

Contribution

It demonstrates the implementation of Wilson renormalization in simple quantum systems and discusses potential extensions and analogues in this context.

Findings

Inverse square potential has a nontrivial ultraviolet fixed point

Hulthen potential shows crossover behavior

Discussion of potential $C$ function analogue

Abstract

We implement the concept of Wilson renormalization in the context of simple quantum mechanical systems. The attractive inverse square potential leads to a $\b$ function with a nontrivial ultraviolet stable fixed point and the Hulthen potential exhibits the crossover phenomenon. We also discuss the implementation of the Wilson scheme in the broader context of one dimensional potential problems. The possibility of an analogue of Zamolodchikov's $C$ function in these systems is also discussed.