Renormalization in Nonrelativistic Quantum Mechanics

TL;DR

This paper discusses how renormalization techniques are applied to nonrelativistic quantum mechanics problems involving singular potentials, leading to finite, consistent solutions for bound states and scattering, and highlights the role of renormalization group equations.

Contribution

It demonstrates the application of renormalization in nonrelativistic quantum mechanics, providing a method to handle ultraviolet divergences in two-body problems with singular potentials.

Findings

Renormalization yields finite results for bound states and scattering.

Perturbative and exact solutions are consistent after renormalization.

Renormalization group equations are useful in this context.

Abstract

The importance and usefulness of renormalization are emphasized in nonrelativistic quantum mechanics. The momentum space treatment of both two-body bound state and scattering problems involving some potentials singular at the origin exhibits ultraviolet divergence. The use of renormalization techniques in these problems leads to finite converged results for both the exact and perturbative solutions. The renormalization procedure is carried out for the quantum two-body problem in different partial waves for a minimal potential possessing only the threshold behavior and no form factors. The renormalized perturbative and exact solutions for this problem are found to be consistent with each other. The useful role of the renormalization group equations for this problem is also pointed out.