Semiclassical analysis of constrained quantum systems

TL;DR

This paper investigates how quantum particles constrained near a submanifold behave in the semiclassical limit, showing that their wave functions closely follow classical constrained trajectories with quantifiable approximation errors.

Contribution

It provides a rigorous semiclassical analysis of constrained quantum systems, establishing norm approximation of quantum evolution by classical constrained dynamics.

Findings

Wave functions are approximated by classical trajectories with error of order hbar^(1/2).

The analysis applies to quantum particles constrained by strong potentials near submanifolds.

The results bridge quantum dynamics and classical constrained systems in the semiclassical regime.

Abstract

We study the dynamics of a quantum particle in R^(n+m) constrained by a strong potential force to stay within a distance of order hbar (in suitable units) from a smooth n-dimensional submanifold M. We prove that in the semiclassical limit the evolution of the wave function is approximated in norm, up to terms of order hbar^(1/2), by the evolution of a semiclassical wave packet centred on the trajectory of the corresponding classical constrained system.