Multiple-Scale Analysis of Quantum Systems

TL;DR

This paper extends multiple-scale analysis to quantum anharmonic oscillators, providing exact operator solutions and connecting perturbative and semiclassical approaches, thus overcoming limitations of traditional perturbation theory.

Contribution

It introduces a quantum extension of multiple-scale analysis, offering exact solutions and insights into quantum anharmonic oscillators beyond conventional methods.

Findings

Exact operator differential equations solved for quantum anharmonic oscillator

Operator mass renormalization derived from the solution

Connection established between perturbative and semiclassical wave functions

Abstract

Conventional weak-coupling Rayleigh-Schr\"odinger perturbation theory suffers from problems that arise from resonant coupling of successive orders in the perturbation series. Multiple-scale analysis, a powerful and sophisticated perturbative method that quantitatively analyzes characteristic physical behaviors occurring on various length or time scales, avoids such problems by implicitly performing an infinite resummation of the conventional perturbation series. Multiple-scale perturbation theory provides a good description of the classical anharmonic oscillator. Here, it is extended to study (1) the Heisenberg operator equations of motion and (2) the Schr\"odinger equation for the quantum anharmonic oscillator. In the former case, it leads to a system of coupled operator differential equations, which is solved exactly. The solution provides an operator mass renormalization of the theory. In the latter case, multiple-scale analysis elucidates the connection between weak-coupling perturbative and semiclassical nonperturbative aspects of the wave function.