Multiple-Scale Analysis of the Quantum Anharmonic Oscillator

TL;DR

This paper extends multiple-scale perturbation theory to the quantum anharmonic oscillator, providing an exact operator solution and revealing operator mass renormalization, thus improving understanding of quantum nonlinear systems.

Contribution

It introduces a novel application of multiple-scale analysis to quantum operators, yielding an exact solution and insights into renormalization effects.

Findings

Exact operator differential equations solved

Operator mass renormalization identified

Improved description of quantum anharmonic oscillator

Abstract

Conventional weak-coupling perturbation theory suffers from problems that arise from resonant coupling of successive orders in the perturbation series. Multiple-scale perturbation theory avoids such problems by implicitly performing an infinite reordering and resummation of the conventional perturbation series. Multiple-scale analysis provides a good description of the classical anharmonic oscillator. Here, it is extended to study the Heisenberg operator equations of motion for the quantum anharmonic oscillator. The analysis yields a system of nonlinear operator differential equations, which is solved exactly. The solution provides an operator mass renormalization of the theory.