Generalized ladder operators for the perturbed harmonic oscillator

TL;DR

This paper develops a formalism to construct corrected ladder operators for a quantum harmonic oscillator with polynomial perturbations, enabling the analysis of states and expectation values beyond the unperturbed case.

Contribution

It introduces the first method to systematically derive ladder operator corrections for any polynomial perturbation of the harmonic oscillator to arbitrary order.

Findings

Explicit forms of corrected ladder operators for q and p4 perturbations

Computed expectation values of position and momentum for perturbed oscillators

Established a foundation for defining coherent and squeezed states in perturbed systems

Abstract

In this paper, we construct corrections to the raising and lowering (i.e. ladder) operators for a quantum harmonic oscillator subjected to a polynomial type perturbation of any degree and to any order in perturbation theory. We apply our formalism to a couple of examples, namely q and p 4 perturbations, and obtain the explicit form of those operators. We also compute the expectation values of position and momentum for the above perturbations. This construction is essential for defining coherent and squeezed states for the perturbed oscillator. Furthermore, this is the first time that corrections to ladder operators for a harmonic oscillator with a generic perturbation and to an arbitrary order of perturbation theory have been constructed.