Generalised quantum anharmonic oscillator using an operator ordering approach

TL;DR

This paper develops a generalized operator ordering method to analyze quantum anharmonic oscillators, deriving energy spectra and perturbative solutions, thereby extending previous theoretical results in quantum mechanics.

Contribution

It introduces a new operator ordering approach for the quantum anharmonic oscillator, enabling generalized energy and frequency calculations and perturbative solutions.

Findings

Derived general expressions for energy eigenvalues and frequency shifts.

Provided a closed-form first-order perturbative operator solution.

Extended recent theoretical results in quantum oscillator analysis.

Abstract

We construct a generalised expression for the normal ordering of (a+a^{\dagger})^{m} for integral values of m and use the result to study the quantum anharmonic oscillator problem in the Heisenberg approach. In particular, we derive generalised expressions for energy eigen values and frequency shifts for the Hamiltonian H=\frac{x^{2}}{2}+\frac{\dot{x}^{2}}{2}+\frac{\lambda}{m}x^{m}. We also derive a closed form first order multi scale perturbation theoretic operator solution of this Hamiltonian with a view to generalise some recent results of Bender and Bettencourt .