Explicit Calculations for Anharmonic Oscillators Using Lie Algebras

TL;DR

This paper develops a Lie algebra-based method to explicitly calculate first-order perturbation energies for anharmonic oscillators, providing a new algebraic approach to quantum perturbation theory.

Contribution

It introduces a novel Lie algebra framework to derive explicit formulas for perturbation energies in anharmonic oscillators, expanding analytical tools in quantum mechanics.

Findings

Derived explicit first-order energy formulas for anharmonic oscillators.

Established a Lie algebra approach to transform Hamiltonians under perturbation.

Provided a general method applicable to Hamiltonians of the form H = a†a + 1/2 + λx^n.

Abstract

We construct Lie algebras arising from commutators of the harmonic Hamiltonian and the perturbed anharmonic Hamiltonian. From there we form a very specific element of the associated Lie group and transform the unperturbed Hamiltonian into the perturbed Hamiltonian minus perturbation energies. An explicit formula is given for the first order perturbation energy of all Hamiltonians of the form H = a^{\dagger}a + 1/2 + \lambda x^n.