A Closed Form Solution for Quantum Oscillator Perturbations Using Lie Algebras

We give a new solution to a well-known problem, that of computing perturbed for quantum oscillators. This article is nearly self contained and with all the necessary algebraic tools to make the subsequent. We define a new family of Lie algebras relevant to making for perturbed (anharmonic) oscillators, and show that the only two closed solutions are indeed harmonic oscillators themselves. Through combinatorics and noncanonical forms of well-known Lie algebras, we able to obtain a fully closed form solution for perturbed eigenvalues to order.