Journal of Physical Mathematics

A closed form solution for quantum oscillator perturbations using Lie algebras

Clark Alexander

Full-text: Open access


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

Article information

J. Phys. Math., Volume 3 (2011), Article ID P101201, 12 pages.

First available in Project Euclid: 22 September 2011

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

Primary: 37K30: Relations with infinite-dimensional Lie algebras and other algebraic structures 70G65: Symmetries, Lie-group and Lie-algebra methods 81Q05: Closed and approximate solutions to the Schrödinger, Dirac, Klein- Gordon and other equations of quantum mechanics

Dynamical systems Ergodic theory Hamiltonian systems Lie algebras Infinite-dimensional systems Symmetries Lie-group Quantum theory Schroedinger equation Klein-Gordon equation


Alexander, Clark. A closed form solution for quantum oscillator perturbations using Lie algebras. J. Phys. Math. 3 (2011), Article ID P101201, 12 pages. doi:10.4303/jpm/P101201.

Export citation