Explicit Calculations for Anharmonic Oscillators Using Lie Algebras

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.