Calculation of low-lying energy levels in quantum mechanics

Abstract

This paper proposes a very simple perturbative technique to calculate the low-lying eigenvalues and eigenstates of a parity-symmetric quantum-mechanical potential. The technique is to solve the time-independent Schroedinger eigenvalue problem as a perturbation series in which the perturbation parameter is the energy itself. Unlike nearly all perturbation series for physical problems, for the ground state this perturbation expansion is convergent and, even though the ground-state energy is in general not small compared with 1, the perturbative results are numerically accurate. The perturbation series is divergent for higher energy levels but can be easily evaluated by using methods such as Pade summation.