A new series solution associated with the quantum harmonic oscillator

Abstract

In the non-relativistic case, the problem of quantum harmonic oscillator is solved by the aid of the Schrödinger equation. An additional term of potential energy that is related with the square of the spatial coordinate is also included in the equation. A second order linear differential equation is obtained in the stationary state. The direct solution gives the wave function in terms of the parabolic cylinder functions. This solution leads to the Hermite polynomials when the quantization of energy is considered. However, the derived expression of the wave function does not yield the correct functions for the limiting case, in which the potential energy is zero. We put forth an alternative solution for the second order linear differential equation. First of all, a new series expression is suggested as an ansatz for the power series solution of differential equations. The coefficients of the series are obtained in terms of recurrence relations. Some computational simulations are given.