Taking the Road Less Traveled: Solving the One-Dimensional Quantum Oscillator using the Parabolic-Cylinder Equation

TL;DR

This paper presents a novel approach to solving the 1D quantum harmonic oscillator by transforming the Schrödinger equation into a parabolic cylinder equation, providing new solutions and applications to systems in electric fields.

Contribution

The authors introduce a variable transformation that solves the 1D harmonic oscillator using parabolic cylinder functions, extending the method to oscillators in electric fields and Lennard-Jones potentials.

Findings

Solutions expressed in parabolic cylinder functions.

Eigenvalues modified by electric field, with specific restrictions.

Application to Lennard-Jones potential bound states.

Abstract

The single well 1D harmonic oscillator is one of the most fundamental and commonly solved problems in quantum mechanics. Traditionally, in most introductory quantum mechanics textbooks, it is solved using either a power series method, which ultimately leads to the Hermite polynomials, or by ladder operators methods. We show here that, by employing one straightforward variable transformation, this problem can be solved, and the resulting state functions can be given in terms of parabolic cylinder functions. Additionally, the same approach can be used to solve the Schr\"odinger equation for the 1D harmonic oscillator in a uniform electric field. In this case, the process yields two possible solutions. One is the well-known result where the 1D oscillator eigenvalues are reduced by a frequency-dependent term, which can have any positive value. The other is where the field term is restricted to be an integer and the eigenvalues are in the same form as for the field-free case. We show how the results can be used to create a harmonic approximation for the bound states of a Lennard-Jones potential.