Green's Function for the Quartic Oscillator

TL;DR

This paper derives a quantum mechanical Green's function for the quartic oscillator by building on previous work that linearized the system and integrated its classical action, expanding the analytical tools available for this nonlinear quantum system.

Contribution

It presents a new form of the quartic oscillator's action function in harmonic oscillator variables, enabling the derivation of its Green's function within quantum mechanics.

Findings

Derived the Green's function for the quartic oscillator.

Connected quartic oscillator analysis with harmonic oscillator methods.

Expanded the analytical framework for nonlinear quantum systems.

Abstract

In this paper, a quantum mechanical Green's function $G_{qo}(y_b,t_b;$ $y_a,t_a)$ for the quartic oscillator is presented. This result is built upon two previous papers: first [1], detailing the linearization of the quartic oscillator $(qo)$ to the harmonic oscillator $(ho)$, second [2], the integration of the classical action function for the quartic oscillator. Here an equivalent form for the quartic oscillator action function $S_{qo}(y_b,t_b;$ $y_a,t_a)$ in terms of harmonic oscillator variables is derived in order to facilitate the derivation of the quartic oscillator Green's Function amplitude. Thus, the papers [1] and [2] and this paper, taken together, result in the incorporation of the quartic oscillator into the non-relativistic quantum mechanical physics literature consisting of those single particle systems whose properties are described in terms of trig functions, their quadratures or both.