Non-perturbative Solution to the Quantum Interaction Problem via Schwinger's Action Principle

TL;DR

This paper presents an exact non-perturbative solution to the quantum interaction problem of coupled harmonic oscillators using Schwinger's Action Principle, introducing new operators and dressed states.

Contribution

It introduces a novel method to solve the quantum interaction problem exactly, employing Schwinger's Variational Principle and new creation-annihilation operators for dressed states.

Findings

Exact solutions for eigenstates and spectra of interacting oscillators

Introduction of dressed states and normal mode operators

Solution obtained without approximation

Abstract

The most realistic situations in quantum mechanics involve the interaction between two or more systems. In the most of reliable models, the form and structure of the interactions generate differential equations which are, in the most of cases, almost impossible to solve exactly. In this paper, using the Schwinger Quantum Action Principle, we found the time transformation function that solves exactly the harmonic oscillator interacting with a set of other harmonic coupled oscillators. In order to do it, we have introduced a new special set of creation and annihilation operators which leads directly to the \emph{dressed states} associated to the system, which are the real quantum states of the interacting \emph{\textquotedblleft field-particle\textquotedblright} system. To obtain the closed solution, it is introduced in the same foot a set of \emph{normal mode} creation and annihilation operators of the system related to the first ones by an orthogonal transformation. We find the eigenstates, amplitude transitions and the system spectra without any approximation. At last, we show that the Schwinger Variational Principle provides the solutions in a free representation way.