Formulation of general dynamical invariants and their unitary relations for time-dependent three coupled quantum oscillators

TL;DR

This paper derives a general dynamical invariant for three coupled time-dependent quantum oscillators, transforming it into a simpler form to obtain exact quantum solutions without approximations.

Contribution

It introduces a new invariant operator for coupled oscillators and uses unitary transformations to find exact solutions for their quantum states.

Findings

Invariant operator formulated for coupled oscillators

Exact quantum solutions obtained without approximations

Invariant characterizes quantum properties under various parameters

Abstract

A general dynamical invariant operator for three coupled time-dependent oscillators is derived. Although the obtained invariant operator satisfies the Liouville-von Neumann equation, its mathematical formula is somewhat complicated due to arbitrariness of time variations of parameters. The parametric conditions required for formulating this invariant are definitely specified. By using the unitary transformation method, the invariant operator is transformed to the one that corresponds to three independent simple harmonic oscillators. Inverse transformation of the well-known quantum solutions associated with such a simplified invariant enables us to identify quantum solutions of the coupled original systems. These solutions are exact since we do not use approximations not only in formulating the invariant operator but in the unitary transformation as well. The invariant operator and its eigenfunctions provided here can be used to characterize quantum properties of the systems with various choices of the types of time-dependent parameters.