# Time dependent coupled harmonic oscillators

## TL;DR

This paper presents a method to solve coupled time-dependent harmonic oscillators using quantum invariants, enabling the analysis of arbitrary frequency variations through unitary transformations and generalized invariants.

## Contribution

It introduces a novel approach employing quantum orthogonal functions invariants and unitary transformations to solve complex coupled oscillators with arbitrary time-dependent frequencies.

## Key findings

- Successfully solves coupled oscillators with arbitrary time-dependent frequencies.
- Derives a generalized Ermakov-Lewis invariant for N coupled oscillators.
- Provides a framework for analyzing complex quantum harmonic systems.

## Abstract

We show that, by using the quantum orthogonal functions invariant, we are able to solve a coupled of time dependent harmonic oscillators where all the time dependent frequencies are arbitrary. We do so, by transforming the time dependent Hamiltonian of the interaction by a set of unitary operators. In passing, we show that $N$ time dependent and coupled oscillators have a generalized orthogonal functions invariant from which we can write a Ermakov-Lewis invariant.