# Partial Integrability of 3-d Bohmian Trajectories

## TL;DR

This paper investigates the conditions under which 3-dimensional Bohmian trajectories of quantum harmonic oscillators are integrable, highlighting the role of initial quantum numbers and exploring implications for Bohmian Mechanics.

## Contribution

It demonstrates how initial quantum numbers determine the existence of integrals of motion in 3D Bohmian trajectories and connects these findings to lower and higher-dimensional cases.

## Key findings

- Existence of integrals depends on initial quantum numbers.
- Examples of trajectories with and without integrals are provided.
- Partial integrability impacts the structure of Bohmian trajectories.

## Abstract

In this paper we study the integrability of 3-d Bohmian trajectories of a system of quantum harmonic oscillators. We show that the initial choice of quantum numbers is responsible for the existence (or not) of an integral of motion which confines the trajectories on certain invariant surfaces. We give a few examples of orbits in cases where there is or there is not an integral and make some comments on the impact of partial integrability in Bohmian Mechanics. Finally, we make a connection between our present results for the integrability in the 3-d case and analogous results found in the 2-d and 4-d cases.