Position-Momentum Uncertainty Relation for an Open Macroscopic Quantum   System
  In this study, we explore the validity of the original Heisenberg position- momentum uncertainty relation for a macroscopic harmonic oscillator interacting with environmental micro particles. Our results show that, in the quasi-classical situation, the original uncertainty relation does not hold when the number of particles in the environment is small. Nonetheless, increasing the environmental degrees of freedom resolves the violation in the region of our investigation. 
Hamid Reza Naeij and Afshin Shafiee
quant-ph