Position–Momentum Uncertainty Relation for an Open Macroscopic Quantum System

Abstract

The macroscopic quantum systems are considered as a bridge between quantum and classical systems. 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 removes the violation bounds in the regions of our investigation.

Appendix

Here, we derive the violation range of uncertainty relation, when the system is coupled to three particles of the environment. Considering a given violation range as

$$\begin{aligned} \Delta {q^2} \Delta {p^2}<\frac{\bar{h}^2}{4} \end{aligned}$$

(43)

One can show that the product of $\langle q^2\rangle \langle p^2\rangle$ in relations (30) and (31) leads to:

$$\begin{aligned} \frac{1}{4\pi ^2}\frac{\big [\big (a^2b^2+1\big )d^2+\big (1-2a^2b^2\big )d\big ]^3}{\big [\big (3a^4d^2+6a^2b^2d+3b^4\big )\big (3a^4+6a^2b^2d+3d^2b^4\big )\big ]^\frac{3}{2}} \end{aligned}$$

(44)

where $a^2+b^2=1$ and d is defined. The above relation is simplified to $1/4\pi ^2(3)^3$, after some mathematical manipulation. This concludes the relation (38) for $N=3$. The same method could be used to prove similar results for other Ns.