Exact lower bound of the uncertainty principle product for the harmonic oscillator with position-momentum coupling

Abstract

We show that the uncertainty principle product for the position and momentum operators for a system described by the Hamiltonian $\hat{H}=\frac{{\hat{p}}^2}{2m}+\frac{1}{2}m{\omega }^2{\hat{x}}^2+\frac{\mu }{2}(\hat{x}\hat{p}+\hat{p}\hat{x})$ where $\mu <\omega$ reads $\Delta x\Delta p\ge \frac{\hslash \omega }{2\sqrt{{\omega }^2-{\mu }^2}}$. All the values bellow this lower bound are thus quantum-mechanically forbidden. We construct the annihilation and creation operators for this system and we calculate the expectation values of the operators $\hat{p}$ and $\hat{x}$ with respect to the corresponding coherent states.