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

We show that the uncertainty principle product for the position and momentum operators for a system described by the Hamiltonian $\widehat{H}=\frac{{\widehat{p}}^2}{2m}+\frac{1}{2}m{\omega }^2{\widehat{x}}^2+\frac{\mu }{2}(\widehat{x}\widehat{p}+\widehat{p}\widehat{x})$ where $\mu <\omega$ reads $\mathrm{\Delta }x\mathrm{\Delta }p\ge \frac{\mathrm{\hslash }\omega }{2\sqrt{{\omega }^2-{\mu }^2}}$. 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 $\widehat{p}$ and $\widehat{x}$ with respect to the corresponding coherent states.