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 $\Delta x \Delta p\ge\frac{ \hbar \omega}{ 2\sqrt{ \omega^2-\mu^2}}$. 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.
Categories: quant-ph math-ph math.MP