Let $A=(m \omega X+i P) / \sqrt{2 m \hbar \omega}$ be the lowering operator of a one dimensional quantum harmonic oscillator of mass $m$ and frequency $\omega$, and let $|0\rangle$ be the ground state defined by $A|0\rangle=0$.

a) Evaluate the commutator $\left[A, A^{\dagger}\right]$.

b) For $\gamma \in \mathbb{R}$, let $S(\gamma)$ be the unitary operator $S(\gamma)=\exp \left(-\frac{\gamma}{2}\left(A^{\dagger} A^{\dagger}-A A\right)\right)$ and define $A(\gamma)=S^{\dagger}(\gamma) A S(\gamma)$. By differentiating with respect to $\gamma$ or otherwise, show that

$$A(\gamma)=A \cosh \gamma-A^{\dagger} \sinh \gamma$$

c) The ground state of the harmonic oscillator saturates the uncertainty relation $\Delta X \Delta P \geqslant \hbar / 2$. Compute $\Delta X \Delta P$ when the oscillator is in the state $|\gamma\rangle=S(\gamma)|0\rangle$.