[QUE/QM-09007]

(a) For harmonic oscillator \[H=\frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2
\hat{x}^2\] set up and solve up the equations of motion for position and
momentum operators in the Heisenberg picture. Hence show that  \[\begin{eqnarray}    \hat{x}(t) = \tilde{x} \cos\omega t +
                   \frac{1}{m\omega} \tilde{p} \sin\omega t \\ \hat{p}(t) = \tilde{p} \cos\omega t - m\omega \tilde{x} \sin\omega t          \end{eqnarray}\]
where \(\tilde{x},\tilde{p}\) in the right hand sides of the above  equations are the  Schr\"{o}dinger picture operators.        
(b) Calculate the commutators   \[\begin{eqnarray*} [\hat{x}(t), \hat{x}(t^\prime))],\qquad  [\hat{x}(t),  \hat{p}(t^\prime))], \qquad  [\hat{p}(t), \hat{p}(t^\prime))],  \end{eqnarray*}\]   for \(t\ne t^\prime\). How do these commutators at unequal times
compare with equal time  commutators? Are \(\hat{x}(t),  \hat{x}(t^\prime)\) compatible observables?       
(c) Verify that at equal times the commutators reduce to canonical  commutation rule \([\hat{x}, \hat{p}] =i\hbar.\)