Problem QM-09011
For Harmonic oscillator problem \[H = \frac{p^2}{2m} + \frac{1}{2}m  \omega^2x^2\]
(A) Write and solve the Heisenberg equations of motion for \(a, a^\dagger\) operators.

(B) Obtain the expectation values of unequal time commutators

1.  \(\big[a(t_1), a(t_2)\big]\)
2. \(\big[a(t_1), a^\dagger(t_2)\big]\)
3. \(\big[a^\dagger(t_1), a^\dagger(t_2)\big]\)

        in the groundstate \(\vert0\rangle\).

(C) Use  your answers to compute the vacuum expectation values of time ordered products

1. \(T\{ q(t_1)\,q(t_2)\}\)
2. \(T\{q(t_1)\, p(t_2)\}\)
3. \(T \{p(t_1)\,p(t_2)\}\).