Consider the simple harmonic oscillator

$$\hat{H} = {{p^2} \over {2m}} + {{ m \omega^2 x^2} \over {2}}$$

Write it in ``second quantized'' form , by expressing x and p in terms of creation and annihilation operators.

$$\hat{H} = \hbar\omega(a^{\dagger} a + \frac{1}{2}).$$

(i) Derive the equation of motion for $a$ and show that

$$a(t) = a(0)e^{- i \omega t}.$$

Repeat this for $a^{\dagger}$.

(ii) Using the relationship

$$a^{\dagger} = \frac{1}{\sqrt{2\hbar}} \left[ \frac{\hat{p}}{\sqrt{m \omega}} + i\hat{x} \sqrt{m \omega} \right].$$

use the results of (i) to show that

$$\begin{eqnarray*}\nonumber \hat{p}(t) & =& \hat{p}(0) \cos \ \omega t - m\omega ... ...omega t + \frac{\hat{p}(0)}{m \omega}sin \ \omega t.\\ \nonumber \end{eqnarray*}$$

(iii) The above operator expressions appear identical to the classical equations of motion.

Why?