Consider a particle of mass $m$ in a harmonic oscillator potential $V(x)=\frac{1}{2}m\omega^2x^2$. I can show that the expectation values for the position and momentum operator are
$<x>=\sqrt{\frac{2\hbar}{m\omega}}\alpha_0cos(\omega t)$
and $<p>=-\sqrt{2\hbar m\omega}\alpha_0sin(\omega t)$
but I am trying to verify that these expectation values are a solution of the classical Hamilton equations of motion. I am confused here since Hamilton's equations are $\dot{q}=\frac{\partial H}{\partial p}$ and $\dot{p}=-\frac{\partial H}{\partial q}$, so I guess we think of $\frac{d<x>}{dt}$ as $\dot{q}$ and $\frac{d<p>}{dt}$ as $\dot{p}$. But I'm not sure what is the equivalent of the $2$ partial derivatives here?
Find the movement of a 50 g mass attached to a spring moving in air with initial conditions y (0) = 4 cm and y’(0)= 40 cm/s. The spring is such that a 30 g mass stretches it 6 cm. Approximate the acceleration of gravity by 1000 cm/s2.