Why don't we use the concept of force in quantum mechanics?

I'm a quarter of the way towards finishing a basic quantum mechanics course, and I see no mention of force, after having done the 1-D Schrodinger equation for a free particle, particle in an infinitely deep potential well, and the linear harmonic oscillator.

There was one small mention of the fact that expectation values obey classical laws. I was wondering why we don't make more use of this fact. For example, in the linear harmonic oscillator problem, one could obtain the temporal evolution of $\langle x \rangle$ using the classical expression $\left(-\frac{dV(x)}{dx}=m\frac{d^2\langle x\rangle}{dt^2}\right)$, and if we could get the time-evolution of $\sigma$ and tack this on, we could re-create the Gaussian and get back $|\Psi(x,t)|^2$. Of course, that last part may not be very easy.

I was just wondering if anybody has tried doing something like this, or if there an obvious flaw in thinking about it this way.