What is the analog of the equation $frac{d^2vec{r}(t)}{dt^2}=frac{vec{r}(t)}{left|vec{r}(t)right|}f(vec{r}(t))$ in Quantum Mechanics?

The easiest way to connect classical and quantum mechanics is to work with the energy, or Hamiltonian, rather than the equations of motion.

You can write the Hamiltonian for your system as

begin{equation} H = frac{1}{2} m dot{vec{r}}^2 + V(|vec{r}|) end{equation} where, to recover your equation of motion, begin{equation} nabla V = -frac{vec{r}}{|vec{r}|}f(|vec{r}|). end{equation}

Then, the conceptually simplest way to represent the corresponding quantum mechanical system is write the Schrodinger equation for wavefunction, $Psi(vec{r},t)$. Explicitly:

begin{equation} i hbar frac{partial Psi }{partial t} = H psi = -frac{hbar^2}{2m}nabla^2 Psi + V(|vec{r}|) Psi end{equation}

Newtonian-equations-like description is not possible in quantum mechanics, since the position and the momentum are not simultaneously measurable. However, the quantum mechanics, obviously contains the classical mechanics and the Newton's laws as a limiting case, as is stated by the correspondence principle. The correspondence between the classical and quantum equations-of-motion is express by the Ehrenfest theorem: $$mfrac{dlangle hat{x}rangle}{dt}=langle hat{p}rangle, frac{dlangle hat{p}rangle}{dt}=-langle V'(hat{x})rangle.$$

Thus, for the central force we could write: $$frac{d^2langle hat{vec{r}}rangle(t)}{dt^2} = frac{1}{m}langle nabla_{vec{R}}V(R)|_{vec{R}=hat{vec{r}}(t)}rangle,$$ where $V(R)$ is the potential corresponding to the force $f(R)$: $$nabla_{vec{R}}V(R) = -frac{vec{R}}{R}f(R).$$ The difficulty in writing literally the relation in the question is notational: since one replaces $vec{r}$ by the position operator, it is impossible to express the radial unit vector as $vec{r}/r$ without ambiguity.

A couple of things to keep in mind is:

• the intricacies with defining time derivatives in quantum mechanics
• the fact that $langle V'(hat{x})rangle neq V'(langlehat{x}rangle)$, which means that the equations are not closed in respect to $langlehat{x}rangle$, which makes them impossible to solve.