Well this is just a random perhaps stupid question but: say I want to study the motion of a tennis ball, hence using classical mechanics, but for some reason I want to start with Schrodinger equation.
Is there a way to use Schrodinger equation to study the motion of a tennis ball?
You can sort of do that. Fully understanding how classical mechanics emerges from quantum mechanics is very complicated, and I've even heard some physicists speculating that it will require a full understanding of quantum gravity (although I do not claim that this is the majority opinion). I will ignore all of this difficulties and focus in a very specific particular case.
Suppose you chose your potential and you solved the Schrödinger equation. There is a result, know as Ehrenfest's Theorem, that states that the expectation value of the observables behave as the classical observables would. For example, $$\langle \hat{p} \rangle = m \frac{\mathrm{d}\left\langle \hat{x}\right\rangle}{\mathrm{d}t},$$ so you could take the expectation values to find the classical result.
The problem, of course, is choosing the initial condition for the Schrödinger equation. What is your initial state? For at least some systems, such as the quantum harmonic oscillator, classical states can be understood as coherent states, which are eigenstates of the annihilation operator. Hence, your initial state would be some coherent state. I'm guessing, but you probably can then pick the correct coherent state by imposing the correct initial position and momentum.
This answer is sort of a guess. All of these steps make sense to me and I expect them to lead to the correct answer, but I didn't carry out the calculation. Also, notice that I used the notion of coherent state, which I don't believe to be applicable for an arbitrary potential. Understanding which states look classical is complicated. The general problem for understanding how classical physics emerges from quantum physics is an open problem.
