Is classical physics crucial for quantum physics? When we solve Schrödinger equation we start by determine what is the potential $V(r,t)$. This usually done by looking at the problem from classical point of view, find $V(r,t)$ using classical physics and than plug it into Schrödinger equation.
Does it mean that classical physics is crucial for quantum physics? Is there any way to replace classical physics with quantum physics completely?
 A: Presume you are asking about the general case and not just about finding an expression for $V(r,t)$...
Note that most interpretations are "philosophically light" - many are silent on the topic, however many also seek to preserve particular macroscopic (classical) behaviours or concepts at the quantum level, so at some level classical physics inspires these interpretations.
Copenhagen is based on Positivism originally and does have something to say about this topic. (Please don't down vote because you are not a fan of Copenhagen) Bohr was quite clear that QM uses classical concepts such as time, momentum, energy etc; he believed that some form of Classical physics was pre-requisite for QM.
I personally regard this requirement for pre-existing classical physics as the greatest failure of QM interpretations at the moment (greater than measurement problem). There are ways around the issue but these interpretatons are not mainstream.
A: Let's say I run a zoo that also sells animals to rich people. You come and you want to get an animal that corresponds to the plastic toy-replica of one of the animals that you bought from our gift-shop. Would I need to look at your plastic replica to figure out which animal should I hand to you? Of course. Does it mean that the real animals are somehow dependent on their plastic replicas? Of course, not.

Quantum mechanics stands on its own and one can arrive at its classical limits by various methods ($\hbar\to 0$, $N\to\infty$, etc.). One can often find that a quantum system has no classical limit or one classical limit or multiple classical limits. What we are usually interested in is finding a quantum system that produces a classical system that we already know in one of its classical limits. An easy (although not full-proof) way of doing this is the famous canonical quantization procedure:

*

*You replace the classical canonical variables with operators such that the commutators of the operators reproduce the Poisson-bracket relations that the original canonical variables followed. This is, roughly speaking, to ensure that the quantum system has the same degrees of freedom and the same symmetries among those degrees of freedom that the classical system has.

*This is precisely where the classical potential comes in. You are replacing the $r$ in the classical Hamiltonian with $\hat{r}$ to produce a quantum Hamiltonian -- in the hopes that the quantum system that you generate will be such that its classical limit will be the classical system that you started out with.

