How/why does the Newtonian approach fail for Quantum mechanics?

Question

Newtonian mechanics, as formulated by Newton, works with vectors and forces.

Later, re-formulations by Lagrange and Hamilton were discovered, which work with action, potential, energy, etc.

The latter approach is what's adopted in quantum mechanics. Is the former approach impossible or impractical for quantum mechanics? If so, how exactly? Can we try to formulate quantum mechanics in the former approach?

Answer 1

It's a matter of working in a different mathematical arena. In classical mechanics,

1. Physical Space $\rightarrow $ Newtonian Formalism
2. Configuration Space $\rightarrow $ Lagrangian Formalism
3. Phase Space $\rightarrow $ Hamiltonian Formalism
4. Hilbert Space $\rightarrow $ Koopman-von-Neumann Formalism

In Quantum Mechanics,

1. Physical Space $\rightarrow $ Pilot Wave/ De Broglie-Bohm Formalism
2. Configuration Space $\rightarrow $ Path Integral/ Feynman Formalism
3. Phase Space $\rightarrow $ Phase-space/ Wigner-Moyal Formalism
4. Hilbert Space $\rightarrow $ Wave-function/ Schrodinger Formalism

We can describe any given system -classical or quantum- using any of the mathematical arenas.

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Edit: I'm not intended to give the full account of how different formalism works. But on @Andrea's comment

The two key equations of Pilot wave theory given by $$\partial_t\rho +\nabla \left(\rho\frac{\nabla S}{m}\right)=0$$ $$\partial_t S+\frac{(\nabla S)^2}{2m}+V+Q=0$$ where $Q$ is known as the quantum potential that take account for quantum behaviors. The first equation is the usual continuity equation the second one, is completely analogous to the classical Hamilton-Jacobi equation.

The key idea is that by introduction of this new potential we get a new force on a particle, We can then calculate the trajectories of particles using Newton's usual second equation $$m\vec{a}=\vec{F}=-\nabla (V+Q)$$ where we $V$ is normal potential and $Q$ is the quantum potential.