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

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?

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.

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.

• This is an interesting answer. Perhaps it’s worth mentioning to OP that in the Pilot Wave theory, the particle is not subjected to forces and instead it is the velocity of the particle that is determined by the wave. That is, while the mathematical arena is the same as the classical Newtonian formalism, the main equation of Newtonian physics $F=ma$ has no counterpart in Pilot Wave. So I don’t think it is correct to say that Pilot Wave is the “Newtonian approach to quantum mechanics”. Jul 18, 2021 at 7:03
• @Andrea Please find the edit. Jul 18, 2021 at 7:14
• Cool! I didn’t know you could formulate Pilot Wave theory like that. I was only aware of the guiding equation. Thanks (I can’t upvote the answer because I had already upvoted it) Jul 18, 2021 at 7:20