Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations
F=dtdP,
P=mdtdV, and
[X,P]=i. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.
Born was convinced that the ‘indeterministic foundations’ of quantum mechanics were here to stay. He also left a calling, that has been largely ignored, to show that the ‘apparently deterministic’ laws of classical physics were derivable from the indeterminacy of quantum mechanics. Here, we attempt to answer the calling left to us by Born and we do so first by showing that the beautiful laws of classical optics are directly derivable from the N -slit quantum probability equation. Secondly, we use the quantum probability dependent intensity to arrive, via Heisenberg's uncertainty principle, first at Newton's second law and then to Newton's law of universal gravitational. What this shows is that there is an explicit cohesiveness in physics with quantum mechanics at the foundations. The only ‘price to pay’ is that the laws of classical physics become approximate rather than ‘exact.’ No more determinism. This is entirely consistent with the fields of experimental physics, quantum or classical, where no measurement is exact.
The derivation becomes possible when we find a new formalism which connects the relativistic mechanics with the quantum mechanics. In this paper, we explore the quantum wave nature from the Newtonian mechanics by using a concept: velocity field. At first, we rewrite the relativistic Newton's second law as a field equation in terms of the velocity field, which directly reveals a new relationship connecting to the quantum mechanics. Next, we show that the Dirac equation can be derived from the field equation in a rigorous and consistent manner.