  The Heisenberg equations of motion for a quantum particle of mass $m$ are
deduced from the infinitesimal qr-number equations of motion for the particle.
The infinitesimal qr-number equations, and hence the standard quantum
mechanical equations, are related to the qr-number equations in much the same
way as the equations of geometric optics are related to those of wave optics.
The qr-number equations of motion for a quantum particle of mass $m$ describe
the motion of a lump, given by on open set in the qr-number space of the
particle, while the infinitesimal qr-number equations describe the motion of a
point-like particle. The qr-number equations of motion are the Hamiltonian
equations of motion for a classical particle of mass $m$ expressed in
qr-numbers. The proof requires that the particle's position operators have only
continuous spectrum and the force functions are smooth.
