  The Newtonian dynamics of particles in brane gravity is investigated. Due to
the coupling of the particles' energy-momentum tensor to the tension of the
brane, the particle is semi-confined and oscillates along the extra dimension.
We demonstrate that the frequency of these oscillations is proportional to the
kinetic energy of the particle in the brane. We show that the classical
stability of particle trajectories on the brane gives us the Bohr--Sommerfeld
quantization condition. The particle's motion along the extra dimension allows
us to formulate a geometrical version of the uncertainty principle.
Furthermore, we exhibited that the particle's motion along the extra dimension
is identical to the time-independent Schr\"odinger equation. The dynamics of a
free particle, particles in a box, a harmonic oscillator, a bouncing particle,
and tunneling are re-examined. We show that the particle's motion along the
extra dimension yields a quantized energy spectrum for bound states.
