  We show that the first order form of the Schrodinger equation proposed in [1]
can be obtained from the Dirac equation in the non-relativistic limit. We also
show that the Pauli Hamiltonian is obtained from this equation by requiring
local gauge invariance. In addition, we study the problem of a spin up particle
incident on a finite potential barrier and show that the known quantum
mechanical results are obtained. Finally, we consider the symmetric potential
well and show that the quantum mechanical expression for the quantized energy
levels of a particle is obtained with periodic boundary conditions. Based on
these conclusions, we propose that the equation introduced in [1] is the
non-relativistic limit of the Dirac equation and more appropriately describes
spin 1/2 particles in the non-relativistic limit.
