  We investigate the ballistic electron transport in a two dimensional Quantum
Wire under the action of an electric field ($E_y$). We demonstrate how the
presence of a Spin Orbit coupling, due to the uniform electric confinement
field gives a non-commutative effect as in the presence of a transverse
magnetic field.
  We discuss how the non commutation implies an edge localization of the
currents depending on the electron spins also giving a semi-classical spin
dependent Hall current.
  We also discuss how it is possible obtain a quantized Spin Hall conductance
in the ballistic transport regime by developing the Landauer formalism and show
the coupling between the spin magnetic momentum and the orbital one due to the
presence of a circulating current.
