Electrons in a curvilinear quantum wire

The adiabatic motion of electrons in curvilinear quantum wires was studied. It was assumed that the cross section of a wire was constant along its length. The potential that limited electron motion across a wire and the shape of the cross section of the wire were considered arbitrary, while the curvature and the torsion (defined as the derivative of the cross section rotation angle with respect to the length) were assumed to be small. An effective nonrelativistic Hamiltonian for the motion of electrons along a wire with the conservation of transverse quantum numbers was obtained. The spin-orbit coupling Hamiltonian related to the curvature and torsion of a wire was found. Particular cases of a rectilinear twisted quantum wire with a noncircular cross section and a curvilinear quantum wire on a plane were studied. Various transverse potential models limiting the motion of electrons were considered. In particular, the coefficients of the effective Hamiltonian for quantum wires with rectangular and circular cross sections and hard walls and for wires with a parabolic potential were found.