Geometric effects resulting from square and circular confinements for a particle constrained to a space curve

Investigating the geometric effects resulting from the detailed behaviors of the confining potential, we consider square and circular confinements to constrain a particle to a space curve. We find a torsion-induced geometric potential and a curvature-induced geometric momentum just in the square case, while a geometric gauge potential solely in the circular case. In the presence of electromagnetic field, a geometrically induced magnetic moment couples with magnetic field as an induced Zeeman coupling only for the circular confinement also. As spin-orbit interaction is considered, we find some additional terms for the spin-orbit coupling, which are induced not only by torsion, but also curvature. Moreover, in the circular case, the spin also couples with an intrinsic angular momentum, which describes the azimuthal motions mapped on the space curve. As an important conclusion for the thin-layer quantization approach, some substantial geometric effects result from the confinement boundaries. Finally, these results are proved on a helical wire.

Figure 1

Sketches of the square case. (a) A twisted tube with square cross section. (b) A cross section with squeezing forces sketched. (c) The double confining potential vs $(q_2,q_3)$. (d) The confining potential vs $q_2$. The confining potential is regarded as a dimensionless quantity because its unit is being taken as the excited energy in $s$ dimension.

Figure 2

Sketches of the circular case. (a) A twisted tube. (b) A cross section with squeezing forces sketched. (c) The radial confining potential vs the variables $(\rho ,\theta )$ sketched. (d) The radial confining potential vs $\rho$ sketched. The confining potential is regarded as a dimensionless quantity because its unit is being taken as the excited energy in $s$ dimension.