Pauli equation for a charged spin particle on a curved surface in an electric and magnetic field

We derive the Pauli equation for a charged spin particle confined to move on a spatially curved surface $\mathcal{S}$ in an electromagnetic field. Using the thin-layer quantization scheme to constrain the particle on $\mathcal{S}$, and in the transformed spinor representations, we obtain the well-known geometric potential $V_g$ and the presence of $e^{-i\phi }$, which can generate additive spin connection geometric potentials by the curvilinear coordinates derivatives, and we find that the two fundamental evidences in the literature [Giulio Ferrari and Giampaolo Cuoghi, Phys. Rev. Lett. 100, 230403 (2008)] are still valid in the present system without source current perpendicular to $\mathcal{S}$. Finally, we apply the surface Pauli equation to spherical, cylindrical, and toroidal surfaces, in which we obtain expectantly the geometric potentials and new spin connection geometric potentials, and find that only the normal Pauli matrix appears in these equations.

Figure 1

The surface $S$ and two auxiliary surfaces $S_1$ and $S_2$. The two surfaces close a subspace $V_N$ embedded in the ordinary 3D space $\Omega$. On the surface $S$, the point is described by $\vec{r}(q_1,q_2)$. In the subspace $V_N$, the point is described by $\vec{R}(q_1,q_2,q_3)$.

Figure 2

A spherical surface of radius $r$ and its coordinate system $(\theta ,\varphi ,\rho )$. The green arrow denotes the direction of the magnetic field $\vec{B}$, which is perpendicular to the plane at $\theta =\frac{\pi }{2}$.

Figure 3

A cylindrical surface of radius $r$ and its coordinate system $(\theta ,y,\rho )$. The magnetic field $\vec{B}$ consists of $B_0$ parallel to the $y$ axis and $B_1$ perpendicular to the $y$ axis at $\theta =0$.

Figure 4

A toroidal surface of radius $r$ and its coordinate system $(\theta ,\varphi ,\rho )$. $R_0$ is the distance from the center of the tube to the center of the torus, $r$ is the radius of the tube, and $R$ is the distance from the center of the tube to one point which lies on the toroidal surface. The magnetic field $\vec{B}$ can be decomposed into two parts: $B_1$ lies in the plane of $R_0$ and $\varphi$, and $B_0$ is perpendicular to the circle defined by $R_0$ and $\varphi$.