正交曲线坐标系中曲面上磁场中粒子的哈密顿量
Hamiltonian for a particle in a magnetic field on a curved surface in orthogonal curvilinear coordinates
摘要：
构造了电磁场中自旋为零的粒子的薛定谔哈密顿量和自旋为二分之一的粒子的包含自旋轨道耦合的泡利哈密顿量.一种新的方法，基于这样的物理论点，即在通过势将粒子挤压到表面时，应该从哈密顿量中删除横向于表面的物理规范协变运动学动量算符（速度算符）。在这两种情况下，所得到的厄米规范不变的表面上的哈密顿量是免费的，从任何参考的分量的矢势横向的表面，和该方法是完全规范无关的。特别是，对于泡利哈密顿量，这些结果是在没有任何进一步假设或近似的情况下精确获得的。导出了圆柱面、球面和环面上薛定谔哈密顿量的显式协变即插即用公式。
英文摘要：
The Schrödinger Hamiltonian of a spin zero particle as well as the Pauli Hamiltonian with spin-orbit coupling included of a spin one-half particle in electromagnetic fields that are confined to a curved surface embedded in a three-dimensional space spanned by a general Orthogonal Curvilinear Coordinate (OCC) are constructed. A new approach, based on the physical argument that upon squeezing the particle to the surface by a potential, then it is the physical gauge-covariant kinematical momentum operator (velocity operator) transverse to the surface that should be dropped from the Hamiltonian(s). In both cases,the resulting Hermitian gauge-invariant Hamiltonian on the surface is free from any reference to the component of the vector potential transverse to the surface, and the approach is completely gauge-independent. In particular, for the Pauli Hamiltonian these results are obtained exactly without any further assumptions or approximations. Explicit covariant plug-and-play formulae for the Schrödinger Hamiltonians on the surfaces of a cylinder, a sphere and a torus are derived.
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