Title: Quantum Mechanics on Manifolds Embedded in Euclidean Space Show Chinese title

Title: 欧几里得空间中嵌入流形上的量子力学

Abstract: Quantum particles confined to surfaces in higher dimensional spaces are acted upon by forces that exist only as a result of the surface geometry and the quantum mechanical nature of the system. The dynamics are particularly rich when confinement is implemented by forces that act normal to the surface. We review this confining potential formalism applied to the confinement of a particle to an arbitrary manifold embedded in a higher dimensional Euclidean space. We devote special attention to the geometrically induced gauge potential that appears in the effective Hamiltonian for motion on the surface. We emphasize that the gauge potential is only present when the space of states describing the degrees of freedom normal to the surface is degenerate. We also distinguish between the effects of the intrinsic and extrinsic geometry on the effective Hamiltonian and provide simple expressions for the induced scalar potential. We discuss examples including the case of a 3-dimensional manifold embedded in a 5-dimensional Euclidean space. Show Chinese abstract

Abstract: 受限于高维空间曲面的量子粒子受到的作用力仅由曲面几何和系统的量子力学性质决定。 当限制作用是由垂直于曲面的力实现时，动力学尤为丰富。 我们回顾了将粒子限制在一个嵌入高维欧几里得空间中的任意流形上的限制势能形式化方法。 我们特别关注出现在曲面上运动的有效哈密顿量中的几何诱导规范势。 我们强调规范势仅在描述曲面法向自由度的状态空间退化时才存在。 我们还区分了内在几何和外在几何对有效哈密顿量的影响，并提供了诱导标量势的简单表达式。 我们讨论了包括三维流形嵌入五维欧几里得空间在内的例子。