纳米结构中的自伴环形偶极算符
The self-adjoint toroidal dipole operator in nanostructures
摘要：
核反应中的宇称破坏导致了新的环形多极子的发现。从那时起，人们观察到环形多极子存在于所有尺度的系统的电磁结构中，从基本粒子到固态系统和超材料。环形偶极子${\bf T}$（最低阶多极）是最常见的。在量子系统中，这对应于环形偶极算符$\hat{\bf T}$，在坐标轴上有投影$\hat{T}_i$（$i=1,2,3$）。在这里，我们分析了一个系统中的量子粒子与圆柱对称，这是一个典型的系统中出现的环矩。我们发现的哈密顿量，动量，和环形偶极子运营商在适当的曲线坐标，这使我们能够找到本征函数的动量运营商的解析表达式的表达式。虽然环形偶极子是厄米的，但它不是自伴的，但在新的坐标系中，算子$\hat{T}_3$分裂成两个分量，其中一个（仅）是厄米的，而另一个是自伴的。自伴分量是物理上有意义的分量，代表一个可观测量。此外，我们数值对角化的哈密顿量和环形偶极子运营商，并找到他们的本征函数和本征值。我们写出了环面上理想粒子系统的配分函数并计算了热力学量。除了证明环形偶极子是自伴的，因此是可观察的（一个基本相关的发现），这样的系统开辟了利用环形偶极子的量子化和量子特性制造超材料的可能性。
英文摘要：
The parity violation in nuclear reactions led to the discovery of the new class of toroidal multipoles. Since then, it was observed that toroidal multipoles are present in the electromagnetic structure of systems at all scales, from elementary particles, to solid state systems and metamaterials. The toroidal dipole ${\bf T}$ (the lowest order multipole) is the most common. In quantum systems, this corresponds to the toroidal dipole operator $\hat{\bf T}$, with the projections $\hat{T}_i$ ($i=1,2,3$) on the coordinate axes. Here we analyze a quantum particle in a system with cylindrical symmetry, which is a typical system in which toroidal moments appear. We find the expressions for the Hamiltonian, momenta, and toroidal dipole operators in adequate curvilinear coordinates, which allow us to find analytical expressions for the eigenfunctions of the momentum operators. While the toroidal dipole is hermitian, it is not self-adjoint, but in the new set of coordinates the operator $\hat{T}_3$ splits into two components, one of which is (only) hermitian, whereas the other one is self-adjoint. The self-adjoint component is the one that is physically significant and represents an observable. Furthermore, we numerically diagonalize the Hamiltonian and the toroidal dipole operator and find their eigenfunctions and eigenvalues. We write the partition function and calculate the thermodynamic quantities for a system of ideal particles on a torus. Besides proving that the toroidal dipole is self-adjoint and therefore an observable (a finding of fundamental relevance) such systems open up the possibility of making metamaterials that exploit the quantization and the quantum properties of the toroidal dipoles.
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