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首页> 外文期刊>Physical review >Quantum mechanics on a Moebius ring: Energy levels, symmetry, optical transitions, and level splitting in a magnetic field
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Quantum mechanics on a Moebius ring: Energy levels, symmetry, optical transitions, and level splitting in a magnetic field

机译：Moebius环上的量子力学：磁场中的能级，对称性，光学跃迁和能级分裂

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We investigate the quantum mechanical energy levels of an electron constrained to motion on a nanoscale Moebius ring by solving the Schroedinger equation on the curved surface. The dimensions of the ring in terms of the lateral and transverse parameters {u,v} for the Moebius ring allow us to identify the quantum numbers for the levels by (n_u,n_v). We show that the energy levels can still be labeled using the quantum numbers of the cylindrical ring of the same dimensions. While the Hamiltonian has invariance under parity in parameter space, the rotational symmetry about any axis in configuration space is lost, so that the double degeneracy of energy levels for azimuthal quantum number n_u ≥ 1, that exists in cylindrical rings, is lifted by a small amount in the Moebius ring. The pattern of level splitting has been identified in terms of the number of twists σ to be 2n_u = sσ where s is an integer. The scaling properties of the energy levels with respect to the dimensions of the ring are derived; using these properties, our numerical results which are given for a specific geometry can be extended to rings of other commensurate dimensions. The absence of rotational invariance for the Mobius ring manifests itself through the orbital angular momentum L_z not commuting with the Hamiltonian. Its expectation values are found to have nearly integral as well as half-integral values of h, and its variances are small. The energy levels with half-integral azimuthal quantum numbers (n_u) are also close to the approximate formula for the equivalent cylindrical ring, provided such half-integral quantum numbers are allowed for the cylindrical geometry. The Zeeman splitting of the energy levels in an external magnetic field is displayed, together with wave functions at a level anticrossing. The optical transitions between electronic states on the Moebius ring are obtained, and a table of oscillator strengths is provided. The results for energy levels for rings with multiple twists are presented. In view of recent technological advances in the production of graphene sheets, we may anticipate the making of such twisted rings with graphene strips of finite width. Graphene strips of finite width have an open band gap at the K points in the Brillouin zone, so that a nonrelativistic treatment with a small effective mass is appropriate. For Moebius rings of graphene, our results would be directly relevant, and we may anticipate their experimental verification in the near future.
机译：我们通过求解曲面上的Schroedinger方程，研究了受约束在纳米Moebius环上运动的电子的量子机械能级。根据Moebius环的横向参数和横向参数{u，v}，环的尺寸使我们可以通过（n_u，n_v）来标识能级的量子数。我们表明，仍然可以使用相同尺寸的圆柱环的量子数来标记能级。虽然哈密顿量在参数空间的奇偶性下具有不变性，但失去了在配置空间中围绕任何轴的旋转对称性，从而使存在于圆柱环中的方位量子数n_u≥1的能级的双重简并性提升了一个小Moebius戒指中的金额。已根据扭曲数σ来确定电平分裂的模式，即2n_u =sσ，其中s是整数。推导了能级相对于环尺寸的缩放特性；利用这些特性，可以将针对特定几何形状给出的数值结果扩展到其他相应尺寸的环。 Mobius环没有旋转不变性，这是通过不与哈密顿量交换的轨道角动量L_z表现出来的。发现其期望值几乎具有h的半整数值和半整数值，并且其方差很小。具有半积分方位角量子数（n_u）的能级也接近等效圆柱环的近似公式，前提是圆柱几何形状允许这种半积分量子数。显示了外部磁场中能级的塞曼分裂，以及处于反交叉能级的波函数。获得了Moebius环上电子态之间的光学跃迁，并提供了一个振荡器强度表。给出了具有多个扭曲的环的能级结果。鉴于石墨烯片材生产中的最新技术进步，我们可以预期使用有限宽度的石墨烯带材制造这种扭曲环。有限宽度的石墨烯带在布里渊区的K点处具有开放带隙，因此采用相对质量较小的有效质量的非相对论性处理是合适的。对于石墨烯的Moebius环，我们的结果将直接相关，并且我们可以预期在不久的将来对其进行实验验证。

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