磁场中椭圆导线薛定谔方程的渐近解
Asymptotic solution of the Schrodinger equation for the elliptic wire in the magnetic field
摘要：
在均匀纵向磁场作用下，得到了无限长椭圆柱势阱中粒子的不可分变量薛定谔方程的渐近解.用标准问题法计算了磁场长度大于边界椭圆焦点距离一半时的量子本征值。在半经典近似下，受限电子（空穴）态被分为边界态（BS）、环态（RS）、双曲焦散态（HCS）和谐振子态（HOS）。对于大的角动量量子数和小的径向量子数，BS和RS被分组为回音壁模式。它们与沿导线截面边界移动的粒子相关联。椭圆焦散线从线芯开始限制运动。跳球模由HCS和HOS两种模式组成，对应于粒子在角动量量子数远小于径向量子数时沿丝径运动的状态。在这种情况下，运动受到双曲焦散线和两个边界椭圆弧的限制。分析了Bi线中空穴激发态的能谱和空间几率分布。
英文摘要：
The asymptotic solution of the Schrodinger equation with non-separable variables is obtained for a particle confined to an infinite elliptic cylinder potential well under applied uniform longitudinal magnetic field. Using standard-problem method, dimension quantized eigenvalues have been calculated when the magnetic length is large enough in comparison with the half of the distance between the boundary ellipse focuses. In semi-classical approximation, the confined electron (hole) states are divided into the Boundary States (BS), Ring States (RS), Hyperbolic Caustic States (HCS), and Harmonic Oscillator States (HOS). For large angular momentum quantum numbers and small radial quantum numbers, the BS and RS are grouped into the Whispering Gallery mode. They associate with particles moving along the wire cross section boundary. The motion is limited from the wire core by the elliptic caustic. Consisting of the HCS and HOS, the Jumping Ball modes correspond to the states of particle moving along a wire diameter when the angular momentum quantum number is much less than the radial quantum number. In this case, the motion is restricted by the hyperbolic caustics and two boundary ellipse arcs. For excited hole states in Bi wire, the energy spectrum and space probability distribution are analyzed.
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