粒子在二维开放型四分之一圆形微腔中的逃逸研究

利用半经典理论对粒子在开放型四分之一圆形微腔中的逃逸过程进行了研究，推导出了逃逸几率密度的计算公式。我们研究了一簇从四分之一圆形微腔的左下方的入口出射、并从该微腔右边界逃逸的粒子轨迹。对于粒子的每一条逃逸轨迹，记录下它的传播时间和逃逸的位置。结果发现逃逸时间图随着逃逸点的位置的变化曲线呈现出振荡结构。随着碰撞次数的增加，逃逸点的位置越靠近该腔的右顶端。对一系列的探测点，找到从源点出发到达探测点的轨迹，然后应用半经典理论来构造波函数，进而给出逃逸几率密度的计算公式。研究结果标明，逃逸几率密度与探测平面上逃逸点的位置、粒子的动量、初始出射角及与微腔的碰撞次数有关。为了更清楚的看出量子力学和经典力学之间的联系，我们对体系的半经典波函数进行傅里叶变换，给出了粒子的路径长度谱。路径长度谱的每个峰值对应于一条粒子逃逸轨迹的长度。本文的研究对理解量子力学和经典力学之间的联系以及研究粒子在微腔中的的逃逸和输运过程可以提供一定的参考价值。

Study of the escape of particle from an open quarter-circular microcavity

The escape of particle from an open quarter-circular microcavity has been examined by the seimiclassical theory, the escape probability density of the particle has been derived. We consider a family of trajectories launched from the left bottom lead of the quarter-circular cavity and escape from the right boundary. For each escaping trajectories, we record the propagation time and the position of the escape point. We find that the escape time graph exhibits an oscillating structure. With the increase of the bouncing number with the surface of the cavity, the position of the detector point becomes closer to the right apex. For a set of detector points, we search for the trajectories from the source point to the detector points. Then we use semiclassical theory to construct the wave function and further put forward the formula for calculating the escape probability density. The calculation results suggest that the escape probability density depends on the detector position, the momentum of the particle, the initial outgoing angle and the bouncing number with the cavity sensitively. In order to see the relation between the quantum mechanics and the classical mechanics clearly, we perform the Fourier transform of the semiclassical wave function and derive the path length spectrum. Each peak in the path length spectrum corresponds to the length of one escape trajectory of the particle. This study provides some reference value for the studying of the correspondence between the classical mechanics and quantum mechanics and may guide the future study of the escape and transport process of particles inside a microcavity.