非对称无限深方势阱中微扰理论与微扰理论的比较
Comparing periodic-orbit theory to perturbation theory in the asymmetric infinite square well
摘要：
具有不连续台阶的无限深方势阱是在半经典极限下表现出非牛顿射线分裂周期轨道的最简单系统之一。用与时间无关的微扰理论（PT）和微扰轨道理论对该系统进行了分析，并比较了由这两种方法导出的能量本征值的近似公式。系统的周期轨道可以根据它们从势阶反射的次数来划分。不同类型的轨道对PT的阶数有不同的贡献。在二阶PT校正中的主导项是由于非牛顿轨道，从步骤正好反射一次。在PT收敛的极限内，π-轨道理论的结果与PT的结果一致，但在此极限之外，π-轨道理论对高于势能阶跃的能量给出了更精确的结果。
英文摘要：
An infinite square well with a discontinuous step is one of the simplest systems to exhibit non-Newtonian ray-splitting periodic orbits in the semiclassical limit. This system is analyzed using both time-independent perturbation theory (PT) and periodic-orbit theory and the approximate formulas for the energy eigenvalues derived from these two approaches are compared. The periodic orbits of the system can be divided into classes according to how many times they reflect from the potential step. Different classes of orbits contribute to different orders of PT. The dominant term in the second-order PT correction is due to non-Newtonian orbits that reflect from the step exactly once. In the limit in which PT converges the periodic-orbit theory results agree with those of PT, but outside of this limit the periodic-orbit theory gives much more accurate results for energies above the potential step.
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