路径积分与索末菲量子化
Path integral and Sommerfeld quantization
摘要：
路径积分公式能再现谐振子势的正确能谱，但不能解决库仑势问题。这是因为路径积分不能正确地考虑边界条件，这是由于库仑势系统中散射态的存在。另一方面，Sommerfeld量子化由于有效地考虑了边界条件，可以再现谐振子和库仑势情形的正确能谱。这两种方案的基本区别在于，在路径积分中对波函数没有任何约束，而索末菲量子化规则是通过要求状态矢量是单值函数来推导的。本文还用方阱和$δ(x)$函数势模型阐明了半经典方法的局限性。
英文摘要：
The path integral formulation can reproduce the right energy spectrum of the harmonic oscillator potential, but it cannot resolve the Coulomb potential problem. This is because the path integral cannot properly take into account the boundary condition, which is due to the presence of the scattering states in the Coulomb potential system. On the other hand, the Sommerfeld quantization can reproduce the right energy spectrum of both harmonic oscillator and Coulomb potential cases since the boundary condition is effectively taken into account in this semiclassical treatment. The basic difference between the two schemes should be that no constraint is imposed on the wave function in the path integral while the Sommerfeld quantization rule is derived by requiring that the state vector should be a single-valued function. The limitation of the semiclassical method is also clarified in terms of the square well and $δ(x)$ function potential models.
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