We explore the semi-classical Sommerfeld method of quantization as applied to states with nonzero angular momentum, and show that it leads to qualitatively and quantitatively useful information about systems with spherically symmetric potentials. We begin by reviewing the traditional application of this model to hydrogen, and discuss the way Einstein–Brillouin–Keller (EBK) quantization resolves a mismatch between Sommerfeld states and true quantum mechanical states. We then analyze systems with logarithmic and Yukawa potentials, and compare the results of Sommerfeld and EBK quantization to those from solving Schrdinger’s equation. We show that the semi-classical quantization techniques provide insight into the spread of energy levels associated with a given principle quantum number, as well as giving quantitatively accurate approximations for the energies. We also argue that analyzing systems in this manner involves an interesting application of numerical methods, as well as providing insight into the connections between classical and quantum mechanical physics.

我们探索了半经典的Sommerfeld量化方法，该方法应用于具有非零角动量的状态，并表明它可得出有关具有球对称电势的系统的定性和定量有用信息。我们首先回顾该模型对氢的传统应用，并讨论爱因斯坦-布里渊-凯勒（EBK）量化解决Sommerfeld状态与真实量子力学状态之间不匹配的方式。然后，我们分析具有对数势和Yukawa势的系统，并将Sommerfeld和EBK量化的结果与求解Schrdinger方程的结果进行比较。我们表明，半经典量化技术可以深入了解与给定的主量子数相关的能级分布，并给出能量的定量精确近似值。我们还认为，以这种方式分析系统涉及数值方法的有趣应用，以及提供对经典力学与量子力学物理之间联系的洞察力。