Semi-classical methods, angular momentum, and non-analytic problems

Abstract

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 Schrödinger’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.