Path integral and Sommerfeld quantization

TL;DR

This paper compares the path integral and Sommerfeld quantization methods, highlighting that the latter effectively incorporates boundary conditions, allowing it to accurately reproduce energy spectra for both harmonic oscillator and Coulomb potentials.

Contribution

It clarifies the fundamental difference between the two schemes and discusses the limitations of semiclassical methods using simple potential models.

Findings

Path integral fails for Coulomb potential due to boundary condition issues.

Sommerfeld quantization successfully reproduces spectra for harmonic oscillator and Coulomb potentials.

Limitations of semiclassical methods are analyzed with square well and delta function potentials.

Abstract

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 $\delta(x)$ function potential models.