The semiclassical limit of eigenfunctions of the Schr\"odinger equation and the Bohr-Sommerfeld quantization condition, revisited

TL;DR

This paper rigorously analyzes the semiclassical limit of eigenfunctions in a 1D quantum potential well, confirming the Bohr-Sommerfeld quantization rule through asymptotic analysis.

Contribution

It provides a rigorous justification for the semiclassical asymptotics of eigenfunctions and validates the Bohr-Sommerfeld quantization condition in this setting.

Findings

Semiclassical eigenfunction asymptotics are justified.

Bohr-Sommerfeld quantization condition is recovered.

Results apply to bound states in 1D potential wells.

Abstract

Consider the semiclassical limit, as the Planck constant $\hbar\ri 0$, of bound states of a quantum particle in a one-dimensional potential well. We justify the semiclassical asymptotics of eigenfunctions and recover the Bohr-Sommerfeld quantization condition.