Semiclassical Quantization Rule for Bound-State Spectrum in Quantum Dots: Scattering Phase Approximation

TL;DR

This paper introduces a new semiclassical quantization rule that incorporates scattering phase effects, enabling easier computation of bound-state energies in quantum dots with both smooth and sharp boundaries.

Contribution

A novel quantization rule that accounts for scattering phases, applicable to systems with sharp and smooth confining potentials, simplifying energy calculations in quantum dots.

Findings

Derived a modified Van Vleck's formula including scattering phases

Established relations among quantum statistics, gauge symmetry, and boundary conditions

Provided a practical method for energy level computation in quantum systems

Abstract

We study the quantum propagator in the semiclassical limit with sharp confining potentials. Including the energy-dependent scattering phase due to sharp confining potential, the modified Van Vleck's formula is derived. We also discuss the close relations among quantum statistics, discrete gauge symmetry, and hard-wall constraints. Most of all, we formulate a new quantization rule that applies to {\it both} smooth and sharp boundary potentials. It provides an easy way to compute quantized energies in the semiclassical limit and is extremely useful for many physical systems.