Homogeneous Field and WKB Approximation In Deformed Quantum Mechanics with Minimal Length

TL;DR

This paper investigates the effects of a minimal length deformation in quantum mechanics on a particle in a homogeneous field, deriving WKB connection formulas and quantization rules within this framework.

Contribution

It introduces a method to derive WKB connection formulas and Bohr-Sommerfeld quantization in deformed quantum mechanics with minimal length, highlighting limitations near steep potential slopes.

Findings

Derived asymptotic wave functions using steepest descent.

Established WKB connection formula up to first order in deformation parameter.

Proved Bohr-Sommerfeld quantization rule within the deformed framework.

Abstract

In the framework of the deformed quantum mechanics with minimal length, we consider the motion of a non-relativistic particle in a homogeneous external field. We find the integral representation for the physically acceptable wave function in the position representation. Using the method of steepest descent, we obtain the asymptotic expansions of the wave function at large positive and negative arguments. We then employ the leading asymptotic expressions to derive the WKB connection formula, which proceeds from classically forbidden region to classically allowed one through a turning point. By the WKB connection formula, we prove the Bohr-Sommerfeld quantization rule up to $\mathcal{O}(\beta)$. We also show that, if the slope of the potential at a turning point is too steep, the WKB connection formula fall apart around the turning point.