# Langer Modification, Quantization condition and Barrier Penetration in Quantum Mechanics

## TL;DR

This paper introduces a uniform asymptotic approximation method for solving Schrödinger equations, improving accuracy over WKB, explaining Langer modification, and accurately calculating wave functions, transmission coefficients, and quantization conditions.

## Contribution

The paper presents a new analytical approximation method that enhances accuracy and provides clear insights into Langer modification in quantum mechanics.

## Key findings

- The method accurately reproduces wave functions for exactly solvable potentials.
- It yields precise transmission coefficients for potential barriers.
- The approach clarifies the origin of Langer modification in quantum problems.

## Abstract

The WKB approximation plays an essential role in the development of quantum mechanics and various important results have been obtained from it. In this paper, we introduce another method, {\it the so-called uniform asymptotic approximations}, which is an analytical approximation method to calculate the wave functions of the Schr\"odinger-like equations, and is applicable to various problems, including cases with poles (singularities) and multiple turning points. An distinguished feature of the method is that in each order of the approximations the upper bounds of the errors are given explicitly. By properly choosing the freedom introduced in the method, the errors can be minimized, which significantly improves the accuracy of the calculations. A byproduct of the method is to provide a very clear explanation of the Langer modification encountered in the studies of the hydrogen atom and harmonic oscillator. To further test our method, we calculate (analytically) the wave functions for several exactly solvable potentials of the Schr\"odinger equation, and then obtain the transmission coefficients of particles over potential barriers, as well as the quantization conditions for bound states. We find that such obtained results agree with the exact ones extremely well. Possible applications of the method to other fields are also discussed.