Langer Modification, Quantization condition and Barrier Penetration in Quantum Mechanics

Understand

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.