Alternating Wentzel-Kramers-Brillouin Approximation to the Schr\"{o}dinger Equation: Rediscover the Bremmers series and beyond

TL;DR

This paper extends the WKB approximation for the Schrödinger equation, rediscovering the Bremmer series and introducing an iterative method to improve phase accuracy and derive a general quantization formula.

Contribution

It introduces an alternating perturbation method that decouples coupled differential equations, refining wave function phases and deriving a comprehensive quantization condition.

Findings

Reproduces the Bremmer series through decoupling

Provides a recursive method to improve wave phase accuracy

Derives a general quantization formula consistent with all-order WKB resummation

Abstract

We propose an extension of Wenzel-Kramers-Brillouin (WKB) approximation for solving the Schr\"odinger equation. A set of coupled differential equations is obtained by considering an ansatz of the wave function with an auxiliary condition on gauging its first derivative. It is shown that the alternating perturbation method can decouple the set of differential equations, yielding the well know Bremmer series, and in addition, by virtue of improvement on amplitudes, can refine the phase of the wave function in a sequence of recursive diagonalizations. We therefore find a general quantization formula in which the geometric-optical-like physics is encoded. Whenever the ratio of the differential reflection coefficient and the classical momentum remains constant, we show that our general quantized formula will reduce to the closed-form quantization condition that agrees with the result obtained by re-summation the perturbative WKB series to all orders.