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

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.