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We propose an extension of Wenzel-Kramers-Brillouin (WKB) approximation forsolving the Schr\"odinger equation. A set of coupled differential equations isobtained by considering an ansatz of the wave function with an auxiliarycondition on gauging its first derivative. It is shown that the alternatingperturbation method can decouple the set of differential equations, yieldingthe well know Bremmer series, and in addition, by virtue of improvement onamplitudes, can refine the phase of the wave function in a sequence ofrecursive diagonalizations. We therefore find a general quantization formula inwhich the geometric-optical-like physics is encoded. Whenever the ratio of thedifferential reflection coefficient and the classical momentum remainsconstant, we show that our general quantized formula will reduce to theclosed-form quantization condition that agrees with the result obtained byre-summation the perturbative WKB series to all orders.

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