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arXiv · 2207.00935

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

Yu-An Tsai·Sheng D. Chao
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Abstract

We propose an extension of the Wentzel-Kramers-Brillouin (WKB) approximation for solving the Schr\"{o}dinger equation. Based on an ansatz for the wave function, subject to an auxiliary condition on its first derivative, we obtain a set of coupled differential equations that decouple, via an alternating perturbation method, into the well-known Bremmer series. The perturbation improves the wave function amplitudes alternately, and its phase is refined through recursive diagonalization. From this construction, we derive a general quantization formula that encodes the geometric-optics-like physics of the system. When the ratio of the differential reflection coefficient to the classical momentum remains constant, the formula reduces to a closed-form quantization condition consistent with that obtained by re-summing the WKB series to all orders.

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Yu-An Tsai, Sheng D. Chao. 2022-07-03. Alternating Wentzel-Kramers-Brillouin Approximation to the Schr\"{o}dinger Equation: Rediscover the Bremmers series and beyond. https://arxiv.org/abs/2207.00935

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