薛定谔方程的交替Wentzel-Kramers-Brillouin近似：重新发现Bremmer级数及其推广
Alternating Wentzel-Kramers-Brillouin Approximation to the Schrödinger Equation: Rediscover the Bremmers series and beyond
AI总结：
提出一种交替WKB近似方法，通过ansatz和辅助条件得到耦合方程，经交替微扰解耦为Bremmer级数，并推导出通用量子化公式，在特定条件下简化为闭式量子化条件。
摘要：
我们提出了一种用于求解薛定谔方程的Wentzel-Kramers-Brillouin（WKB）近似的扩展方法。基于波函数的一个ansatz，并对其一阶导数施加辅助条件，我们得到一组耦合微分方程，通过交替微扰方法，这些方程解耦为众所周知的Bremmer级数。该微扰交替地改进波函数的振幅，而其相位通过递归对角化得到精化。从这一构造中，我们推导出一个通用的量子化公式，该公式编码了系统的几何光学类物理。当微分反射系数与经典动量之比保持恒定，该公式简化为一个闭式量子化条件，与通过对WKB级数进行全阶重求和所得结果一致。
英文摘要：
We propose an extension of the Wentzel-Kramers-Brillouin (WKB) approximation for solving the Schrödinger 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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