The WKB approximation plays an essential role in the development of quantum mechanics and various important results have been obtained from it. In this paper, we introduce another method, the so-called uniform asymptotic approximations, which is an analytical approximation method to calculate the wave functions of the Schrödinger-like equations, and it is applicable to various problems, including cases with poles (singularities) and multiple turning points. A distinguished feature of the method is that in each order of the approximations the upper bounds of the errors are given explicitly. By properly choosing the freedom introduced in the method, the errors can be minimized, which significantly improves the accuracy of the calculations. A byproduct of the method is to provide a very clear explanation of the Langer modification encountered in the studies of the hydrogen atom and harmonic oscillator. To further test our method, we calculate (analytically) the wave functions for several exactly solvable potentials of the Schrödinger equation, and then obtain the transmission coefficients of particles over potential barriers, as well as the quantization conditions for bound states. We find that such obtained results agree with the exact ones extremely well. Possible applications of the method to other fields are also discussed.

中文翻译：

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量子力学中的Langer修改，量化条件和势垒穿透

WKB近似在量子力学的发展中起着至关重要的作用，并已从中获得了各种重要的结果。在本文中，我们介绍了另一种方法，即所谓的均匀渐近逼近法，这是一种计算Schrödinger方程的波函数的解析逼近方法，适用于各种问题，包括极点（奇点）情况和多个转折点。该方法的一个显着特征是，在近似的每个顺序中，明确给出了误差的上限。通过适当地选择该方法引入了自由，错误可以被最小化，这显著提高了计算的准确性。该方法的副产品是对氢原子和谐波振荡器研究中遇到的Langer修饰提供非常清晰的解释。为了进一步检验我们的方法，我们（解析地）计算了薛定ding方程的几个可精确求解的电势的波函数，然后获得了势垒上粒子的透射系数以及束缚态的量化条件。我们发现，这样获得的结果与确切结果非常吻合。还讨论了该方法在其他领域的可能应用。然后获得粒子在势垒上的透射系数，以及束缚态的量化条件。我们发现，这样获得的结果与确切结果非常吻合。还讨论了该方法在其他领域的可能应用。然后获得粒子在势垒上的透射系数，以及束缚态的量化条件。我们发现，这样获得的结果与确切结果非常吻合。还讨论了该方法在其他领域的可能应用。