In 1926, Erwin Schrodinger wrote a series of papers that invented wave mechanics and set the foundation for much of the single-particle quantum mechanics that we teach today. In his first paper, he solved the Schrodinger equation using the Laplace method, which is a technique that is quite powerful, but rarely taught. This is unfortunate, because it opens the door to examining quantum mechanics from a complex-analysis perspective. Gaining this experience with complex analysis is a useful notion to consider when teaching quantum mechanics, as these techniques can be widely used outside of quantum mechanics, unlike the standard Frobenius summation method, which is normally taught, but rarely used elsewhere. The Laplace method strategy is subtle and no one has carefully gone through the arguments that Schrodinger did in this first paper, instead it is often just stated that the solution was adopted from Schlesinger's famous differential equation textbook. In this work, we show how the Laplace method can be used to solve for the quantum-mechanical energy eigenfunctions of the hydrogen atom, following Schrodinger's original solution, with all the necessary details, and illustrate how it can be taught in advanced instruction; it does require familiarity with intermediate-level complex analysis, which we also briefly review.

中文翻译：

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薛定谔最初的氢量子力学解决方案

1926 年，埃尔温·薛定谔写了一系列论文，发明了波力学，并为我们今天教授的大部分单粒子量子力学奠定了基础。在他的第一篇论文中，他使用拉普拉斯方法求解薛定谔方程，这是一种非常强大但很少教授的技术。这是不幸的，因为它打开了从复分析角度研究量子力学的大门。在教授量子力学时，通过复分析获得这种经验是一个有用的概念，因为这些技术可以在量子力学之外广泛使用，这与通常教授的标准 Frobenius 求和方法不同，但在其他地方很少使用。拉普拉斯方法策略很微妙，没有人仔细研究过薛定谔在第一篇论文中所做的论证，相反，通常只是说该解决方案是从施莱辛格著名的微分方程教科书中采用的。在这项工作中，我们展示了如何使用拉普拉斯方法来求解氢原子的量子力学能量本征函数，遵循薛定谔的原始解决方案，包含所有必要的细节，并说明如何在高级教学中教授它；它确实需要熟悉中级复杂分析，我们也简要回顾一下。并说明如何在高级教学中教授它；它确实需要熟悉中级复杂分析，我们也简要回顾一下。并说明如何在高级教学中教授它；它确实需要熟悉中级复杂分析，我们也简要回顾一下。