Schrodinger's original quantum-mechanical solution for hydrogen

TL;DR

This paper revisits Schrodinger's original 1926 solution for the hydrogen atom using the Laplace method, highlighting its pedagogical value and complex analysis techniques often overlooked in standard quantum mechanics teaching.

Contribution

It provides a detailed, accessible reconstruction of Schrodinger's original solution using the Laplace method, emphasizing its educational potential in advanced quantum mechanics courses.

Findings

Laplace method effectively solves hydrogen atom eigenfunctions

Complex analysis techniques enhance understanding of quantum solutions

Original Schrodinger solution can be pedagogically valuable

Abstract

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.