Schrödinger’s original quantum–mechanical solution for hydrogen

• Published in: European journal of physics Vol. 42; no. 3; p. 35405
• Main Authors: Galler, Anna, Canfield, Jeremy, Freericks, James K
• Format: Journal Article
• Language: English
• Published: 01.05.2021
• ISSN: 0143-0807; 1361-6404
• DOI: 10.1088/1361-6404/abb9ff

Abstract In 1926, Erwin Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger’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.