  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.
