In order to be able to illustrate the central concepts in subsequent developments in the main body of this case study, we establish a certain technical background against which we can discuss the origins of Wheeler's and of Heisenberg's scattering matrix. This outline will provide a brief and selective summary of nonrelativistic Schrödinger scattering theory, classical electromagnetic wave theory and the theory of the interaction of a quantized atomic system with the radiation field. The material presented here I take to be given and unproblematic by the mid to late 1930s. As a justification for this assumption let me point out that the essentials of all these topics can be found in the 1933 edition of Mott and Massey's The Theory of Atomic Collisions, in the 1933 edition of Slater and Frank's Introduction to Theoretical Physics and in the 1936 edition of Heitler's The Quantum Theory of Radiation. These were all standard reference works at the time so that their contents can be assumed to have been known to practicing theoretical physicists of that era. The order of presentation here is not necessarily the historical one and very few specific references will be given for this background material. Also, a unified notation and a fairly modern presentation will be used to facilitate reading and subsequent reference for illustrations. The technical details given for this introductory background will be more complete than for most of the historical presentation in the main body of the text.