Wave mechanics, from Louis de Broglie to Schrödinger: a comparison

Abstract: Erwin Schrödinger’s work on wave mechanics started in late 1925, stimulated by his study of Louis de Broglie’s thesis. It is well known that in his initial attempts to formulate a quantum theory of the atom Schrödinger tried to develop a relativistic theory, following de Broglie’s ideas, and only afterwards he looked for a non-relativistic wave equation. It is straightforward to derive the wave equation corresponding to de Broglie’s phase waves. both in the relativistic and non-relativistic realms. In the case of his relativistic attempt, Schrödinger did indeed follow a simple approach, using de Broglie’s theory. In the non-relativistic approach, he attempted to produce an independent derivation of the wave equation, following several different lines, instead of using de Broglie’s results in the classical limit. This paper analyses Schrödinger’s derivations of the wave equation, showing the differences and similarities between his theory and de Broglie’s. It will be shown that, although it is formally possible to derive the wave equation from de Broglie’s theory, there is an incompatibility between the two theories: it would be impossible to make any sense of de Broglie’s ideas in the case of the rigid rotator, for instance. Schrödinger’s approach was, in this sense, independent and incompatible with de Broglie’s theory, and it could be easily applied to many different physical situations. This heuristic value of Schrödinger’s wave equation is another very important distinction between the two theories, since de Broglie’s theory only led to a single new prediction: the wave behaviour of electrons in diffraction experiments.

Keywords: Schrödinger, Erwin; de Broglie, Louis; wave mechanics; wave equation; quantum mechanics; history of physics