Credit: FRANCIS SIMON/AIP EMILIO SEGRÈ VISUAL ARCHIVES

Werner Heisenberg's trip to Heligoland in June 1925 is a legend. Plagued by hay fever, the 23-year old escaped to the pollen-free island in the North Sea, to return with deep insight that would change the way we think about quantum mechanics. Electrons in atoms, he came to realize, do not move in sharp orbits with definite radii and periods of rotation. As a consequence, their motion should not be described by a coordinate that depends on time, but by an array of transition amplitudes. Heisenberg, Max Born and Pascual Jordan — Paul Dirac made independent contributions — expanded the approach into what would become known as the matrix-mechanics formulation of quantum mechanics.

Only a year later, Erwin Schrödinger (pictured) presented a different formalism: wave mechanics, which uses a vastly different mathematical language — differential equations rather than the algebraic approach of matrix mechanics. Already Schrödinger was considering the relation between his own theory and the quantum mechanics of Heisenberg, Born and Jordan. In 1926, in a paper originally published in Annalen der Physik, he presented arguments leading to the conclusion that the two so different approaches are indeed equivalent.

But to what degree Schrödinger proved the equivalence between the two frameworks has been the subject of some recent debate — is it actually a 'myth' that Schrödinger established the equivalence of matrix mechanics and wave mechanics, that is, that they describe the same physics? Contributing to the discussion, Slobodan Perovic argues that providing a fully fledged general proof was never the goal of Schrödinger's paper (Studies in History and Philosophy of Modern Physics doi: 10.1016/j.shpsb.2008.01.004; 2008).

Rather, in Perovic's view, the case has to be discussed in a specific context — the context of Niels Bohr's model of the atom. Both matrix mechanics and wave mechanics were constructed against the background of Bohr's model, and Schrödinger's main goal was, according to Perovic, to establish the coherence of the two approaches with that model. This served, not least, to underline the significance of wave mechanics. Matrix mechanics, after all, had been more successful in explaining the spectral lines of the hydrogen atom — a fact that explains, in part, why Schrödinger focused on showing explicitly (and successfully) how matrices can be constructed from eigenfunctions, whereas he only sketches rather than proves the reciprocal equivalence.

The full proof of the mathematical equivalence of matrix mechanics and wave mechanics followed only a couple of years later, notably after the Copenhagen interpretation was framed. This influential interpretation of quantum mechanics is rooted in the equivalence of the two approaches — but an equivalence, Perovic argues, in the context of Bohr's model, rather than the full proof of the isomorphism of the mathematical frameworks underlying the approaches.