[fgh]

> Von Neumann, who saw the underlying mathematics better than almost anyone, showed how wave and matrix mechanics were essentially the same, and how one could be expressed in the other’s language.

Maybe he did that, but Dirac was certainly the first to do so.

[n4r9]

> By the time von Neumann started his investigations on the formal framework of quantum mechanics this theory was known in two different mathe- matical formulations: the "matrix mechanics" of Heisenberg, Born and Jordan, and the "wave mechanics" of Schrödinger. The mathe- matical equivalence of these formulations had been established by Schrödinger, and they had both been embedded as special cases in a general formalism, often called "transformation theory," developed by Dirac and Jordan. This formalism, however, was rather clumsy and it was hampered by its reliance upon ill-defined mathematical objects, the famous delta-functions of Dirac and their derivatives. Although von Neumann himself attempted at first, in collaboration with Hilbert and Nordheim [l], to edify the quantum-mechanical formalism along similar lines, he soon realized that a much more natural framework was provided by the abstract, axiomatic theory of Hilbert spaces and their linear operators [2], This mathematical formulation of quantum mechanics, whereby states of the physical system are described by Hilbert space vectors and measurable quan- tities by hermitian operators acting upon them, has been very suc- cessful indeed. Unchanged in its essentials it has survived the two great extensions which quantum theory was to undergo soon: the relativistic quantum mechanics of particles (Dirac equation) and the quantum theory of fields.

Leon van Hove, "Von Neumann's Contributions to Quantum Mechanics", 1958

https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958...

[fgh]

Thanks, that clarifies his actual contribution.

[n4r9]

I remember taking quantum mechanics as a second year mathematics undergrad in the mid 2000s and - as a mostly pure mathematician - balking at the Dirac delta function. Wasn't until the third year course on principles of QM did things start feeling steady.