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Initially by Werner Heisenberg, offered the first theoretical calculation to reproduce the hydrogen emission spectrum experimental data.
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It is apparently more closely related to the ladder operator method, which is a more algebraic than the more analytical Schrodinger equation.
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It appears that this formulation makes the importance of the Poisson bracket clear, and explains why physicists are so obsessed with talking about position and momentum space. This point of view also apparently makes it clearer that quantum mechanics can be seen as a generalization of classical mechanics through the Hamiltonian.
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QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga by Silvan Schweber (1994) mentions however that relativistic quantum mechanics broke that analogy, because some 2x2 matrix had a different form, TODO find that again.
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Ancestors

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