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Nonrelativistic Quantum Mechanics (QM) relies in a number of postulates and although some authors may disagree about the exact set, there are a few which are quite indisputable:

  1. States are represented by vectors in a Hilbert space.
  2. Observables are represented by Hermitian operators acting on the Hilbert space.
  3. If the system is in state $|\psi\rangle$, then the measurement of the observable $A$ yields one of the eigenvalue of the operator associated to $A$.
  4. (Born Postulate) The probability of measuring $A$ and obtaining $a$, given that the system is in the state $|\psi\rangle$, is $|\langle a|\psi\rangle |^2$. After the measurement, the system is found to be in state $|a\rangle$.
  5. The state vector of the system obeys the Schrodinger equation.

Who did formulate the first three of these principles (or any other you may regard as fundamental) as fundamental postulates of QM?

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All these principles gradually evolved, in a series of papers starting from Heisenberg, Schrodinger and followed by Born, Jordan and Dirac, some of these papers are joint, the contribution of each author is impossible to separate.

In the form that you state them all of them were formulated for the first time in Dirac's book. However these postulates lack mathematical rigor (and Dirac understood this).

I mean the use of the words "Hermitian" and "eigenvalue". The operators representing observables must be self-adjoint. The exact meaning of these things was clarified for the first time by von Neumann. (Technically speaking, observables may not have any eigenvalues or eigenstates, they may not be "acting on the Hilbert space" etc., in the case of unbounded observables and/or counituius spectrum).

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