Postulates, like axioms, are established or widely accepted principles. Moreover, in the branch of mathematics of geometry, they are self-evident truths that require no proofs. They are assumed as necessary as fundamental prerequisite conditions for a basis of mathematical reasoning.

In the foundations of quantum mechanics there are at least four fundamental postulates. (1) State vectors|yñ’s of complex Hilbert mathematical space must be used to represent the states of all physical quantum systems. (2) Hermitian operators H’s represent physical measured observables like energy and linear momentum. These can act on state vectors. Their eigenvectors form an orthonormal basis of the entire state space of the particular physical system under investigation. (3) All probable real measurements can only be given by the eigenvalues of a particular Hermitian operator. (4) The probability of a particular measurement is given by the Born Rule. For example: if the state of a quantum system is given |yñ=(Ö2/3)|Añ+(Ö3/3)|Bñ+(2/3)|Cñ where |Añ, |Bñ, and |Cñ represent an orthonormal basis. These are eigenvectors of the Hermitian Hamiltonian operator such that H|Añ=E|Añ, H|Bñ=2E|Bñ, and H|Cñ=3E|Cñ. Since áy|=(Ö2/3)áA|+(Ö3/3)áB|+(2/3)áC|, the norm of the state is áy|yñ=(2/9)áA|Añ+(3/9)áB|Bñ+(4/9)áC|Cñ=1 and áA|Bñ=áA|Cñ=áB|Cñ=0. Therefore, áA|yñ=Ö2/3 and |áA|yñ|²=2/9 is the probability of measuring E, 3/9 is the probability of measuring 2E, and 4/9 is the probability of measuring 3E. Since 2/9+3/9+4/9=9/9=1, the probability is complete.