Postulates of Quantum mechanics

Few postulates of Quantum mechanics are stated below:

Postulate 1 (describes the state of quantum mechanical system): At a given time $t_o$ the state of quantum mechanical system is defined by specifying a ket $|\psi(t_o)>$ belonging to state space $\varepsilon$ .

Postulate 2 (description of physical quantity): Every measurable quantity A is described by an observable $\hat{A}$ acting in the state space $\varepsilon$ .

Postulate 3 (Measurement of physical quantity): The only possible result of a measurement of a physical quantity A is one of the Eigen values of observable $\hat{A}$ .

Postulate 4 (Principle of spectral decomposition) Statement 1 (Discrete non-degenerate spectrum): When a physical quantity A is measured on a quantum mechanical system in the normalized state $|\psi>$ , the probability $P(a_n)$ of obtaining the non degenerate Eigen values $a_n$ of the observable $\hat{A}$ is $P(a_n) = |u_n|\psi>|^2$ here $|u_n> = a_n|u_n>$ and $|u_n>$ is normalized.

Postulate 4 Statement 2 (Discrete degenerate spectrum): When a physical quantity A is measured on a quantum mechanics system in the normalized state $|\psi>$ , the probability of obtaining the degenerate Eigen value $a_n$ of the observable $\hat{A}$ is
$P(a_n) = \sum\limits_{i=1}^{g_n}|u_n^{(i)}|\psi>|^2$ where $\hat{A}|u_n^{(i)} = a_n|u_n>$ and $g_n$ is degeneracy of $a_n$ .

Postulate 5 (Reduction of wave packet): If the measurement of a physical quantity A on a quantum mechanical system in the state $|\psi>$ , gives the result $a_n$ , the state immediately after the measurement is given by the normalized projection of $|\psi>$ onto the Eigen space associated with $a_n$ that is $latex \frac{P_n|\psi>}{\sqrt{<\psi|P_n|\psi>}} $

Postulate 6 (Time evolution of quantum mechanical state): The time evolution of state vector is governed by the Schrodinger equation.
$i\hbar\frac{d|\psi(t)>}{dt} = H(t)|\psi(t)>$
here $\hat{H}(t)$ is Hamiltonian of system.