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.

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