(1) A general state vector is represented by |Ψ> (a ket)

(2) A physical observable is represented by an operator A that acts on |Ψ>

(3) The possible measurements of an observable are the eigenvalues aₙ of the operator A

(4) The probability of measuring eigenvalue aₙ is given by Paₙ=|<aₙ|Ψ>|²

(5) After a measurement A that yields aₙ, the quantum system is in a new state that is the normalized projection of the original system ket onto the ket (or kets) corresponding to the result of the measurement:
    |Ψ'> = Pn|Ψ> / √(<Ψ|Pn|Ψ>)

(6) The time-evolution of the quantum system is determined by the Hamiltonian operator H(t) through the Schrodinger equation
    ιħ(d/dt)|Ψ(t)> = H(t)|Ψ(t)>

