1. The state of a quantum mechanical system is described mathematically by a normalized ket $|\psi\rangle$ that contains all the information we can know about the state.
2. A physical observable is described mathematically by an operator A that acts on kets.
3. The only possible result of a measurement of an observable is one of the eigenvalues $a_n$ of the corresponding operator A.
4. The probability of obtaining the eigenvalue $a_n$ in a measurement of the observable A on the system in the state $|\psi\rangle$ is $$P(a_n)=\left| \langle a_n | \psi \rangle \right|^2$$ where $|a_n\rangle$ is the normalized eigenvector of A corresponding to the eigenvalue $a_n$.
5. After a measurement of A that yields the result $a_n$, 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 results of the measurement: $$|\psi'\rangle = \frac{P_n |\psi\rangle}{\sqrt{\langle\psi|P_n|\psi\rangle}}$$
6. The time evolution of a quantum system is determined by the Hamiltonian or total energy operator $H(t)$ through the Schrödinger equation $$i\hbar\frac{d}{dt} |\psi(t)\rangle = H(t)|\psi(t)\rangle$$

Note: A common misconception among students first learning quantum mechanics is that a measurement is carried out by operating on a ket with the operator associated with observable one wants to measure.