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Quantum mechanics can be built on some axioms also known as the postulates of Quantum Mechanics. For a clear presentation the postulates are stated only for a discrete basis, they are easily generalized for continuous bases.

1. Postulate: Every physical system at a certain instant of time $t_0$ is described by a state vector $| \psi (t_0) \rangle$, that is an element of $\mathcal{H}$ the Hilbert Space.

2. Postulate: Each measurable physical quantity is associated with an operator $A$ of $\mathcal{H}$; this operator is an observable.

remark: Let $A$ be an element of the Hilbert Space $\mathcal{H}$ with eigenvectors $|\phi_1 \rangle,|\phi_2 \rangle,\dots$ and eigenvalues $\lambda_1,\lambda_2, \dots$, that $A$ satisfies the eigenvalue equation

\begin{align} A |\phi_n \rangle= \lambda_n |\phi_n \rangle \end{align}

3. Postulate: The result of a measurement of any physical observable is always an eigenvalue of the corresponding operator $A$.

remark: Any state vector $|\psi_n \rangle$ can be represented as a combination of basis vectors given by the eigenvectors of the operator $A$

\begin{align} |\psi \rangle= \sum_n c_n |\phi_n \rangle \end{align}

than the coefficients are given by the overlap of $|\phi_n \rangle$ and $|\psi \rangle$

\begin{align} c_n =\langle \phi_n |\psi \rangle \end{align}

$c_n$ is called the probability amplitude (not the probability!).

4. Postulate: For the measurement of an observable $A$ of a physical system in the normed state $| \psi (t) \rangle$, the probability to find the system in eigenstate $|\phi_n \rangle$ is $P_n=|c_n|^2 =|\langle \phi_n |\psi \rangle|^2$

remark: By the completeness relation is immediately clear that

\begin{align} \langle \psi(t) |\psi(t) \rangle=1 \Rightarrow \langle \psi(t) |\psi(t) \rangle=\sum_n \langle \phi_n |c_n^*(t) c_n(t) |\phi_n \rangle=\sum_n |c_n(t)|^2\langle \phi_n |\phi_n \rangle=\sum_n |c_n(t)|^2=1 \end{align}

the sum of the probabilities add to one.
The mean of the obervable $\langle A \rangle (t)$ is given by

\begin{align} \langle A \rangle (t)&=\frac{\sum_n \lambda_n \, |c_n(t)|^2}{\sum_n |c_n(t)|^2} =\frac{\sum_n \lambda_n \, \langle \psi(t) |\phi_n \rangle \langle \phi_n |\psi(t) \rangle}{\sum_n \langle \psi(t) |\phi_n \rangle \langle \phi_n |\psi(t) \rangle}=\frac{\langle \psi(t) |\, \{ \sum_n \lambda_n |\phi_n \rangle \langle \phi_n | \} \, |\psi(t) \rangle}{\langle \psi(t) | \, \{ \sum_n |\phi_n \rangle \langle \phi_n | \} \, |\psi(t) \rangle}=\frac{\langle \psi(t) |\, A \, |\psi(t) \rangle}{\langle \psi(t) |\psi(t) \rangle}\\ \end{align}

For a normed state with $\langle \psi(t) |\psi(t) \rangle=1$ the average value of the observable is thus

\begin{align} \langle A \rangle (t)&=\langle \psi(t) |\, A \, |\psi(t) \rangle\\ \end{align}

5. Postulate: If the measurement of an observable of a physical system in state $|\psi(t) \rangle$ gives the result $\lambda_n$, then immediately after the measurement, the same eigenstate $|\phi_n \rangle$ corresponding to $\lambda_n$, will be measured with probability $P(\lambda_n)=1$ (the wave function collapsed).
remark: This is how a system can be prepared in a certain eigenstate.

6. Postulate: The evolution of the state vector $|\psi(t) \rangle$ with time is given by the Schrödinger Equation \begin{align} i \, \hbar \frac{\partial}{ \partial t} |\psi(t) \rangle & = H \, |\psi(t) \rangle\\ \end{align}

remark: The formal solution to the Schrödinger equation with initial condition $|\psi(t_0) \rangle$ is

\begin{align} |\psi(t) \rangle & = e^{- \frac{i H }{\hbar}\,(t-t_0)} |\psi(t_0) \, \rangle\\ \end{align}

provided the Hamiltonian operator $H$ is time independent (autonomous). We call

\begin{align} U(t,t_0)=e^{- \frac{i H }{\hbar}\,(t-t_0)} \end{align}

the evolution or time development operator, which takes the system from the state $|\psi(t_0) \rangle$ to the state $|\psi(t) \rangle$. To solve a problem explicitly one needs to choose a representation.