My name is Alexander FufaeV and here I write about:

What are the 4 Postulates of Quantum Mechanics?

Postulate #1
The normalized wave function $ \mathit{\Psi}(\boldsymbol{r},t) $ completely describes the state of a quantum mechanical system, for example the state of an electron.

A normalized wave function encloses the area = 1. For this to be fulfilled, it must approach zero for $ |\boldsymbol{r}| \rightarrow \infty $.

Postulate #2
The temporal evolution (dynamics) of a wave function $ \mathit{\Psi}(\boldsymbol{r},t) $ is described by the Schrödinger equation:

$$ \begin{align} i\hbar \, \frac{\partial}{\partial t} \mathit{\Psi}(\boldsymbol{r},t) ~=~ \hat{H} \, \mathit{\Psi}(\boldsymbol{r},t) \end{align} $$

Here, $ \hat H $ is a Hamilton operator that describes the total energy of the quantum mechanical system.

Postulate #3
Measurements in quantum mechanics are described by Hermitian operators $ \hat H $. The mean value (of many individual measurements of a measurand $ H $, which were measured on the system with the state $ \mathit{\Psi} $) is given by:

$$ \begin{align} \langle H \rangle ~:=~ \langle \mathit{\Psi} ~|~ \,\hat{H} \, \mathit{\Psi} \rangle ~=~ \int \mathit{\Psi}^*\,\hat{H} \, \mathit{\Psi} ~ \text{d}^3x \end{align} $$

Possible measurement results $ h_n $ that belong to the quantity $ H $ (e.g. momentum, position, energy etc.) are the eigenvalues of the operator $ \hat{H} $ (for example momentum, position, energy operator) with the associated quantum numbers $ n $ (for example $n$-th energy level) and eigenfunctions $ \mathit{\Psi}_n $:

$$ \begin{align} \hat{H} \, \mathit{\Psi}_n ~=~ h_n \, \mathit{\Psi}_n \end{align} $$

Postulate #4
The probability $ P(q_n) $ of measuring the measured value $ q_n $ (e.g. momentum value) is given by

$$ \begin{align} P(q_n) ~=~ |\langle\mathit{\Psi}_n ~|~ \mathit{\Psi}\rangle|^2 ~=~ \int \mathit{\Psi}_{n}^* \, \mathit{\Psi} ~ \text{d}^3 x \end{align} $$

Here, $ \langle \mathit{\Psi}_n ~|~ \mathit{\Psi} \rangle $ is a scalar product that tells you how much of the state $ \mathit{\Psi}_n $ is contained in the total state $ \mathit{\Psi} $ (which should of course be normalized).

Overlap of Two Wave Functions