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.

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

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:

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 $$\mathit{\Psi}_n$$:

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

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).