Alexander Fufaev
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:

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).

Overlap of Two Wave Functions