# 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).