Schrödinger Gleichung

The Schrödinger equation determining the probabilities of a system’s possible states and how they evolve.

It is described by a wavefunction that distributes the probabilities for all possible states (outcomes / possibilities).
It also determines the allowed energy levels that a particle can occupy, indicating that a particle can only use specific, quantized amounts of energy. These discrete energy levels dictate the possible transitions and interactions of particles within quantum systems.

Schrödinger equation

$i\hslash \frac{\partial }{\partial t}\Psi (\mathbf{r},t)=\left(-\frac{{\hslash }^2}{2m}{\nabla }^2+V(\mathbf{r},t)\right)\Psi (\mathbf{r},t)$

Schrädinger equation (time-independent)

$E\Psi (\mathbf{r})=\frac{-{\hslash }^2}{2m}{\nabla }^2\Psi (\mathbf{r})+V(\mathbf{r})\Psi (\mathbf{r})$

Formula	What is it	Variation
$E$	energy eigenvalue
$\Psi (r)$	wave function	$\Psi (x)$
$\frac{-{\hslash }^2}{2m}{\nabla }^2$	kinetik energy	$\frac{-{\hslash }^2}{2m}\frac{d^{²}\Psi (x)}{d{x}^2}$
$V(\mathbf{r})$	potential energy	$V\Psi (x)$