# Formula Time-dependent Schrödinger equation (3d)

## Wave function

Three-dimensional probability amplitude, with which the you can calculate the probability for finding a quantum mechanical particle at a certain position. The wave function depends in general on the location $$\boldsymbol{r}$$, and on the time $$t$$.

## Laplace-Operator

The Laplace operator is applied to the wave function. It contains the second partial derivatives with respect to the spatial coordinates:$\nabla^2 ~=~ \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$

## Potential energy

Unit
Potential energy function which gives the potential energy of a quantum mechanical particle at the location $$\boldsymbol{r}$$ at the time $$t$$. Thus, in general, the potential energy is dependent on location and time.

## Imaginary unit

Unit
Imaginary unit is a complex number for which the following relation holds: $$\textbf{i}^2 ~=~ -1$$.

## Reduced Planck constant

Unit
Reduced Planck constant is a natural constant and has the value: $$\hbar ~=~ \frac{h}{2 \pi} ~=~ 1.054 \, 572 ~\cdot~ 10^{-34} \, \text{Js}$$

## Mass

Unit
Mass of the quantum mechanical particle (e.g. an electron).