Formula Time-dependent Schrödinger equation (3d)
$$i \, \hbar \, \frac{\partial \mathit{\Psi}}{\partial t} ~=~ - \frac{\hbar^2}{2m} \, \nabla^2 \, \mathit{\Psi} ~+~ W_{\text{pot}} \, \mathit{\Psi}$$
Wave function
$$ \mathit{\Psi}(\boldsymbol{r}, t) $$ 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
$$ \nabla^2 $$ 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
$$ W_{\text{pot}} $$ Unit $$ \mathrm{J} $$ 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
$$ i $$ Unit $$ - $$ Imaginary unit is a complex number for which the following relation holds: \( \textbf{i}^2 ~=~ -1 \).
Reduced Planck constant
$$ \hbar $$ Unit $$ \mathrm{Js} $$ 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
$$ m $$ Unit $$ \mathrm{kg} $$ Mass of the quantum mechanical particle (e.g. an electron).