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
\( \mathit{\Psi}(\boldsymbol{r}, t) \) Unit \( \frac{1}{\sqrt{\text{m}^3}} \) 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
\( \nabla^2 \) Unit \( \frac{1}{\text{m}^2} \) 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 \( \text{J} \) 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 \( \text{Js} \) 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 \( \text{kg} \) Mass of the quantum mechanical particle (e.g. an electron).