Formula: Time independent Schrödinger equation (3d)
$$W \, \mathit{\Psi} ~=~ - \frac{\hbar^2}{2\class{brown}{m}} \, \nabla^2 \, \mathit{\Psi} ~+~ W_{\text{pot}} \, \mathit{\Psi}$$
Wave function
$$ \mathit{\Psi} $$
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 on the location \( \boldsymbol{r} \).
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:
\[ \nabla^2 ~=~ \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \]
Total energy
$$ W $$ Unit $$ \mathrm{J} = \mathrm{Nm} = \frac{ \mathrm{kg} \, \mathrm{m^2} }{ \mathrm{s}^2 } $$
Total energy of a quantum mechanical particle described by the stationary state \( \mathit{\Psi} \).
Potential energy
$$ W_{\text{pot}} $$ Unit $$ \mathrm{J} = \mathrm{Nm} = \frac{ \mathrm{kg} \, \mathrm{m^2} }{ \mathrm{s}^2 } $$
Potential energy can depend on location \( \boldsymbol{r} \) in the case of stationary Schrödinger equation, but not on time \( t \).
Reduced Planck constant
$$ \hbar $$ Unit $$ \mathrm{Js} = \frac{ \mathrm{kg} \, \mathrm{m}^2 }{ \mathrm{s} } $$
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
$$ \class{brown}{m} $$ Unit $$ \mathrm{kg} $$
Mass of the quantum mechanical particle (e.g. an electron).