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Formula Time independent Schrödinger equation (3d)

Formula
Formula: Time independent Schrödinger equation (3d)
Plane wave in complex plane
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Wave function

Unit
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

Unit
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

Unit
Total energy of a quantum mechanical particle described by the stationary state \( \mathit{\Psi} \).

Potential energy

Unit
Potential energy can depend on location \( \boldsymbol{r} \) in the case of stationary Schrödinger equation, but not on time \( t \).

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).