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Formula Time-dependent Schrödinger equation (3d)

Formula
Formula: Time-dependent 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 in general on the location \( \boldsymbol{r} \), and on the time \( t \).

Laplace-Operator

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

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).